Gibbs by Elysio Ruggeri

STAT

05tcentennial

VECTOR ANALYSIS

Bicentennial publications With

approval of the President and Fellows of Yale University, a series of volumes has been prepared by a number of the Professors and In the

structors,

issued

connection

with

the

Bicentennial Anniversary, as a partial indica tion of the character of the studies in which the University teachers are engaged.

This

series

of volumes

raDuate$ of

respectfully

tfc

dedicated to

VECTOR ANALYSIS^ A TEXT-BOOK FOR THE USE OF STUDENTS OF MATHEMATICS AND PHYSICS

FOUNDED UPON THE LECTURES OF J.

WILLARD

GIBBS, PH.D., LL.D.

Formerly Professor of Mathematical Physics in Yale University

EDWIN BIDWELL WILSON,

PH.D.

Professor of Vital Statistics in

Harvard School of Public Health

NEW HAVEN YALE UNIVERSITY PRESS

BY YALE UNIVERSITY Published, December, 1901 Second Printing, January 19/3 Third Printing, July, 1916 fourth Printing^ April, 1922 ,

Fifth Printing, October, 1925 Sixth Printing, April, 1020

Seventh Printing, October, 1951 Eighth Printing, April,

1943

PRINTED

from

the publishers.

THE UNITED STATES OF AMERICA

PKEFACB BY PROFESSOR GIBBS SINCE the printing of a short pamphlet on the Elements of never published, but Vector Analysis in the years 1881-84, somewhat widely circulated among those who were known to the desire has been expressed be interested in the subject, in

more than one quarter, that the substance of that trea

tise,

perhaps in fuller form, should be made accessible to

the public.

As, however, the years passed without my finding the meet this want, which seemed a real one, I was to have one of the hearers of my course on Vector very glad leisure to

Analysis in the year 1899-1900 undertake the preparation of a text-book on the subject. I have not desired that Dr. Wilson should aim simply at the reproduction of lectures, but rather that he should use his own judgment in all respects for the production of a text-book in which the subject should be so illustrated by an

adequate number of examples as to meet the wants of stu dents of geometry and physics. J.

YALE UNIVERSITY,

September, 1901.

WILLARD

GIBBS.

GENERAL PREFACE WHEN

undertook to adapt the lectures of Professor Gibbs on VECTOR ANALYSIS for publication in the Yale Bicenten nial Series, Professor Gibbs himself was already so fully I

engaged upon his work to appear in the same

series,

Elementary

Principles in Statistical Mechanics, that it was understood no material assistance in the composition of this book could be

expected from him.

For

this reason

entirely free to use my own of the topics to be treated It has been

he wished

to feel

discretion alike in the selection

and

in the

mode

of treatment.

endeavor to use the freedom thus granted

only in so far as was necessary for presenting his method in text-book form.

used in the follow ing pages has been taken from the course of lectures on Vector Analysis delivered annually at the University by Professor Gibbs. Some use, however, has been made of the chapters on Vector Analysis in Mr. Oliver Heaviside s Elec

far the greater part of the material

tromagnetic Theory (Electrician Series, 1893) and in Professor Foppl s lectures on Die Maxwell sche Theorie der Electricitdt

(Teubner, 1894).

previous study of Quaternions has

also been of great assistance.

The material thus obtained has been arranged in the way which seems best suited to easy mastery of the subject. Those Arts, which it seemed best to incorporate in the text but which for various reasons may well be omitted at the first reading have been marked with an asterisk (*). Nu merous illustrative examples have been drawn from geometry, mechanics, and physics. Indeed, a large part of the text has to do with applications of the method. These applications have not been set apart in chapters by themselves, but have

GENERAL PREFACE

been distributed throughout the body of the book as fast as the analysis has been developed sufficiently for their adequate treatment. It is hoped that by this means the reader may be better enabled to

make

practical use of the book.

Great care

has been taken in avoiding the introduction of unnecessary ideas, and in so illustrating each idea that is introduced as

make its necessity evident and its meaning easy to grasp. Thus the book is not intended as a complete exposition of

the theory of Vector Analysis, but as a text-book from which much of the subject as may be required for practical appli

may be learned. Hence a summary, including a list more important formulae, and a number of exercises, have been placed at the end of each chapter, and many less essential points in the text have been indicated rather than fully worked out, in the hope that the reader will supply the details. The summary may be found useful in reviews and

cations of the

for reference.

The subject of Vector Analysis naturally divides itself into three distinct parts. First, that which concerns addition and the scalar and vector products of vectors. Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function. The a necessary introduction to both other parts.

first

part

The second

and third are mutually independent. Either may be taken up first. For practical purposes in mathematical physics the second must be regarded as more elementary than the third. But a student not primarily interested in physics would nat urally pass from the first part to the third, which he would probably find more attractive and easy than the second. Following the book

this division of the subject, the

main body of

divided into six chapters of which two deal with each of the three parts in the order named. Chapters I. and II.

treat of addition, subtraction, scalar multiplication, and and vector products of vectors. The exposition

the scalar

has been

made

stood by and

It can quite elementary. readily be under especially suited for such readers as have a

knowledge of only the elements of Trigonometry and Ana-

GENERAL PREFACE

Those who are well versed in Quaternions lytic Geometry. or allied subjects may perhaps need to read only the sum maries.

Chapters

III.

and IV. contain the treatment

those topics in Vector Analysis which, though of less value to the students of pure mathematics, are of the utmost impor

tance to students of physics. Chapters V. and VI. deal with To students of physics the linear the linear vector function. is of particular importance in the mathemati treatment of phenomena connected with non-isotropic media and to the student of pure mathematics this part of the book will probably be the most interesting of all, owing

vector function

cal

;

to the fact that it leads to Multiple Algebra or the Theory of Matrices. concluding chapter, VII., which contains the

development of certain higher parts of the theory, a of applications, and a short sketch of imaginary or

number

complex been added. In the treatment of the integral calculus, Chapter IV., questions of mathematical rigor arise. Although modern theorists are devoting much time and thought to rigor, and vectors, has

although they will doubtless criticise this portion of the book adversely, it has been deemed best to give but little attention to the discussion of this subject.

And

the more so for the

reason that whatever system of notation be employed ques tions of rigor are indissolubly associated with the calculus

and occasion no new difficulty to the student of Vector Analysis, who must first learn what the facts are and may postpone until later the detailed consideration of the restric tions that are put

upon those

Notwithstanding the

facts.

efforts

more than half a century

which have been made during

introduce Quaternions into

physics the fact remains that they have not found wide favor. On the other hand there has been a growing tendency espe

decade toward the adoption of some form of The works of Heaviside and Foppl re ferred to before may be cited in evidence. As yet however no system of Vector Analysis which makes any claim to cially in the last

Vector Analysis.

completeness has been published.

In fact Heaviside says

"I am

which

in hopes that the chapter

now

finish

may

GENERAL PREFACE

x ii

serve as a stopgap till regular vectorial treatises come to be written suitable for physicists, based upon the vectorial treat ment of vectors" (Electromagnetic Theory, Vol. I., p. 305). Elsewhere in the same chapter Heaviside has set forth the

claims of vector analysis as against Quaternions, have expressed similar views.

The keynote, be

and others

must was Professor

then, to any system of vector analysis

This, I feel confident,

its

practical utility. He uses it s point of view in building up his system. and in courses on his Electricity Magnetism and on entirely

Gibbs

Electromagnetic Theory of Light. In writing this book I have tried to present the subject from this practical stand point, tions:

and keep clearly before the reader

What

mind the ques

combinations or functions of vectors occur in

physics and geometry ? And how may these be represented symbolically in the way best suited to facile analytic manip ulation ? The treatment of these questions in modern books

on physics has been too

much

subtraction of vectors.

This

confined to the addition and

is scarcely enough. It has been the aim here to give also an exposition of scalar and

vector products, of the operator y, of divergence and curl which have gained such universal recognition since the ap

pearance of Maxwell

Treatise on Electricity

and Magnetism,

of slope, potential, linear vector function, etc., such as shall be adequate for the needs of students of physics at the

present day and adapted to them. It has been asserted by some that Quaternions, Vector Analysis, and all such algebras are of little value for investi

gating questions in

mathematical physics.

assertion shall prove true or not, one vectors are to mathematical

may

still

Whether

this

maintain that

physics what invariants are to geometry. As every geometer must be thoroughly conver sant with the ideas of invariants, so every student of physics should be able to think in terms of vectors. And there is

no way in which he, especially at the beginning of his sci entific studies, can come to so true an appreciation of the importance of vectors and of the ideas connected with them as by working in Vector Analysis and dealing directly with

GENERAL PREFACE the vectors themselves.

xiii

those that hold these views the

of Professor Foppl s Vorlesungen uber Technische Mechanik (four volumes, Teubner, 1897-1900, already in a second edition), in which the theory of mechanics is devel oped by means of a vector analysis, can be but an encour

success

aging sign. I

and

colleagues, Dr. M. B. Porter Prof. H. A. Bumstead, for assisting me with the manu

take pleasure in thanking

script.

The good

services of the latter have been particularly

valuable in arranging Chapters III. and IV* in their present form and in suggesting many of the illustrations used in the

under obligations to my father, Mr. Edwin H. Wilson, for help in connection both with the proofs and the manuscript. Finally, I wish to express my deep indebt For although he has been so edness to Professor Gibbs.

work.

also

preoccupied as to be unable to read either manuscript or proof, he has always been ready to talk matters over with me, and it is he who has furnished me with inspiration suf ficient to carry through the work.

EDWIN BIDWELL WILSON. YALE UNIVERSITY,

October, 1901.

PREFACE TO THE SECOND EDITION THE

only changes which have been made in this edition are

a few corrections which

readers have been kind

enough

point out to me.

E. B.

TABLE OF CONTENTS PAGE

PREFACE BY PROFESSOR GIBBS

vii

GENERAL PREFACE

CHAPTER

ADDITION AND SCALAR MULTIPLICATION ARTS.

1-3

4 5

6-7

8-10 11

13-16 17

18-19

20-22

23-24 25

SCALARS AND VECTORS EQUAL AND NULL VECTORS THE POINT OF VIEW OF THIS CHAPTER SCALAR MULTIPLICATION. THE NEGATIVE SIGN ADDITION. THE PARALLELOGRAM LAW SUBTRACTION LAWS GOVERNING THE FOREGOING OPERATIONS COMPONENTS OF VECTORS. VECTOR EQUATIONS THE THREE UNIT VECTORS 1, j, k APPLICATIONS TO SUNDRY PROBLEMS IN GEOMETRY. VECTOR RELATIONS INDEPENDENT OF THE ORIGIN CENTERS OF GRAVITY. BARYCENTRIC COORDINATES THE USE OF VECTORS TO DENOTE AREAS SUMMARY OF CHAPTER i EXERCISES ON CHAPTER i

4 6

....

.... ....

12 14 18

...

CHAPTER

46 51

DIRECT AND SKEW PRODUCTS OF VECTORS 27-28 29-30

31-33 34-35 36

THE DIRECT, SCALAR, OR DOT PRODUCT OF TWO VECTORS THE DISTRIBUTIVE LAW AND APPLICATIONS THE SKEW, VECTOR, OR CROSS PRODUCT OF TWO VECTORS THE DISTRIBUTIVE LAW AND APPLICATIONS THE TRIPLE PRODUCT A* B C

55 58

60 63

CONTENTS

XVI

PAGE

ARTS.

37-38

39-40

41-42 43-45 46-47

48-50 51

52 53

THE SCALAR TRIPLE PRODUCT A* B X C OR [ABC] THE VECTOR TRIPLE PRODUCT A X (B X C) PRODUCTS OF MORE THAN THREE VECTORS WITH APPLI CATIONS TO TRIGONOMETRY RECIPROCAL SYSTEMS OF THREE VECTORS SOLUTION OF SCALAR AND VECTOR EQUATIONS LINEAR IN AN UNKNOWN VECTOR SYSTEMS OF FORCES ACTING ON A RIGID BODY KINEMATICS OF A RIGID BODY CONDITIONS FOR EQUILIBRIUM OF A RIGID BODY RELATIONS BETWEEN TWO RIGHT-HANDED SYSTEMS OF THREE PERPENDICULAR UNIT VECTORS PROBLEMS IN GEOMETRY. PLANAR COORDINATES SUMMARY OF CHAPTER n EXERCISES ON CHAPTER n .

68 71

....

...

101

CHAPTER

104 106 109

113

III

THE DIFFERENTIAL CALCULUS OF VECTORS 55-56 57

58-59 60 61

63-67 68 69 70 71

72 73

74-76

77 78

DERIVATIVES AND DIFFERENTIALS OF VECTOR FUNCTIONS WITH RESPECT TO A SCALAR VARIABLE CURVATURE AND TORSION OF GAUCHE CURVES KINEMATICS OF A PARTICLE. THE HODOGRAPH THE INSTANTANEOUS AXIS OF ROTATION INTEGFATION WITH APPLICATIONS TO KINEMATICS SCALAR FUNCTIONS OF POSITION IN SPACE THE VECTOR DIFFERENTIATING OPERATOR THE SCALAR OPERATOR A VECTOR FUNCTIONS OF POSITION IN SPACE THE DIVERGENCE V* AND THE CURL INTERPRETATION OF THE DIVERGENCE INTERPRETATION OF THE CURL X LAWS OF OPERATION OF V> * X THE PARTIAL APPLICATION OF V- EXPANSION OF A VEC TOR FUNCTION ANALOGOUS TO TAYLOR S THEOREM. APPLICATION TO HYDROMECHANICS THE DIFFERENTIATING OPERATORS OF THE SECOND ORDER GEOMETRIC INTERPRETATION OF LAPLACE S OPERATOR AS THE DISPERSION V* SUMMARY OF CHAPTER in EXERCISES ON CHAPTER in

....

V >

115 120 125 131

133 136 138

147 149

150 152 155 157

159

166

170 172

177

CONTENTS

xvii

CHAPTER IV THE INTEGRAL CALCULUS OF VECTORS PAGE

ARTS.

79-80

LINE INTEGRALS OF VECTOR FUNCTIONS WITH APPLICA TIONS

83 84

179

THEOREM STOKES S THEOREM GAUSS

184

187

CONVERSE OF STOKES S THEOREM WITH APPLICATIONS TRANSFORMATIONS OF LINE, SURFACE, AND VOLUME IN

TEGRALS. GREEN S THEOREM REMARKS ON MULTIPLE-VALUED FUNCTIONS " POTENTIAL. THE INTEGRATING OPERATOR " POT COMMUTATIVE PROPERTY OF POT AND REMARKS UPON THE FOREGOING THE INTEGRATING OPERATORS "NEW," "LAP," " MAX " RELATIONS BETWEEN THE INTEGRATING AND DIFFER

THE POTENTIAL

86-87

88 89 90

ENTIATING OPERATORS "

POT "

A SOLUTION OF POISSON

MAXWELLIANS

211

215

222

230 234 240

CERTAIN BOUNDARY VALUE THEOREMS SUMMARY OF CHAPTER iv EXERCISES ON CHAPTER iv

205

197

200

228 is

EQUATION SOLENOIDAL AND IRROTATIONAL PARTS OF A VECTOR FUNCTION. CERTAIN OPERATORS AND THEIR INVERSE MUTUAL POTENTIALS, NEWTONIANS, LAPLACIANS, AND

93-94

193

243

249

255

CHAFIER V LINEAR VECTOR FUNCTIONS 97-98 99

100

LINEAR VECTOR FUNCTIONS DEFINED 260 DYADICS DEFINED 264 ANY LINEAR VECTOR FUNCTION MAY BE REPRESENTED BY A DYADIC. PROPERTIES OF DYADICS 266 THE NONION FORM OF A DYADIC 269 THE DYAD OR INDETERMINATE PRODUCT OF TWO VEC TORS IS THE MOST GENERAL. FUNCTIONAL PROPERTY OF THE SCALAR AND VECTOR PRODUCTS 271 PRODUCTS OF DYADICS 276 DEGREES OF NULLITY OF DYADICS 282 THE IDEMFACTOR 288

....

101

102

108-104 105-107 108

CONTENTS

XV111

PAGE

ARTS.

109-110 111

112-114 115-116 117

118-119 120 121

122

POWERS AND ROOTS OF DYADICS SELF-CONJUGATE AND ANTICONJUGATE DYADICS. SELF-CONJUGATE PARTS OF A DYADIC RECIPROCAL DYADICS.

ANTI-SELF-CONJUGATE

DYADICS.

290 294

THE VECTOR PROD

UCT. QUADRANTAL VER8ORS 297 REDUCTION OF DYADICS TO NORMAL FORM 302 DOUBLE MULTIPLICATION OF DYADICS 306 THE SECOND AND THIRD OF A DYADIC 310 CONDITIONS FOR DIFFERENT DEGREES OF NULLITY 313 NONION FORM. DETERMINANTS 315 INVARIANTS OF A DYADIC. THE HAMILTON-CAYLEY .319 EQUATION SUMMARY OF CHAPTER v 321 EXERCISES ON CHAPTER v 329

....

...

CHAPTER VI ROTATIONS AND STRAINS 123-124 125-126 127 128

129

130 131

132

HOMOGENEOUS STRAIN REPRESENTED BY A DYADIC ROTATIONS ABOUT A FIXED POINT. VERSORS THE VECTOR SEMI-TANGENT OF VERSION BlQUADRANTAL VERSORS AND THEIR PRODUCTS .

332

334

343

339

CYCLIC DYADICS RIGHT TENSORS TONICS AND CYCLOTONICS REDUCTION OF DYADICS TO CANONICAL FORMS, TONICS, CYCLOTONICS, SIMPLE AND COMPLEX SHEARERS SUMMARY OF CHAPTER vi .

CHAPTER

347 351

353 356

368

VII

MISCELLANEOUS APPLICATIONS 136-142

QUADRIC SURFACES

143-146

THE PROPAGATION OF LIGHT

147-148

VARIABLE DYADICS CURVATURE OF SURFACES HARMONIC VIBRATIONS AND BIVECTORS

149-157 158-162

IN CRYSTALS

....

372 392

403

....

411

426

VECTOR ANALYSIS

VECTOR ANALYSIS CHAPTER

ADDITION AND SCALAR MULTIPLICATION 1.]

IN mathematics and especially in physics two very

different kinds of quantity present themselves.

Consider, for

example, mass, time, density, temperature, force, displacement

and acceleration. Of these quantities some can be represented adequately by a single number temperature, by degrees on a thermometric scale time, by mass and density, by numerical valyears, days, or seconds ues which are wholly determined when the unit of the scale On the other hand the remaining quantities are not is fixed. of a point, velocity,

;

Force to be sure

capable of such representation. of so

many pounds

grams

them must be considered

magnitude.

said to be

weight; velocity, of so

feet or centimeters per second.

But

many

in addition to this each

having direction as well as

force points North, South, East, West, up,

down, or in some intermediate direction. The same is true of displacement, velocity, and acceleration. No scale of num bers can represent

them adequately.

It can represent only

their magnitude, not their direction. 2.]

Definition

vector is a quantity

which

considered

as possessing direction as well as magnitude.

Definition

A scalar is a quantity which is considered as pos

sessing magnitude but no direction.

VECTOR ANALYSIS

The positive and negative numbers of ordinary algebra are the For this reason the ordinary algebra is called typical scalars. scalar algebra when necessary to distinguish it from the vector algebra or analysis which

the subject of this book. the displacement of translation in space.

The typical vector is first a point

Consider

line

straight

Let P be displaced and take a new position

(Fig. 1).

PP. The

ment it is

1 .

move

f .

the length of ; the direction of to the direction of the line from is

Next consider a displacement not

but of points

represented by the magnitude of the displace

This change of position line

in a

all

the points in space.

in straight lines in the

Let

same direction and

of one, all

the

for the

same distance D. body

rigid

This is equivalent to shifting space as a in that direction through the distance without

Such a displacement is and magnitude.

rotation.

possesses direction

called a translation.

When

space undergoes

a translation T, each point of space undergoes a displacement equal to T in magnitude and direction; and conversely if

which any one particular point P suf T is known, then that of any other known for Q Q must be equal and parallel

the displacement

fers in the translation

point to

also

PP. The

T is represented geometrically or graphically by an arrow T (Fig. 1) of which the magnitude and direction are equal to those of the translation. The absolute position translation

of this arrow in space is entirely immaterial. Technically the arrow is called a stroke. Its tail or initial point is its origin;

and

its

head or

final point, its terminus.

In the figure the

and the terminus by T. This geo designated by metric quantity, a stroke, is used as the mathematical origin

symbol

for all vectors, just as the ordinary positive and negative bers are used as the symbols for all scalars.

num

ADDITION AND SCALAR MULTIPLICATION *

As examples

3.] sity,

mass, time, den

of scalar quantities

and temperature have been mentioned. volume, moment

tance,

however,

Others are dis

of inertia, work, etc.

by no means the

distinguishing characteristics, as

its

dimensions in the sense well

cited.

distance

Magnitude,

sole property of these quantities.

Each implies something besides magnitude.

own

a time

known 3,

Each has

its

an example of which to physicists

a work

may

are very

3, etc.,

3 is, however, a property common the only one. Of all scalar quantiperhaps is the number It implies nothing but tities pure simplest. magnitude. It is the scalar par excellence and consequently

The magnitude

different.

them

it is

all

used as the mathematical symbol for

As examples

all scalars.

of vector quantities force, displacement, velo

Each of these has city, and acceleration have been given. other characteristics than those which belong to a vector pure

The concept

and simple. two alone vector.

But

to a rigid

two ideas and

magnitude of the vector and direction of the force is more complicated. When it is applied

body the

consideration;

And

of vector involves

line in

which

acts

must be taken

into

magnitude and direction alone do not suf

applied to a non-rigid body the point of application of the force is as important as the magnitude or Such is frequently true for vector quantities other direction.

fice.

than force.

in case

it is

Moreover the question of dimensions

as in the case of scalar quantities.

the stroke, which

The mathematical

present vector,

the primary object of consideration in this book, abstracts from all directed quantities their magni is

tude and direction and nothing but these

mathe matical scalar, pure number, abstracts the magnitude and Hence one must be on his guard lest from that alone. ;

just as the

analogy he attribute some properties to the mathematical vector which do not belong to it ; and he must be even more careful lest he obtain erroneous results

by considering the

VECTOR ANALYSIS

vector quantities of physics as possessing no properties other

than those of the mathematical vector.

never do to consider force and

its

For example

would

effects as unaltered

This warning may not be parallel shifting it may possibly save some confusion. necessary, yet Inasmuch as, taken in its entirety, a vector or stroke 4.] to

itself.

but a single concept, it may appropriately be designated by one letter. Owing however to the fundamental difference is

between scalars and vectors, it is necessary to distinguish Sometimes, as in mathe carefully the one from the other. matical physics, the distinction is furnished by the physical

Thus

interpretation.

must be scalars

scalar

but

;

n be the index

m, the mass, and the

force,

and

refraction

the time, are also

the

acceleration,

are

When, however,

the letters are regarded merely no with particular physical significance some symbols difference must be relied upon to distinguish typographical

vectors. as

vectors from scalars.

Hence

in this

book Clarendon type

used for setting up vectors and ordinary type for scalars. This permits the use of the same letter differently printed 1 to represent the vector and its scalar Thus if magnitude.

C be the

electric current in

magnitude and

direction,

C may

be used to represent the magnitude of that current if g be the vector acceleration due to gravity, g may be the scalar value of that acceleration ; if v be the velocity of a ;

moving

mass, v

may be

the magnitude of that velocity. The use of Clarendons to denote vectors makes it possible to pass from directed quantities to their scalar magnitudes by a mere

change in the appearance of a letter without any confusing change in the letter itself. Definition

Two vectors

the same magnitude 1

are said to be equal

and the same

direction.

This convention, however,

instances

when they have

is by no means invariably followed. In some would prove just as undesirable as it is convenient in others. It is

chiefly valuable in the application of vectors to physics.

ADDITION AND SCALAR MULTIPLICATION The

equality of two vectors

Thus

usual sign =.

and B

denoted by the

A=B

Evidently a vector or stroke is not altered by shifting it about parallel to itself in space. Hence any vector A = PP r

may

(Fig. 1) for the

segment

the point

In this

P falls

way

all

drawn from any assigned point

may

moved

vectors in space

may

;

parallel to itself until

and

upon the point

as origin

upon some point

be replaced by directed

segments radiating from one fixed point

Equal vectors

in space will of course coincide, when placed with their ter mini at the same point 0. Thus (Fig. 1) A = PP\ and B = Q~Q

both

fall

upon T

For the numerical determination of a vector are necessary. If r,

with

</>,

its

~OT. three scalars

These may be chosen in a variety of ways.

be polar coordinates in space any vector r drawn origin at the origin of coordinates may be represented

by the three scalars

</>,

6 which determine the terminus of

the vector.

r~(r,*,0). Or if #, y9 z be Cartesian coordinates in space a vector r may be considered as given by the differences of the coordinates a/, y

of its terminus

and those

#, y, z

x,y

of its origin.

y,z

z).

If in particular the origin of the vector coincide

with the

origin of coordinates, the vector will be represented three coordinates of its terminus r

When two

-(*

,*,

by the

* )

vectors are equal the three scalars which repre

sent them must be equal respectively each to each. one vector equality implies three scalar equalities.

Hence

VECTOR ANALYSIS

A vector A is said

Definition

magnitude A is Such a vector equal to

to be equal to zero

when

its

zero.

A is

called a null or zero vector

naught in the usual manner.

and

written

Thus

A = 0.

All null vectors are regarded as equal to each other without

any considerations

of direction.

In fact a null vector from a geometrical standpoint would that is to be represented by a linear segment of length zero a It would have a consequently wholly inde point. say, by terminate direction

or,

what amounts to the same

thing,

none at

be regarded as the limit approached by a If, however, vector of finite length, it might be considered to have that it

all.

direction

which

finite vector,

the limit approached by the direction of the the length decreases indefinitely and ap

when

proaches zero as a limit.

The

justification for disregarding

and looking upon all null vectors as equal is that when they are added (Art. 8) to other vectors no change

this direction

occurs and

when

the product

multiplied (Arts. 27, 31) by other vectors

is zero.

In extending to vectors the fundamental operations

5.]

and arithmetic, namely, addition, subtraction, and multiplication, care must be exercised riot only to avoid self-

of algebra

contradictory definitions but also to lay down useful ones. Both these ends may be accomplished most naturally and

by looking to physics (for in that science vectors con tinually present themselves) and by observing how such easily

If then A be a given displace or what is two, three, or in ment, force, velocity, general x times A? What, the negative of A? And if B be another,

quantities are treated there.

what

the

of A and B ? That and B taken together ?

sum

equivalent of

to say,

what

the

The obvious answers

to these questions suggest immediately the desired definitions.

ADDITION AND SCALAR MULTIPLICATION

Scalar Multiplication Definition:

6.]

positive scalar

and

Thus v

when

direction

its

said to be multiplied by a magnitude is multiplied by that scalar

vector

its

unaltered

is left

v be a velocity of nine knots East by North, 2 times a velocity of twenty-one knots with the direction still

East by North.

be the force exerted upon the scalepan by a gram weight, 1000 times f is the force exerted by a kilogram. The direction in both cases is vertically down if f

ward. If

A be

the vector

and x the

denoted as usual by

x It

is,

scalar the product of x

and

A is

however, more customary to place the scalar multiplier

before the multiplicand A. This multiplication by a scalar is called scalar multiplication, and it follows the associative law

x (y A) as in ordinary algebra

mediately obvious

and

(x y)

A=y

arithmetic.

when the

fact

(x A)

This statement

taken into consideration

that scalar multiplication does not alter direction but merely multiplies the length. Definition

Any

vector

vector a in

its

A unit vector is one whose

magnitude

unity.

A may

be looked upon as the product of a unit direction by the positive scalar A, its magni

tude.

A=A The unit vector

by I/A

= a A.

similarly be written as the product of or as the quotient of A and A.

may

A = ^1 A = -I A A

VECTOR ANALYSIS

reverses its

A~B

from

magnitude unchanged.

two feet

be A, the stroke

but which

prefixed to a vector

sign,

its

A be a displacement for two feet to

a displacement for

stroke

The negative

direction but leaves

For example

Definition

7.]

of the negative sign

of motion.

" reaction."

The

B A, which

of the

Another

Newton s

denote an "action," positive sign,

instead of

illustration of the use

be taken from

may

same length

B to A

in the direction from

to 5, will be

the right, Again if the

to the left.

+ may ,

third law

will denote the

be prefixed to a vec

tor to call particular attention to the fact that the direction

The two

has not been reversed.

signs

and

when used

in connection with scalar multiplication of vectors follow the

same laws

These are

of operation as in ordinary algebra.

symbolically

++=+

+- = -;

;

(ra

The

interpretation

+ = -;

= +;

A).

(

obvious.

Addition and Subtraction

The

8.]

addition of two vectors or strokes

may

be treated

most simply by regarding them as defining translations in space (Art. 2), Let S be one vector and T the other. Let P be a point of space (Fig. 2). The trans lation S carries P into P such that the 1

line

carry

into

parallel to FIG.

into

the line

T and equal

P P"

being

to it in magnitude.

Consequently the result of S followed by

P into the point be any other point in space, S will carry Q such that Q~Q = S and T will then carry Q into Q" T

P".

equal to S in magnitude and will then

The transformation T

direction.

to carry the point

now Q

ADDITION AND SCALAR MULTIPLICATION

= T.

such that Q Q"

Thus

S followed f

by T

carries

into Q".

For Q Q Q" the two sides Q Q and Q Q", being equal and parallel to S and T respectively, must be likewise parallel to P P and P P" respectively which are also parallel to S and T. Hence the triangle

Moreover,

equal to

P".

the third sides of the triangles

That

must be equal and

parallel

Q Q"

equal and parallel to

PP".

As Q

is any point in space this is equivalent to saying that means of S followed by T all points of space are displaced by the same amount and in the same direction. This displace

ment

therefore

translations S and

two

Consequently the

translation.

are equivalent to a single translation R.

Moreover S

The

= PP R

stroke

strokes S and

and T

called the

which

P", then resultant

it is

R = PP". or

equivalent.

sum This

of the

two

sum

noted in the usual manner by

R=S + From analogy with

the

sum

or resultant of two translations

the following definition for the addition of any two vectors laid

down.

Definition

The sum

or resultant of

two vectors

found

by placing the origin of the second first

upon the terminus of the and drawing the vector from the origin of the first to the

terminus of the second. Theorem.

9.]

added does not

The order affect the

which two vectors S and T are

sum.

S followed by T gives precisely the same result as T followed into by S. For let S carry (Fig. 3) ; and T, P into P". S + T then carries into P". Suppose now that T carries P into

The

P P " line PP

equal and parallel to

PP".

Con-

VECTOR ANALYSIS

sequently the points P,

P P

and

lie

at the vertices of

a parallelogram.

pm pn

allel

carries

Hence

e q ua l an(J

PP.

into

par-

Hence S P". T fol

lowed by S therefore car

P" through P\ T carries P into P" through P m The final result is in

ries

into

whereas S followed by

This

either case the same.

by writing

It is to

may be

+T=T+

common

origin

through P.

;

PP and T = PP m are the two sides pprpp" which have the point P as

be noticed that S =

of the parallelogram

designated symbolically

and that

This leads

JL=PP"

to another

stating the definition of the

sum

the diagonal

very

two

drawn

common way

vectors.

two vectors be drawn from the same origin and a parallelo gram be constructed upon them as sides, their sum will be that If

diagonal which passes through their common origin. " This is the well-known " parallelogram law according to

which the physical vector quantities force, acceleration, veloc It is important to ity, and angular velocity are compounded. note that in case the vectors

lie

along the same line vector

addition becomes equivalent to algebraic scalar addition. The of the if two vectors to be added are added the vectors lengths

have the same direction site directions.

;

but subtracted

In either case the

sum

they have oppo has the same direction if

as that of the greater vector. 10.]

After the definition of the

been laid down, the

sum

of several

sum

may

be found by adding

two vectors has

together the first two, to this sum the third, to this the fourth, and so on until all the vectors have been combined into a sin-

ADDITION AND SCALAR MULTIPLICATION The final

gle one.

result

the same as that obtained by placing upon the terminus of the

the origin of each succeeding vector

preceding one and then drawing at once the vector from the origin of the first to the terminus of the last. In case these two points coincide the vectors form a closed polygon

and that

sum is zero. Interpreted geometrically this states are such that the number of displacements R, S, T form the sides of a closed polygon taken in R, S, T

their

strokes

order, then the effect of carrying out the displacements

Each point

of space is brought

mechanics

terpreted in

back

is nil.

to its starting point.

states that

any number

of forces

act at a point and if they form the sides of a closed polygon taken in order, then the resultant force is zero and the point is in equilibrium under the action of the forces. of sequence of the vectors in a sum is of no con This may be shown by proving that any two adja

The order sequence. cent vectors

may

be interchanged without affecting the result.

To show

Let

Then Let now

A, B

= A B, C = B C, D =

B C = D. Then C B C D 1

consequently

D = C. Hence OJ = A + B + D +

= D E.

a parallelogram and

which proves the statement. Since any two adjacent vectors may be interchanged, and since the sum may be arranged in any order by successive interchanges of adjacent vectors, the order in which the vectors occur in the sum is immaterial. 11.] is

Definition

added

vector

said to be subtracted

after reversal of direction.

when

Symbolically,

A - B = A + (- B). By

this

means subtraction

reduced to addition and needs

VECTOR ANALYSIS

is however an interesting and difference of two vectors the of representing

no special consideration. important

way

A = OA, B =

Let

geometrically.

There

Complete

0IT(Fig. 4).

the parallelogram of which A and B Then the diagonal are the sides. C is the sum A + B of the ~OG

Next complete the B parallelogram of which A and = OB are the sides. Then the di two

vectors.

agonal 02)

will be the

sum

But the and is equal parallel segment OD Hence BA may be taken as the difference to the two to BA. This leads to the following rule The differ vectors A and B. ence of two vectors which are drawn from the same origin is and the negative of

the vector

drawn from the terminus

tracted

the terminus of the vector from which

of the vector to be it

sub

Thus the two diagonals of the parallelogram, which constructed upon A and B as sides, give the sum and dif

tracted. is

ference of

and

In the foregoing paragraphs addition, subtraction, and 12.] scalar multiplication of vectors have been defined and inter

To make

the development of vector algebra mathe exact and matically systematic it would now become necessary to demonstrate that these three fundamental operations follow preted.

the same formal laws as in the ordinary scalar algebra, al though from the standpoint of the physical and geometrical interpretation

_of

vectors this

may seem

superfluous.

laws are

m II

III,

n (m A)

= A+

(m n} A,

+ (B + C), A + B r, B + A, (m + n) A = m A + n A, m (A + B) = m A + m B, - (A + B) = - A - B.

(A III a

(n A)

These

ADDITION AND SCALAR MULTIPLICATION 1

the so-called law of association and commutation of

the scalar factors in scalar multiplication. I 6 is the law of association for vectors in vector addition.

states that in adding vectors parentheses may be inserted at any points without altering the result. 11 is the commutative law of vector addition.

III a

the distributive law for scalars in scalar multipli

the distributive law for vectors in scalar multipli

cation.

III 6 cation. Ill,

The

the distributive law for the negative sign. proofs of these laws of operation depend upon those

propositions in elementary geometry which have to deal with the first properties of the parallelogram and similar triangles. will not be given here; but

it is suggested that the out for the sake of fixing the fundamental reader work them

They

ideas of addition, subtraction,

The

clearly in mind. in the statement

and scalar multiplication more

result of the laws

may

summed up

The laws which govern addition, subtraction, and

scalar

multiplication of vectors are identical with those governing these operations in ordinary scalar algebra. It is precisely this identity of formal laws

the extension of the use of the familiar signs of arithmetic to the algebra of vectors and

which ensures the correctness of

which

justifies

=, +, and it is

results obtained

also this

by operat

ing with those signs in the usual manner. One caution only need be mentioned. Scalars and vectors are entirely different sorts of quantity.

to each other

For

this reason

they can never be equated where each is

except perhaps in the trivial case

For the same reason they are not to be added together. So long as this is borne in mind no difficulty need be antici pated from dealing with vectors much as if they were scalars. Thus from equations in which the vectors enter linearly with

zero.

VECTOR ANALYSIS

14 coefficients

scalar

unknown

vectors

may

be eliminated or

with the same limita found by solution in the same way and for the eliminations and solu as in ordinary algebra;

tions

scalar coefficients of the equations depend solely on the If for and not at all on what the variables represent.

tions

instance

cC +

aA + &B + then A, B,

D may

dD = 0,

be expressed in terms of the other

three

And two

= --:OA + &B + cC). a

vector equations such as

and yield

A+ 4B=E

A+

by the usual processes the solutions

A=3E-4F B

and

= 3 F - 2 E.

Components of Vectors 13.]

Definition

Vectors are said to be collinear

they are parallel to the same line; coplanar,

when

parallel

same plane. Two or more vectors to which no line can be drawn parallel are said to be non-collinear. Three or more vectors to which no plane can be drawn parallel are

to the

be non-coplanar.

said to

Obviously any two vectors are

coplanar.

vector b collinear with a

may be

expressed as the product of a and a positive or negative scalar which is the ratio of the magnitude of b to that of a. The sign is positive

Any

when b and

a have the same direction

have opposite directions.

then

;

OA =

negative, when they a, the vector r drawn

ADDITION AND SCALAR MULTIPLICATION from the origin either direction

any point of the

line

produced in

= x a.

(1)

x be a variable scalar parameter this equation

there

may

fore be regarded as the (vector) equation of all points in the

Let now be any point not that line produced the line or upon in either direction (Fig. 5).

OA.

line

Let

OB = b.

The

not of the form x a line parallel to point upon

it.

vector b

and

Let

The vector

collinear with a

and

Hence the vector drawn

R is 0~E=0~B + ITR r = b + #a.

This equation all

Flo 5

be any

consequently expressible as #a.

from

surely

Draw through B

may

(2)

be regarded as the (vector) equation of is parallel to a and of which

the points in the line which

B is

one point.

Any

14.]

a and b

and b

may

vector r coplanar with two non-collinear vectors be resolved into two components parallel to a

This resolution

respectively.

may

be accomplished by constructing the par allelogram (Fig. 6) of which the sides are parallel to a and b and of which the di

agonal is r. Of these components one is x a ; the other, y b. x and y are respec tively the

scalar ratios (taken with the

proper sign) of the lengths of these components to the lengths of a

and

Hence r

=xa+

(2)

a typical form for any vector coplanar with a and b. If several vectors r x , r 2 r 3 may be expressed in this form as is

VECTOR ANALYSIS

=x r2 = # r = x rl

their

sum

+ + +

2/ 2

b. 2/3

r is then

(ft

)

the well-known theorem that the components of a

This

sum

of vectors are the

sums

components of those

of the

If the vector r is zero

vectors.

each of

its

components must

be zero. Consequently the one vector equation r equivalent to the two scalar equations

(3)

vector r in space may be resolved into three components parallel to any three given non-coplanar vectors. Let the vectors be a, b, 15.]

Any

and

The

may then

resolution

accom

plished by constructing the parallelepiped (Fig. 7) of

which the edges a, b, and

are parallel to c

and of which the

agonal

FIG.

It.

This par-

allelopiped may be drawn easily by passing three planes parallel re c and a through the origin

b, b and c, and a similar set of three planes through its These six planes will then be parallel in pairs

spectively to a of the vector r

terminus

and

is r.

;

ADDITION AND SCALAR MULTIPLICATION and hence form a obvious.

The

where

and

x, y,

That the

parallelepiped.

which are

the planes are lines

parallel to

three components of r are x

intersections of a,

or b, or c

z are respectively the scalar ratios

and zc;

(taken with

the proper sign) of the lengths of these components to the

length of

and

a, b,

Hence

= #a +

7/b

(4)

a typical form for any vector whatsoever in space. Several vectors r lf r 2 , r 3 may be expressed in this form as

rx r2 1*3

Their sum r 1

= r + r2 + l

=x =# =X

+ + 2 a + Z l

yl b y2 b 2/3

+ +

z l c, *2 c

then F3

a + *2 + X Z + + + )!> (2/i 2/2+ 3/3 + Ol +^2 + ^3+ "O

0*1

If the vector r is zero

each of

its

three components

Consequently the one vector equation r

= 0.

zero.

equivalent to

the three scalar equations

xl 2/i

+ + +

#2 2/2

Should the vectors

+ #3 + + 2/3 + +%+

= =

= /y

(5)

be coplanar with a and b, all the com vanish. In this case therefore the above

all

ponents parallel to c equations reduce to those given before.

If two equal vectors are expressed in terms of the 16.] same three non-coplanar vectors, the corresponding scalar co efficients are equal.

VECTOR ANALYSIS

Let

=x a+y b+z x =x Then y=y = (x x a + (y - y ) b + (* - z ) c. r - r = For x - * = 0, y - y = 0, z - * = 0. Hence 1

)

But

this

would not be true

if a, b,

and

were coplanar. In

that case one of the three vectors could be expressed in terms of the other

two

= m a + n b. = #a + y b + s c = (a + m z) a + (y + = x a + y b + z c = (x + m z a + (y + r = [(x + m z ) (x + m z )] a, c

Then

r r

;

)

Hence the individual components of a and b (supposed different) are zero.

Hence

But

this

tively to

+ mz =

-f

z) b,

nz)

r in the directions

+ mz + nz

by no means necessitates x, y, z to be equal respec In a similar manner if a and b were colx\ y\ z 1

impossible to infer that their coefficients vanish The theorem may perhaps be stated as follows : individually.

linear it

In case two equal vectors are expressed in terms of one vector, or two non-collinear vectors, or three non-coplanar vectors, the .

But this is not ne corresponding scalar coefficients are equal. the two vectors be true collinear or the three vectors, ; if cessarily coplanar.

This principle will be used in the

applications

(Arts. 18 et seq.).

The Three Unit Vectors

i, j,

In the foregoing paragraphs the method of 17.] express ing vectors in terms of three given non-coplanar ones has been explained.

The

simplest set of three such vectors

the rect-

ADDITION AND SCALAR MULTIPLICATION

This in Solid Cartesian Geometry. angular system familiar distinct however be either of two very rectangular system may

In one case (Fig.

types.

8, first

part) the Z-axis

lies

upon

X Y- plane on which rotation through a right

that side of the

to the F-axis appears counterclockwise angle from the X-axis or positive according to the convention adopted in Trigonome

This relation

try.

may be

stated in another form.

If the

axis be directed to the right and the F-axis vertically, the Or if the X^-axis will be directed toward the observer. axis point toward the observer

the ^-axis will point upward.

and the F-axis to the Still another method of

right,

state-

Z ,,k

Left-handed

Right-handed FIG.

ment

common

in mathematical physics

and engineering.

a right-handed screw be turned from the Xaxis to the Faxis it will advance along the (positive) Z-axis. Such a sys

tem

of axes is called right-handed, positive, or counterclock It is easy to see that the F-axis lies upon that side of

wise. 2

the ^X-plane on which rotation from the ^-axis to the Xaxis is counterclockwise ; and the X-axis, upon that side of 1

the X-, Y-, or Z-axis the positive half of that axis

meant.

The

X Y-

0. plane means the plane which contains the X- and Y-axis, i. e., the plane z 2 convenient right-handed system and one which is always available consists

of the thumb, first finger, and second finger of the right hand. If the thumb and first finger be stretched out from the palm perpendicular to each other, and if the

second finger be bent over toward the palm at right angles to first finger, a righthanded system is formed by the fingers taken in the order thumb, first finger,

second

finger.

VECTOR ANALYSTS

the F^-plane on which rotation from the axis

counterclockwise.

Thus

F-axis to the Z-

appears that the relation

between the three axes

is perfectly symmetrical so long as the If a is observed. order right-handed cyclic screw is turned from one axis toward the next it advances

XYZXY

same

along the third. In the other case (Fig. that side of the

angle from the

The

ative.

second part) the ^-axis lies upon F-plane on which rotation through a right 8,

JT-axis to the F-axis appears

F-axis then lies

upon

clockwise or neg

that side of the

^X-plane

on which rotation from the ^-axis to the X-axis appears clockwise and a similar statement may be made concerning the X-axis in

its

as the

same

In this case,

relation to the F^-plane.

the relation between the three axes cyclic order

X YZX Y

S3 mmetrical so long preserved but it is just

the opposite of that in the former case.

If a fe/Mianded

turned from one axis toward the next

Hence

the third.

this

system

too,

screw

advances along

called left-handed, negative,

or clockwise. 1

The two systems are not superposable.

One

metric. mirror.

If the JT-

the

They

other

are

sym

seen in a

image and F-axes of the two different systems be

superimposed, the ^-axes will point in opposite directions. Thus one system may be obtained from the other by reversing the direction of one of the axes. that

little

thought will show

two of the axes be reversed in direction the system will not be altered, but if all three be so reversed it will be. if

Which asmuch

two systems be used, matters little. But in the formulae of geometry and mechanics differ

of the

slightly in the all

which

matter of sign, it is advisable to settle once for shall be In this book the adopted. right-handed or

counterclockwise system will be invariably employed.

A left-handed system may be formed the by one was formed by the right. 1

left

hand

just as

a right-handed

ADDITION AND SCALAR MULTIPLICATION Definition

The

three letters

will be reserved to de

note three vectors of unit length drawn respectively in the directions of the JT-, T-, and Z- axes of a right-handed rectan gular system. In terms of these vectors, any vector

= xi +

The

coefficients

coordinates.

be expressed as

zk.

(6)

are the ordinary Cartesian coordinates

y, z

of the terminus of r

may

if its

origin be situated at the origin of

The components

of r parallel to the X-, F-,

and

^f-axes are respectively

The

rotations about

about k from

By means

from

z k.

to k, about

from k to

and

are all positive.

of these vectors

k such a correspondence is and the analysis in Car

established between vector analysis tesian coordinates that

becomes possible to pass at will There is nothing contradic

from either one to the other.

On the contrary it is often desirable tory between them. or even necessary to translate the formulae obtained by vector methods into

Cartesian coordinates for the sake of

comparing them with results already known and it is still more frequently convenient to pass from Cartesian

on account of the brevity thereby obtained and because the vector expressions show forth the analysis to vectors both

intrinsic

meaning

of the formulae.

Applications *18.J Problems in plane geometry easily

by vector methods.

Any two

may

which

others in that plane

may

be solved

non-collinear vectors in

be taken as the fundamental ones in terms of

the plane all

may frequently

may

also be selected at pleasure.

be expressed.

Often

The

origin

possible

VECTOR ANALYSIS

make such an advantageous choice of the origin and funda mental vectors that the analytic work of solution is materially simplified.

the

same

The

adaptability of the vector

method

as that of oblique Cartesian coordinates

about

with differ

ent scales upon the two axes. Example 1 : The line which joins one vertex of a parallelo gram to the middle point of an opposite side trisects the diag

onal (Fig.

Let

9).

A BCD

be the parallelogram, the line joining the vertex B to the middle point of the side

AD, R

the point in which this line cuts the

diagonal

A R is one third of origin, A B and AD as the

To show

Choose A as two fundamental vectors S and T. Then the sum of S and T. Let AR = R. To show

AC.

FlG 9

T).

where x

the ratio of

R=y

And where y

ER to EB

the scalar ratio of

an unknown

scalar.

T), to

to be

shown equal

to. Hence or

x (S

-i

* S

X) T

=yS+

Hence, equating corresponding coefficients (Art. 16),

2 (1

- x) = y.

ADDITION AND SCALAR MULTIPLICATION From which Inasmuch

y x

also

well as the diagonal

must be

the line j&2?

trisected as

A C.

If through any point within a triangle lines Example 2 drawn parallel to the sides the sum of the ratios of these :

lines to their corresponding sides is

Let

ABC be

as origin,

and

the triangle,

A B and A C as

it. Choose S two fundamental vectors the

the point within

Let

AR = R = w S + m

parallel to

the fraction of

tion (1

A B which is

A C. The

(a)

cut off by the line through B must be the frac

remainder of

m) S. Consequently by

the line parallel to

similar triangles the ratio of

A C itself is A B to the

to the line

Similarly the ratio of the line parallel to itself is

Next express

n ).

third side of the triangle.

R= Hence (m line

+ ri) S

through

ri)

and the theorem

B C.

angles the ratio of this line to the three ratios

- m) +

of S

and T

S the

Evidently from (a)

the fraction of

parallel to

R in terms

ra).

line

- S).

A B which is

cut off by the

Consequently by similar

BC itself is (m + n).

- n) +

(m +

ri)

tri

Adding

= 2,

proved.

If from any point within a parallelogram lines Example 3 be drawn parallel to the sides, the diagonals of the parallelo :

grams thus formed intersect upon the diagonal of the given parallelogram.

Let

and

A B CD

be a parallelogram,

LN two lines through R parallel

a point within respectively to

it,

AB and

VECTOR ANALYSIS

AD,

the points K, Z,

N lying

upon the

sides

DA, AS,

B C, CD respectively. To

show that the diagonals and and LBME meet LM of the two parallelograms on A C. Choose A as origin, A B and A D as the two funda mental vectors S and T. Let

KRND

R= and

P be the

let

AB = m S

KN with LM.

point of intersection of

KN=KR + BN = m S +

Then

=(1 -m)

rc)

= n T + x [m S + (1 n) T], P = m S + y [(1 - m) S + n T].

Hence and Equating

coefficients,

4- ft T,

solution,

m=m+ ;

m+ ~

m m+

Substituting either of these solutions in the expression for P, the result is

P^-^-^S +

T),

is collinear with A C. Problems in three dimensional geometry may be essentially the same manner as those in two dimen

which shows that P *

19.]

solved in sions.

In this case there are three fundamental vectors in all others can be expressed. The method of

terms of which solution

analogous to that in the simpler case.

Two

ADDITION AND SCALAR MULTIPLICATION

The co

expressions for the same vector are usually found.

In this

corresponding terms are equated.

efficients of the

way

the equations between three unknown scalars are obtained from which those scalars may be determined by solution and

then substituted in either of the expressions for the required vector. The vector method has the same degree of adapta the Cartesian

bility as

method

which oblique axes with

The following examples like worked out not so much for

different scales are employed.

those in the foregoing section are

their intrinsic value as for gaining a familiarity with vectors.

Example 1 point within

Let

A B CD

a tetrahedron

and

any

P and produce the lines To faces in A\ B C D

Join the vertices to

it.

until they intersect the opposite

show

A~A Choose

TTB

as origin,

~C~O

and the edges A J?, A C, Let the vector

three fundamental vectors B, C, D.

A =AA =AB+

Also

The

= A P=IE + raC +

vector B.

Hence

coefficients

Hence

(C-B)+y (D~B). m = xv 1

&.,1

= I

and

PA _ JL-1 PA*

ZZ ~~& 7

WC =

A =B+ Equating

B and C coplanar with in of be terms them. may expressed

+m+n "

as the

A P be

VECTOR ANALYSIS

A B = # 2 C + y2 D A B = ^t + B B = B + & 2 (P - B). C + y 2 D = B + 2 (ZB + mC + ^D-B 2 = 1 + *, (J - 1),

In like manner

and

Hence

and

Hence

-i

and In the same

Adding

way

may be shown

.PL

CC*

the four ratios the result i

that

JL vn

-4-

^ _L 7 J_

-I-

Example % : To find a line which passes through a given point and cuts two given lines in space. Let the two lines be fixed respectively by two points A and B, C and D on each. it as origin and let

Let

be the given point.

Any

point

P of A B may be

P= OP= Any

point

If the points

are collinear

0~A

That

= ~OC, D=d~D.

expressed as

= A + x (B- A).

+ xA

CD may likewise

P and Q lie

Choose

in the

be written

same

line

through

P and

ADDITION AND SCALAR MULTIPLICATION

possible to equate coefficients one of the four of the other three.

Before

it is

vectors

must be expressed in terms

Then Tf

=A+ (1

_1_

- A)

x (B

_J_

_1_

1 x=zy x = zy m,

Hence

y (n

Substituting in

P and

= i-

I^^T

- 1)J.

_________

ft I

Either of these

and cutting

A+ m B +m

be taken as defining a line drawn from

may

A B and

CD.

Vector Relations independent of the Origin

Example 1

20.]

divide a line

find the vector

the terminus ratio

= ~OP of

divides

-f-

is,

= B.

which in the

B FIG. 10.

m That

in a given ratio

(Fig. 10).

Choose any arbitrary point Let OA = A and OB origin.

B=A

- A).

=nA+ mB n

(7)

VECTOR ANALYSIS

The components

the line

parallel to

and to the segments divided by the point P. If

PB it

and B are in inverse

into

which the

should so happen that

externally, the ratio

A P / PE

ratio

A B is

line

P divided

would be nega

and the signs of m and n would be opposite, but the formula would hold without change if this difference of sign in m and n be taken into account. tive,

Example 2

find the point of intersection of the

medians

of a triangle.

at random. Choose the origin Let A BC be the given = A, ()B = B, and "00 = C. Let A ,C triangle. Let 0~A be respectively the middle points of the sides opposite the f

vertices A, B, (7. Let be the point of intersection of the medians and M = the vector drawn to it. Then

and

~< = B that has been chosen outside of the plane of the so that C are B, A, triangle non-coplanar, corresponding coeffi

Assuming

cients

Hence

may

be equated.

= 9y

M =4 (A + B + C).

ADDITION AND SCALAR MULTIPLICATION The vector drawn

to the

to one third of the

sum

median point of a triangle

of the vectors

drawn

29 equal

to the vertices.

In the problems of which the solution has just been given the origin could be chosen arbitrarily and the result is in dependent of that choice. Hence it is even possible to disre

gard the origin entirely and replace the vectors A, B, C, etc., by their termini A, B, C, etc. Thus the points themselves

become the subjects of analysis and the formulae read

A +mB m+n

M=~(A

and This

+B+

C).

typical of a whole class of problems soluble

by vector between the

In fact any purely geometric relation must necessarily be independent of the origin assumed for the analytic demonstration. In methods.

different parts of a figure

some

cases,

such as those in Arts. 18, 19, the position of the

origin may be specialized with regard to some crucial point of the figure so as to facilitate the computation but in many ;

other cases the generality obtained by leaving the origin unspecialized and undetermined leads to a symmetry which

renders the results just as easy to compute and more easy to

remember.

: The necessary and sufficient condition that a vector equation represent a relation independent of the origin is that the sum of the scalar coefficients of the vectors on

Theorem

equal to the sum of the coefficients of the vectors upon the other side. Or if all the terms of a vector equation be transposed to one side leaving

one side of the sign of equality

zero on the other, the

sum

of the scalar coefficients

be zero.

Let the equation written in the

latter

form be

must

VECTOR ANALYSIS

to by adding a constant vector origin from The equation to each of the vectors A, B, C, D ----

Change the

B = OO

then becomes

a (A

B) +

If this is to be

must vanish.

That is

+ B) +

d (D

+ R) +

independent of the origin the coefficient of

Hence

two examples

this condition is fulfilled in the

cited

obvious.

m+ m+ l

M = \3 (A

* 21.]

The necessary and

f B

C),

sufficient

condition that two

vectors satisfy an equation, in which the coefficients is zero, is that the vectors

and

of the scalar

be equal in magnitude

in direction.

First let

and

It is of course

vanish.

+ 6B = + 6 = 0.

assumed that not both the

If they did the equation

stitute the value of

the

sum

a and

would mean nothing. Sub

a obtained from the second equation into

first.

-&A + Hence

coefficients

= 0.

ADDITION AND SCALAR MULTIPLICATION Secondly

and B are equal in magnitude and direction

the equation

subsists

A-B = The sum

between them.

The necessaiy and

of the coefficients

is zero.

sufficient condition that three vectors

satisfy an equation, in which the sum of the scalar coefficients is zero, is that when drawn from a common origin they termi nate in the same straight line. 1

aA + 6B +

First let

and

+ c = 0.

Not all the coefficients a, J, c, vanish or the equations would be meaningless. Let c be a non-vanishing coefficient. Substitute the value of a obtained from the second equation into the first.

Hence the vector which

joins the extremities of

C and

joins the extremities of A and B. Hence those three points -4, -B, C lie on a line. Secondly drawn from OB,G= OA, B suppose three vectors

collinear with that

which

the same origin vectors

terminate in a straight

line.

Then

the

AB = B - A and A~C = C - A are collinear.

subsists.

Hence

The sum

the equation

of the coefficients

on the two sides

the same.

The necessary and in which the sum of 1

sufficient condition that

an equation,

the scalar coefficients

zero, subsist

Vectors which have a common origin and terminate in one Hamilton " termino-collinear:

line are called

VECTOR ANALYSIS

between four vectors, is that 1 they terminate in one plane. aA

First let

and

drawn from a common

origin

+ 6B + cC + dV = a + b + c + d = Q.

Let d be a non-vanishing coefficient. Substitute the value of a obtained from the last equation into the first.

- A) = 6

d (D

The

line

termini A, B,

(7,

of A, B, C,

= D - A, ~AC = C - A, One

planar vectors. the other two. /

where

C).

Hence

all

four

one plane. Secondly do lie in one plane.

and ~AB

=B-A

them may be expressed

- A) +

m, and n are certain is

n (D

scalars.

are co

in terms of

- A) = 0,

The sum

of the coeffi

zero.

Between any five vectors there of whose coefficients is zero. B,

This leads to the equation

cients in this equation

Let A,

lie

suppose that the termini of A, B,

Then AZ)

A B and A C.

coplanar with

- B) +

C,D,E be the

five

exists one equation the

sum

Form

the

given vectors.

differences

E-A,

E--B,

E-C, E-D.

One of these may be expressed - or what amounts to the same

in terms of the other three

thing there must exist an

equation between them. ft

The sum 1

A) +

- B) + m

- C) +

of the coefficients of this equation

- D) =

is zero.

Vectors which have a common origin and terminate in one plane are called " by Hamilton termino-complanar."

ADDITION AND SCALAR MULTIPLICATION

The results of the foregoing section afford simple *22.] solutions of many problems connected solely with the geo metric properties of figures. Special theorems, the vector equations of lines and planes, and geometric nets in two and three dimensions are taken up in order.

Example

If a line

be drawn parallel to the base of a

triangle, the line which joins the opposite vertex to the inter section of the diagonals of the

trapezoid thus formed bisects the base (Fig. 11).

ABC

Let

be the triangle,

the line parallel to the base CB, G the point of intersection of the diagonals

pezoid tion of

that

and

CBDE,

of the tra

and Fthe

bisects

CB.

origin at random.

intersec

To show

with CB.

FI(J

Choose the Let the vectors drawn from

to the

various points of the figure be denoted by the corresponding

Then

Clarendons as usual. lel to

CB,

E-D holds true. it

since

by hypothesis paral

the equation

The sum

should be.

- B)

of the coefficients

Rearrange

the

is evidently zero as terms so that the equation

takes on the form

nC =

The vector E n C is coplanar with E and C. It must cut the line EC. The equal vector D 7&B is coplanar with D and B. It must cut the line DB. Consequently the vector represented by either side of this equation must pass through the point A. Hence

7iC

?iB

= #A.

VECTOR ANALYSIS

However the points E, 0, and A lie upon the same straight Hence the equation which connects the vectors E,C, line. is zero. and A must be such that the sum of its coefficients This determines x as 1

Hence

=D-

- w) A.

another rearrangement and similar reasoning

E+ Subtract the

first

=D + 7iC=

cuts

vector

n)Qt.

second equation from the

n (B + C) This

n) G

and AQ.

- n) A.

must therefore be a

a multiple that the sum of the coeffi multiple of F and such cients of the equations which connect B, C, and F or 0, A, and F shall be zero.

n (B + C)

Hence

+ )G F

Hence

)

2 nf.

The proof has covered considerable space because each detail of the reasoning has been given. In reality, however, the actual analysis has con and the theorem has been proved.

sisted of just four equations obtained simply

Example %

To determine

from the

first.

the equations of the line and

plane.

Let the line be fixed by two points A and B upon it. Let be any point of the line. Choose an arbitrary origin.

The

vectors A, B, and

P terminate

in the

aA + 6B a + I + p = 0.

and ,

Therefore

= a

same

line.

Hence

ADDITION AND SCALAR MULTIPLICATION For different points values.

the scalars a and b have different

be replaced by x and

They may

more generally

which are used

Then

to represent variables.

Let a plane be determined by three points -4, B, and C. Let P be any point of the plane. Choose an arbitrary origin.

The

vectors A, B, C, and

terminate in one plane.

Hence

6B + cC

+ c+p = Q. aA + 6B + cC P= a

and Therefore

-f c

a, 6,

vary for different points of the plane,

customary to write in their stead

+ Example 3 ;

The

line

which

it is

joins

one vertex of a com

plete quadrilateral to the intersection of divides the opposite sides har

two diagonals

monically (Fig. 12). be four vertices Let A, B, C,

A B meet

Let

of a quadrilateral.

and

in a fifth vertex E,

meet

Let the two diagonals

intersect in

that

and

by E.

2?" in the

That

and

To show

intersects

such that the lines

in the sixth vertex F.

AB same

in a point

and

i" and

(7I> are

in a point

divided internally at

ratio as they are divided externally

show that the

cross ratios

VECTOR ANALYSIS

Choose the origin at random. The four vectors A, B, C, D drawn from it to the points A, B, C, D terminate in one

Hence

plane.

and

+e+

= 0.

two terms Separate the equations by transposing

a Divide

+c=

+c aA + d D F= In like manner a + d (a + C )G (a + d)F " Form: + c) (a + d) (a (a + c)Q (a + rf)F a

+d 6B + cC b + c b

cC (a

Hence

Separate the equations again and divide

+ c +

aA + EB _

d).

6B +

-f b

(6)

b and CD in the ratio But equation (a) shows that JE divides C D in the c:d. Hence E and E" divide CD internally and

2? divides

in the ratio a

c / d.

ratio

same

externally in the

ratio.

Which

of the

two divisions

and which external depends upon the relative signs and d. If they have the same sign the internal point

internal of c

of division is

and

E may

Example 4

E; -

opposite signs, be shown to divide

1 .

In a similar way

A B harmonically.

discuss geometric nets.

a geometric net in a plane

of points

it is

and straight

Start with a certain

meant a

figure

composed

lines obtained in the following

number

of points all of

which

manner.

lie in

one

ADDITION AND SCALAR MULTIPLICATION Draw

joining these points in pairs. These lines will intersect each other in a number of points.

plane.

all

the

lines

the lines which connect these points in pairs. This second set of lines will determine a still greater number

Next draw of points

all

which may in turn be joined in

pairs

and so

on.

The construction may be kept up indefinitely. At each step the number of points and lines in the figure increases. Probably the most interesting case of a plane geometric net that in

which four points are given

commence

with.

Joining these there are six lines which intersect in three Three new lines may points different from the given four.

now be drawn in the figure. These cut out six new points. From these more lines may be obtained and so on. To treat this net analytically write down the equations =

and

+b+c+ d=

(c)

which subsist between the four vectors drawn from an unde

From

termined origin to the four given points.

these

it is

possible to obtain

Tjl

+ dD c + d Z>B + dD

A + 6B a + b A + cC a + c A + dJ) a + d

+d &B + cC b + c b

two parts and

dividing. Next four vectors such as A, D, E, F may be chosen and the equa tion the sum of whose coefficients is zero may be determined.

splitting the equations into

This would be

aA + dV +

treating this equation as (c) be obtained*

E-f

= 0.

was treated new points may

VECTOR ANALYSIS

H= 1

A + dD a

aA +

+ (a + 2a + b + c 6)E

ft)E __

<?D+ a

4- c

+ +

c)F

+ c)F +d

Equations between other sets of four vectors selected from A, B, C, D, E, F, may be found and from these more points ;

The

process of finding more points goes forward fuller account of geometric nets indefinitely. may be found in Hamilton s " Elements obtained.

of Quaternions,"

Book

regards geometric nets in space just a word may be said. Five points are given. From these new points may be obtained by finding the intersections of planes passed through sets of three of the

remaining

given points with lines connecting the construction may then be carried for

The

pairs.

ward with the points thus obtained. is

similar to

five vectors five points.

the

sum

that in the

The analytic treatment case of plane nets. There are

drawn from an undetermined origin to the given Between these vectors there exists an equation whose

This equation

coefficients is zero.

separated into parts as before and the be obtained.

then

= a

+ 6B +b

A + cC a

new

points

may be may thus

+ dD + +d+e

6B + dV + b

two of the points and others may be found in the same Nets in are also discussed by Hamilton, loc. cit. way. space are

ADDITION AND SCALAR MULTIPLICATION

Centers of Gravity *

The

23.]

center of gravity of a

system of particles

may

be found very easily by vector methods. The two laws of physics which will be assumed are the following: 1. The center of gravity of two masses (considered as situated at points) lies on the line connecting the two masses and divides it into two segments which are inversely pro portional to the masses at the extremities.

In finding the center of gravity of two systems of masses each system may be replaced by a single mass equal in magnitude to the sum of the masses in the system and situated at the center of gravity of the system. Given two masses a and b situated at two points

Their center of gravity

and B.

given by

where the vectors are referred

any origin whatsoever.

This follows immediately from law 1 and the formula (7) for division of a line in a given ratio.

The

center of gravity of three masses

three points

-4,

masses a and

mass a

C may

may

b situated at the

Hence

= (a +

situated at the

be found by means of law

The

be considered as equivalent to a single point

Then

a, J, c

A + &B a + b "

A + 6B a

-f-

= aA-h&B-f-cC a

-f b

VECTOR ANALYSIS

number of masses Evidently the center of gravity of any at the points A, B, C, D, ... may a, &, c, d, ... situated

The

be found in a similar manner.

aA +

+ cO + rfD + + b + c + d + ...

The

lines

which

...

ftB

Theorem 1

result

join the center of gravity of a

divide it into three triangles which triangle to the vertices are proportional to the masses at the op

Let A, B,

posite vertices (Fig. 13).

be the vertices of a triangle weighted with masses a, &, c. Let G be the cen

Join A, B,

ter of gravity.

the opposite sides in

G and

produce the lines until

they intersect

A B\ C

To show

respectively.

that

the areas

G B C G CA G A B A B C = :

The

last

proportion between

from compounding the

ABC

first three.

and a

is,

+ b

comes

however, useful in

the demonstration.

Hence

ABC AA GBC~ GA! ABC GBC

In a similar manner

and

CTA a

GA G~A

b f

BCA GCA~ CAB ~_ GAB

Hence the proportion is proved. Theorem 2 : The lines which

~~c

join the center of gravity of a tetrahedron to the vertices divide the tetrahedron into four

ADDITION AND SCALAR MULTIPLICATION

tetrahedra which are proportional to the masses at the oppo site vertices.

D be

the vertices of the tetrahedron weighted be the center of respectively with weights a, &, c, d. Let to G and produce the lines until Join A, B, C, gravity.

Let

-4,

B, C,

they meet the opposite faces in the volumes

A B\

To show

that

BCDG:CDAG:DABG:ABCG:ABCD

BCDA BCDG

In like manner

and

ABCD

which proves the proportion. * 24.]

a suitable choice of the three masses, a,

J, c

(7, the center of gravity G may be made to coincide with any given point of the triangle. If this be not obvious from physical considerations it cer

cated at the vertices A, B,

tainly becomes so in the light of the foregoing theorems. For in order that the center of gravity fall at P, it is only

necessary to choose the masses a, 6, c proportional to the areas of the triangles PEG, PCA^ and respectively.

PAB

Thus not merely one infinite number of

set of masses a,

which

may

be found, but

from each other only a common factor of These quantities by proportionality. sets

differ

VECTOR ANALYSIS

42 a, 6, c

may

points

therefore be looked

ABC.

the triangle

inside of

as coordinates of the

upon

To each

set there

and to each point corresponds a definite point P,

number

corresponds an infinite

however do not

differ

obtain the points

ABC

of the triangle

which

of sets of quantities,

from one another except for a factor

of proportionality.

there

of the plane

one

may

ABC which lie outside

resort to the conception of

of gravity of the negative weights or masses. The center and 1 situated at the points masses 2 and respectively would be a point G dividing the line externally in the

A AB

ratio 1

That

point of the line a suitable set of masses

Any

any point

a, b

of the

plane suitable set of masses a, 6,

from the other two

ABC.

Inasmuch

produced may be represented by which differ in sign. Similarly

ABC

may be

represented by a

c of which one will differ in sign the point lies outside of the triangle as only the ratios of a, 6, and c are im

portant two

of the quantities may always be taken positive. idea of employing the masses situated at the vertices as coordinates of the center of gravity is due to Mobius and was published by him in his book entitled " Der barycentrische

The

Calcul" in 1827. point of

modern

This

be fairly regarded as the starting

analytic geometry.

The conception in nature

may

of negative masses

which have no existence

be avoided by replacing the masses at the vertices by the areas of the triangles GBC, A, and to which they are proportional. The coordinates of

may

GAB

a point

would then be three numbers proportional

areas of the three triangles of which and the sides of a given triangle

P is

ABC,

of these areas

the

common

the bases.

to the

vertex

The

determined by the following definition.

;

sign

ADDITION AND SCALAR MULTIPLICATION

a triangle is said to be the vertices A, B, C follow each other in the

The area

Definition: positive

ABC

when

positive or counterclockwise

direction

upon the

circle

scribed through them. The area is said to be negative when the points follow in the negative or clockwise direction. letters therefore does not alter Cyclic permutation of the the sign of the area.

Interchange of two letters which amounts to a reversal of the cyclic order changes the sign.

A CB = BA If

P be

= CBA = -A B C.

any point within the triangle the equation

PAB+PBC+PCA=ABC must

The same

hold.

will also hold if

be outside of the

triangle provided the signs of the areas be taken into con The areas or three quantities proportional to sideration. them may be regarded as coordinates of the point P. " coordinates to The extension of the idea of " barycentric

space is immediate. The four points A, B, C, D situated at the vertices of a tetrahedron are weighted with mass a, J, c, d center of gravity G is represented by or four others proportional to them. To quantities obtain points outside of the tetrahedron negative masses

The

respectively.

these

be employed. Or in the light of theorem 2, page 40, may be replaced by the four tetrahedra which are proportional to them. Then the idea of negative vol

may

the masses

umes takes the idea it

here

place of that of negative weights.

this

of considerable importance later, a brief treatment of

not be out of place.

may

Definition

to be positive

The volume A B CD of a tetrahedron is said when the triangle ABC appears positive to

VECTOR ANALYSIS

the eye situated at the point D. The volume is negative if the area of the triangle appear negative. To make the discussion of the signs of the various tetrahedra perfectly clear solid

modeL

A plane

difficult to see

from

it is

almost necessary to have a

drawing is scarcely sufficient. It is which triangles appear positive and

which negative. The following relations will be seen to hold if a model be examined. The interchange of two letters in the tetrahedron A BCD changes the sign.

ACBD = CBAD=BACD=DBCA The sible

sign of the tetrahedron for any given one of the pos twenty-four arrangements of the letters may be obtained

by reducing that arrangement to the order A B C D by means of a number of successive interchanges of two letters. If the

number

as that of

of interchanges

A B CD ;

even the sign

odd, opposite.

the same

Thus

any point inside of the tetrahedron

A B CD

the

equation

ABCP-BCDP+ CDAP-DABP=ABCD P

holds good. It still is true if be without the tetrahedron provided the signs of the volumes be taken into considera tion. The equation may be put into a form more symmetri cal

and more

to one

easily

number.

remembered by transposing

all

the terms

Then

The

proportion in theorem 2, page 40, does not hold true the signs of the tetrahedra be regarded. It should read

BCDG:CDGA:DGAB:GABC:ABCD

ADDITION AND SCALAR MULTIPLICATION

point G- lies inside the tetrahedron a, J, c, d repre sent quantities proportional to the masses which must be If the

A,B,C,D respectively if G is to be the G lies outside of the tetrahedron they may

located at the vertices

center of gravity. If

or be regarded as masses some of which are negative four as numbers ratios determine better whose merely perhaps still

the position of the point Gr. In this manner a set of "bary" centric coordinates is established for space.

The vector P drawn from an indeterminate point of the plane

A B C is

origin to

any

(page 35)

aA + yB + zC x + y + z Comparing

this

with the expression

aA + &B + cC a + b + c it

will be seen that the quantities x, y, z are in reality nothing

more nor

less

than the barycentric coordinates of the point ABO. In like manner from

with respect to the triangle equation

__#A + yB + 2C + x

+z+w

which expresses any vector P drawn from an indeterminate origin in terms of four given vectors A, B, C, D drawn from the

same

origin, it

may

be seen by comparison with

+ &B + c C + rfD +b+c+d x, y, 2, w are precisely

a that the

four quantities

the bary

centric coordinates of P, the terminus of P, with respect to

CD. Thus the vector methods in which the origin is undetermined and the methods of the " Bary " centric Calculus are practically co-extensive.

the tetrahedron

was mentioned before and

may

be well to repeat here

VECTOR ANALYSIS

46 that the origin

may

the vectors replaced by

left

wholly out of consideration and

The

their termini.

vector equations

then become point equations

A + y B 4-

+ xA + yB + zC + wD w. x + y + z x

and

This step brings in the points themselves as the objects of " to the " Barycentrische Calcul analysis and leads still nearer of

Mobius and the "Ausdehnungslehre The Use of Vectors Definition:

25.]

Grassmann.

Areas

denote

area lying in one

bounded by a continuous curve itself

PQR

and

which nowhere cuts

when

said to appear positive from the point

PQR

letters

plane

the

each

other in the counterclockwise

or positive order;

when

they

negative, in the

negative or clockwise order (Fig. 14). It is evident that an area

can have no determined sign per se, but only in reference to that direction in

boundary

supposed to be traced

side of its plane.

PQR;

For the area

and

and an area viewed from

which

some point

its

out

negative relative to negative relative to the is

same area viewed from a point O upon the side of the plane A circle lying in the F-plane and described opposite to 0. f

in the positive trigonometric order appears positive

from every

axis point on that side of the plane on which the positive lies, but negative from all points on the side upon which

ADDITION AND SCALAR MULTIPLICATION the negative ^-axis

and the direction

lies.

For

view must be boundary kept

this reason the point of

of description of the

clearly in mind.

Another method of stating the

definition

as follows

a person walking upon a plane traces out a closed curve, the area enclosed is said to be positive if it lies upon his left-

hand

negative if upon his right. It is clear that if two persons be considered to trace out together the same curve by walking upon opposite sides of the plane the area enclosed side,

will lie

upon the right hand

other.

To one

and the

left

hand

of the to the

;

That side of the plane upon which the area

other, negative.

seems positive

of one

will consequently appear positive

called the

positive

side

;

the side

upon

which appears negative, the negative side. This idea is familiar to students of electricity and magnetism. If an it

around a closed plane curve the lines of magnetic force through the circuit pass from the negative to electric current flow

the positive side of the plane. positive magnetic pole placed upon the positive side of the plane will be repelled by the circuit.

A plane

area may be looked upon as possessing more than or positive negative magnitude. It may be considered to possess direction, namely, the direction of the normal to the Hence a plane positive side of the plane in which it lies. area

areas

a vector quantity.

when looked upon

Theorem 1

magnitude direction

The following theorems concerning as vectors are important.

If a plane area be

denoted by a vector whose

the numerical value of that area and whose

the normal upon the positive side of the plane,

then the orthogonal projection of that area upon a plane will be represented by the component of that vector in the direction normal to the plane of projection (Fig. 15). Let it be projected Let the area lie in the plane MN.

orthogonally upon the plane

Let

M N&nd M* N

inter-

VECTOR ANALYSIS

48 sect in the line

and

angle between these a rectangle PQJRS in SP are respectively parallel

let the diedral

two planes be x. Consider whose sides, PQ, RS and QR,

first

and perpendicular rectangle

will be equal will be equal

to the line

This will project into a

R S in M N The sides P Q and JR S to PQ and US; but the sides Q R and S P to QR and SP multiplied by the cosine of #, f

the angle between the planes.

Consequently the rectangle

FIG. 15.

Hence

rectangles, of which the sides are respectively and parallel perpendicular to I, the line of intersection of the two planes, project into rectangles whose sides are likewise

respectively parallel

equal

and perpendicular

and whose area

the area of the original rectangles multiplied

by the

cosine of the angle between the planes.

From

this it follows that

area which

of the angle

any area

projected into an by the cosine

equal to the given area multiplied

between the planes.

For any area

A may be

vided up into a large number of small rectangles by drawing a series of lines in

MN parallel and perpendicular to the line

ADDITION AND SCALAR MULTIPLICATION

Each of these rectangles when projected is multiplied by the cosine of the angle between the planes and hence the total On the area is also multiplied by the cosine of that angle. other hand the component A of the vector A, which repre sents the given area, in the direction normal to the plane of projection is equal to the total vector A multiplied

MN f

by the cosine of the angle between its direction which is This ^and the normal to the normal to the plane two x to the the normals is for between ; planes angle angle

the same as the angle between the planes. between the magnitudes of A and A is therefore is

A =A 1

The

relation

cos x,

which proves the theorem. 26.]

Definition

said to be

Two

added when

plane areas regarded as vectors are the vectors which represent them are

added.

A vector

area

consequently the

sum

of its three

com

ponents obtainable

by orthogonal projection upon three Moreover in adding two mutually perpendicular planes. areas each

be resolved into

its three components, the added as scalar quantities, and corresponding components these sums compounded as vectors into the resultant area.

may

A generalization

of this statement to the case

planes are not mutually orthogonal

where the three

and where the projection

oblique exists. surface made up of several plane areas may be repre sented by the vector which is the sum of all the vectors

In case the surface be looked upon representing those areas. as forming the boundary or a portion of the boundary of a solid, those sides of the bounding planes which lie outside of the body are The vec conventionally taken to be positive. tors which represent the faces of solids are always directed

out from the

solid,

not into it 4

VECTOR ANALYSIS

50 Theorem 2 surface

The vector which

represents a closed polyhedral

is zero.

be proved by means of certain considerations of Suppose the polyhedron drawn in a body of hydrostatics.

This

fluid

may

assumed

cluded. 1 pressures.

surface

to be free

from

all

external forces, gravity in

The fluid is in equilibrium under its own internal The portion of the fluid bounded by the closed

moves neither one way nor the

other.

Upon each face

of the surface the fluid exerts a definite force proportional

to the area of the face

these forces

must be

and normal

The resultant of all in equilibrium. Hence

it.

zero, as the fluid is

sum of all the vector areas in the closed surface is zero. The proof may be given in a purely geometric manner.

the

Consider the orthogonal projection of the closed surface upon any plane. This consists of a double area. The part of the surface farthest from the plane projects into positive area ; the part nearest the plane, into negative area. Thus the surface projects into a certain portion of the plane which

covered twice, once with positive area and once with negative. These cancel each other. Hence the total projection of a

upon a plane (if taken with regard to sign) is But by theorem 1 the projection of an area upon a plane is equal to the component of the vector representing that area in the direction perpendicular to that plane. Hence closed surface zero.

the vector which represents a closed surface has no component along the line perpendicular to the plane of projection. This, however, was any plane whatsoever. Hence the vector is zero.

The theorem has been proved closed surface consists of planes. 1

for the case in

which the

In case that surface be

Such a state of affairs is realized to all practical purposes in the case of a polyhedron suspended in the atmosphere and consequently subjected to atmos pheric pressure. The force of gravity acts but is counterbalanced by the tension in the suspending string.

ADDITION AND SCALAR MULTIPLICATION curved

may

be regarded as the limit of a polyhedral surface

whose number

of faces increases without limit.

Hence the

vector which represents any closed surface polyhedral or curved is zero. If the surface be not closed but be curved it

be represented by a vector just as if it were polyhedral. That vector is the limit l approached by the vector which represents that polyhedral surface of which the curved surface

may

when

the limit

number

the

of faces

becomes indefinitely

great.

SUMMARY OF CHAPTER

a quantity considered as possessing magnitude and direction. Equal vectors possess the same magnitude and the same direction. vector is not altered by shifting it vector

parallel to itself.

nitude

multiply

its

unchanged.

null or zero vector

one whose

mag

multiply a vector by a positive scalar length by that scalar and leave its direction To multiply a vector by a negative scalar mul

zero.

length by that scalar and reverse its direction. Vectors add according to the parallelogram law. To subtract a vector reverse its direction and add. Addition, subtrac tiply its

tion,

and multiplication of vectors by a

same

scalar follow the

laws as addition, subtraction, and multiplication in ordinary vector may be resolved into three components algebra.

any three non-coplanar vectors. can be accomplished in only one way.

parallel to

The components

= x* + yb +

This resolution

zc.

(4)

of equal vectors, parallel to three given

non-coplanar vectors, are equal, and conversely if the com ponents are equal the vectors are equal. The three unit In vectors i, j, k form a right-handed rectangular system. 1 This limit exists and is unique. It is independent of the the polyhedral surface approaches the curved surface.

method

which

VECTOR ANALYSIS

terms of them any vector

may

Cartesian coordinates #, y,

= xi + yj+zk.

(6)

The

point which divides a line in a given given by the formula

Applications. ratio

be expressed by means of the

m+ The necessary and

(7)

sufficient condition that a vector equation

represent a relation independent of the origin is that the sum Between of the scalar coefficients in the equation be zero.

any four vectors there cients.

If the

sum

exists

an equation with scalar

of the coefficients

coeffi

zero the vectors are

an equation the sum of whose scalar termino-coplanar. coefficients is zero exists between three vectors they are If

termino-collinear.

masses A, B, C

The center situated

a, &, c

gravity of a number of at the termini of the vectors of

supposed to be drawn from a

common

origin

given by the formula

vector

may

be used to denote an area.

If the area is

plane the magnitude of the vector is equal to the magnitude of the area, and the direction of the vector is the direction of the normal upon the positive side of the The vector plane.

representing a closed surface

zero.

EXERCISES ON CHAPTER 1.

Demonstrate the laws stated in Art.

A triangle may be

I 12.

constructed whose sides are and equal to the medians of any given triangle. 2.

parallel

ADDITION AND SCALAR MULTIPLICATION 3.

The

six points in

which the three diagonals of a com*

meet the pairs of opposite sides lie three four upon straight lines. If two triangles are so situated in space that the three 4. points of intersection of corresponding sides lie on a line, then plete quadrangle

by three

the lines joining the corresponding vertices pass through a

common

point and conversely. Given a quadrilateral in space. Find the middle point 5. of the line which joins the middle points of the diagonals.

Find the middle point of the line which joins the middle Show that these two points are points of two opposite sides. the same and coincide with the center of gravity of a system of equal masses placed at the vertices of the quadrilateral. 6.

two opposite

sides of a quadrilateral in space be

divided proportionally and if two quadrilaterals be formed by joining the two points of division, then the centers of gravity of these

two quadrilaterals

lie

on a line with the center of

gravity of the original quadrilateral. By the center of gravity meant the center of gravity of four equal masses placed at the vertices. Can this theorem be generalized to the case is

where the masses are not equal ? 7.

The

bisectors of the angles of a triangle

meet in a

point.

edges of a hexahedron meet four by four in three points, the four diagonals of the hexahedron meet in a point. In the special case in which the hexahedron is a parallelepiped 8.

If the

the three points are at an infinite distance Prove that the three straight lines through the middle 9. points of the sides of

any

face of a tetrahedron, each parallel

to the straight line connecting a fixed point with the mid dle point of the opposite edge of the tetrahedron, meet in a

A complete quadrangle consists of the six straight lines which may he passed through four points no three of which are collinear. The diagonals are the lines which join the points of intersection of pairs of sides 1

VECTOR ANALYSIS

54 point

and

10.

E and

that this point

such that

PE passes

through

the center of gravity of the tetrahedron.

bisected by Show that without exception there exists one vector

equation with scalar coefficients between any four given vectors A, B,

Discuss the conditions imposed upon three, four, or five vectors if they satisfy two equations the sum of the co 11.

efficients in

each of which

is zero.

CHAPTER

DIRECT AND SKEW PRODUCTS OF VECTORS Products of 27.]

THE

Two

Vectors

operations of addition, subtraction,

and

scalar

multiplication have been defined for vectors in the way suggested by physics and have been employed in a few It

applications.

new combinations

now becomes

necessary to introduce two

These will be called products because they obey the fundamental law of products i. e., the distributive law which states that the product of A into the of vectors.

;

sum of B and C is equal to the sum of the products of A into B and A into C. The direct product of two vectors A and B is Definition :

the scalar quantity obtained by multiplying the product of the magnitudes of the vectors by the cosine of the angle be

tween them.

The

direct product

denoted by writing the two vectors

with a dot between them as

A-B. This

read

dot

B and

therefore

may

often be called the

dot product instead of the direct product. It is also called the scalar product owing to the fact that its value is sca lar. If be the magnitude of A and that of B, then by

definition

A-B = ^cos

(A,B).

(1)

Obviously the direct product follows the commutative law

A-B = B

(2)

VECTOR ANALYSIS

multiplied by a scalar the product

If either vector be

That

multiplied by that scalar.

(x A)

(x B)

= x (A

B).

In case the two vectors A and B are collinear the angle be tween them becomes zero or one hundred and eighty degrees and its cosine is therefore equal to unity with the positive or

Hence the scalar product of two parallel negative sign. vectors is numerically equal to the product of their lengths. The sign of the product is positive when the directions of the vectors are the same, negative

product of a vector by

itself is therefore

of its length

A.A=^4 2 if

Consequently vector

when they

The

are opposite.

equal to the square (3)

the product of a vector by itself vanish the

a null vector.

and B are perpendicular the angle between them becomes plus or minus ninety degrees and the cosine vanishes. Hence the product A B vanishes. In case the two vectors

Conversely

Hence

the scalar product

either

A B

vanishes, then

A B cos (A, B) = 0. B or cos (A, B) is zero,

vectors are perpendicular or one of them condition for the perpendicularity of two

which vanishes, 28.]

vectors

The i, j, k

A B=

is null.

Thus

the

vectors, neither of

scalar products of the three

fundamental unit

are evidently

ii = jj = kk = i

and either the

(4)

= 0.

more generally a and b are any two unit vectors the

product a

= cos

(a, b).

DIRECT AND SKEW PRODUCTS OF VECTORS Thus the

scalar product determines the cosine of the angle is in a certain sense equivalent to

between two vectors and

For

it.

this reason

might be better to give a purely which

geometric definition of the product rather than one

depends upon trigonometry. This is easily accomplished as If a and b are two unit vectors, a b is the length

follows

upon the other. If more generally and B are any two vectors A B is the product of the length

of the projection of either

by the length

of either

From

of projection of the other

upon

it.

these definitions the facts that the product of a vector

the square of its length and the product of two perpendicular vectors is zero follow immediately. The trigo itself is

nometric definition can also readily be deduced.

The

scalar product of

two vectors

cosine of the included angle

examples vector

may

The

be cited.

whenever the

will appear

of importance.

projection of a

The following vector B upon a

AB A = AB A a cos

A A where a

(A, B)

= B cos

(A,

a unit vector in the direction of A.

(5)

is itself

unit vector the formula reduces to

(A-B) A If

be a constant force and

by the force

cos

(A,B) A.

a displacement the

during the displacement

work done If

repre

sent a plane area (Art. 25), and if B be a vector inclined to that plane, the scalar prod

uct

A B

which the area

will be the

volume

of the cylinder

the

base and of

which B is the directed slant height. For the volume (Fig. 16) is equal to the base multiplied by the altitude h. This is

FlG

the projection of v

B upon A

= A h = A B cos

cos (A, B).

(A, B)

Hence

VECTOR ANALYSIS

The

29.]

scalar or direct product follows the distributive

That is

law of multiplication.

(A This

B) .C

magnitude

tion.

To show (A

=A c = A

equal to the

the relation (6)

The

+B+

...)

three unit vectors

and then

sum

B But ;

that of

no peculiar

i, j,

of the projections.

If in particular

,A^,A 3

lines

<ey

2 j i

+ ^k)

of (4) this reduces to

A-* = A l

Hence

terms of the

;

By means

difficulties.

are expressed in

A = ^[ 1 i + ^ 2 j + ^ 8 k, B = ^ i + JB2 j + B k, A- B = (A i + A z j + A B k) (^i + = A B i i + A 2 i j + A B% l

B upon

the projection of the

be used just as the product in ordi

A and B k

direc

B. (0o)

+ a+-") = A-P + A.Q+... + B.P + B.Q + ...

may

It has

two vectors

upon

its

By an immediate generalization

proved.

scalar product

nary algebra. If

A + B upon

that of

(0o) c + B

(<7c)

c is the projection of

(6)

multiplied by a unit vector c in

sum A + B

B.C.

of projections. Let C be equal

may be proved by means

to its

= A-C +

and

A and B

+ A2 E, + A B JB,.

and B are unit

B 19 S29 SB

are

vectors, their

(7)

components

the direction cosines of the

referred to X, Y, Z.

DIRECT AND SKEW PRODUCTS OF VECTORS

= cos (A, .# 2 = cos (B,

A = cos (A, JT), ^! = cos (B, JT),

A<i

A B

Moreover

A z = cos = cos 3

F), T),

(A, ^f), (B,

Hence

the cosine of the included angle.

the equation becomes cos (A, B)

= cos

(A,

cos (B,

X) +

+ A and B

In case

known

cos (A,

cos

T) cos

(A,)

(B,

cos (B,Z).

are perpendicular this reduces to the well-

relation

= cos

(A, JT) cos (B,

X) +

cos (A, cos

Y) cos

(A,^)

cos

(B,

(B,)

between the direction cosines of the

A and

line

30.]

the line B.

and B are two sides

and

side

AisG = -B-JL

of a triangle

OAB,

the third

PlG 17

(Fig. 17).

C*C = (B-A). (B-A)=B-B 2 2 2 (7 = A + J5 -2 A^cos(AB).

That

sum

the square of one side of a triangle is equal to the of the squares of the other two sides diminished by twice is,

their product times the cosine of

the angle between them.

Or, the square of one side of a triangle is equal to the sum of the squares of the other two sides diminished by twice the product of either of those sides by the projection of the other

upon If

and D

the generalized Pythagorean theorem. and B are two sides of a parallelogram, C

are the diagonals.

= (A +

Then

B)=A.A + 2A.B + B.B, D.D=(A-B).(A-B)=A-A-2A.B + B.B, C-C + D.D = 2(A-A + BB), a 2 + 7) 2 = 2 (A* C.C

=A+

B).(A +

VECTOR ANALYSIS

sum of

the squares of the diagonals of a parallelo sum of the squares of two sides.

That

is,

the

gram

equal to twice the

In like manner also

C*-D = 4A 2

That

cos (A, B).

the difference of the squares of the diagonals of a parallelogram is equal to four times the product of one of the sides If

is,

by the projection of the other upon

it.

any vector expressed in terms of

A = A i + A 2 j + A B k, A A = A* = A* + A* +

then

But

vectors

A*.

(8)

be expressed in terms of any three non-coplanar unit as

a, b, c

A = a? + 2

This formula

2 J

2 a b cos

bc +

(a, b)

+ +

2 b

cos

2 ca cos

(b, c) (c,

a).

analogous to the one in Cartesian geometry

which gives the distance between two points referred to If the points be x v y v z v and # 2 , y v z% the oblique axes. distance squared

31.]

- x^ + (y a - yi 2 + (z 2 - zj* + 2 (a, - xj (y a - 2/0 cos (X, Y) + 2 (yt - ft) (,-*!> cos (F.S) + 2 (z 2 -24) (x 2 - xj cos (^,-T). (* 2

Definition:

the vector

normal upon

)

The skew product

of the vector

into

the vector quantity C whose direction is the that side of the plane of A and B on which

DIRECT AND SKEW PRODUCTS OF VECTORS

rotation from A to B through an angle of less than one hundred and eighty degrees appears positive or counter clockwise and whose magnitude is obtained by multiplying ;

the product of the magnitudes of

angle from

The

which

sine of the

to B.

direction

and B by the

A x B may

also be defined as that in

ordinary right-handed it turns so as

screw advances as

c= AXB

toward B (Fig. 18). The skew product is denoted by

to carry

a cross as the direct product was by a dot. It is written

C =A

and read

product.

cross B.

For

FIG. 18.

x B

this reason it is often called the cross

frequently, however, it is called the vector prod to the fact that it is a vector quantity and in con

uct,

owing

trast

with the direct or scalar product whose value

The vector product C

when where

x B

definition

= ^J5sin

(A,B)c,

are the magnitudes of A and a unit vector in the direction of

and

c is

is scalar.

(9)

respectively

and

In case

and

skew product A X B reduces to the multiplied by the sine of the angle from A to B.

are unit vectors the

unit vector c

Obviously also if either vector A or B is multiplied by a scalar x their product is multiplied by that scalar.

A) X B If

=AX

(zB)

A and B are parallel the angle

= xC.

between them

either zero

and eighty degrees. In either case the sine vanishes and consequently the vector product A X B is a null vector. And conyersely if A X B is zero or one hundred

A B sin

(A,

= 0.

VECTOR ANALYSIS

B or

sin (A, B) is zero. Thus the condition for vanishes is A X B vectors neither which two of parallelism of of As a corollary the vector product 0. any vector into

Hence

itself vanishes.

The vector product

32.]

two vectors

will appear

wher

ever the sine of the included angle is of importance, just as the scalar product did in the case of the cosine. The two prod ucts are in a certain sense complementary. They have been

denoted by the two

and the

common

signs of multiplication, the dot

In vector analysis they occupy the place held by the trigonometric functions of scalar analysis. They are at the same time amenable to algebraic treatment, as will be

seen

cross.

later.

present a few uses of the vector product

may

be cited. If

and B

(Fig. 18) are the

two adjacent sides

of a parallel

ogram the vector product C

x B

= A B sin

(A, B) c

represents the area of that parallelogram in magnitude direction (Art. 25). This geometric representation of A

and

X B

common

occurrence and importance that it might well be taken as the definition of the product. From it the is

of such

The vector product If A and A are two forces forming a couple, the moment of the couple is A X B provided only that B is a vector drawn from A. The product makes its any point of A to any point of trigonometric definition follows at once.

appears in mechanics in connection with couples.

appearance again in considering the velocities of the individ ual particles of a body which is rotating with an angular ve If R be the locity given in magnitude and direction by A. radius vector

the product

(Art. 51).

of rotation

drawn from any point

AX&

will give the velocity of the extremity of

This velocity

and

of the axis of rotation

perpendicular alike to the axis

to the radius vector B.

DIRECT AND SKEW PRODUCTS OF VECTORS

The vector products A X B and B x A are not the They are in fact the negatives of each other. For if rotation from A to B appear positive on one side of the plane of A and B, rotation from B to A will appear positive on the other. Hence A X B is the normal to the plane of A and B upon that side opposite to the one upon which B x A is the normal. The magnitudes of A X B and B X A are the same. Hence 33.]

same.

AxB = -BxA.

The factors in a vector product can

(10)

interchanged if

and only

if the sign of the product be reversed.

This is the first instance in which the laws of operation in vector analysis differ essentially from those of scalar analy sis. It may be that at first this change of sign which must

accompany the interchange of factors in a vector product will give rise to some difficulty and confusion. Changes similar to this are, however, very familiar. No one would think of inter changing the order of x and y in the expression sin (x without prefixing the negative sign to the result. Thus sin (y

although the sign

sin (x

y),

not required for the case of the cosine,

cos (y

Again

= cos

(

the cyclic order of the letters

y).

ABC in the

area of a

triangle be changed, the area will be changed in sign (Art. 25).

AB C = -ACB.

In the same manner this reversal of sign, which occurs

when will

the order of the factors in a vector product

appear after a little practice

natural and convenient as 34.]

The

distributive

it is

reversed,

and acquaintance just

necessary.

law of multiplication holds in the

case of vector products just as in ordinary algebra

except

VECTOR ANALYSTS

order of the factors must

that the

carefully maintained

when expanding.

A very simple proof may be given by making use of the ideas developed in Art. 26. Suppose that C not coplanar with A and B. Let A

and B be two

sides of a triangle taken

(A + B)

Then

in order.

Form

third side (Fig. 19). of

which

will be the

the prism

this triangle is the base

and

the slant height or edge. which C The areas of the lateral faces of this of

4 FIG. 19.

prism are

Ax The

B x

-f

areas of the bases are

But the sum prism

(A x B) and

- - (A

B).

of all the faces of the prism

AxC + BxC-(A + B)xC = A X

The

zero; for the

Hence

a closed surface.

+BX

= (A +

B) X

(11)

relation is therefore

proved in case C is non-coplanar Should C be coplanar with A and B, choose D, any vector out of that plane. Then C + D also will lie out of with

and

that plane.

A X

Hence by

(11)

+B X

D).

Since the three vectors in each set A, C, D, and B, C, D, and A + B, C, D will be non-coplanar if D is properly chosen, the products may be expanded.

DIRECT AND SKEW PRODUCTS OF VECTORS

AxC + AxD-fBxC + BxD = (A +

+ B) x D. AxD + BxD = (A + B)xD. AxC + BxC = (A + B)xC.

But by (11) Hence

B) x C

The

This completes the demonstration.

The

for a vector product.

(A + B+---)x(P + a +

distributive

generalization

= AxP + Axa +

---)

+BxP+B

The

ixi = jxj =

---

(11)

vector products of the three unit vectors easily seen by means of Art. 17 to be 35.]

law holds

immediate.

i, j,

k are

kxk = 0,

ixj=-j xi = k, jxk = k x = kxi = ixk=j.

(12)

The skew product of two equal l vectors of the system i, j, k is zero. The product of two unequal vectors is the third taken with the positive sign if the vectors follow in the cyclic order i j k but with the negative sign if they do not. If two vectors A and B are expressed in terms of i, j, k,

may be found by expanding according law and substituting.

their vector product to the distributive

A x B = (A

+ -4 + ^ k) x (^i + + 3*) = A ixi + A B ixj + A BzixTt x k, x + AZ B + A 2 j x + AI + A S k x + A BZ k x j + A B k x k. A x B = (A^B^ - A + (A Z B -A,BB )j - A BJ k. 4- (A, z l

2 j

Hence

B j

This follows also from the fact that the sign

factors

reversed.

Hence

Xj=

Xj=0. 5

changed when the order of

VECTOR ANALYSIS

66 This

may

be written in the form of a determinant as

Ax B = The

sum

formulae for the sine and cosine of the

or dif

ference of two angles follow immediately from the dot and Let a and b be two unit vectors lying in the cross products. i

j-plane.

If x be the angle that a

angle b makes with

= cos x +

cos y

Hence

cos (y

cos

Hence

)

= cos

(sin y cos

x).

sin y sin

x),

sin x cos y).

sin x cos y. y cos x b ) = k sin (y + x) k (sin y cos x + sin x cos y).

sin (a,

sin

= k sin

and

x),

x sin

sin (a, b)

sin

+ +

sin

sin (y

= cos

cos y cos x

cos (a, b

axb

sin (y

Hence If

x b x b

sin y

sin y sin x. cos y cos x sin y j, cos y i

a.V: (y + x) a

sin

cos x cos y

Hence

i -f

cos (a, b)

and y the

then

makes with

y cos x

sin # cos y.

are the direction cosines of

unit vectors a and a referred to

JT,

= li + m + k, m = cos (a, a ) = IV + m m + a

two

then

as has already been

shown

in Art. 29.

for the square of the sine of the angle

be found.

The

familiar formula

between a and a may

DIRECT AND SKEW PRODUCTS OF VECTORS a

= sin

x a

(a,

)

= (mn

+ (Jm -f m) where

x a

sin 2 (a, a )

)

x a

= (mn

This leads to an easy

(72

+ w2 +

)

(V*

= sin (a, a m n)*+ (nl 2

)

way

I) j

e is a unit vector perpendicular to a

ri) i

and a

= sin 2 2

(a,

+(lm

). I

m)*.

of establishing the useful identity

(ll

Two

Products of More than

+ mm + n n

)

Vectors

been said concerning products in which the number of vectors is greater than two. If three vectors are combined into a product the result 36.]

to this point nothing has

a triple product. Next to the simple products A-B and AxB the triple products are the most important. All higher products may be reduced to them. called

The

simplest triple product is formed by multiplying the two vectors A and B into a third C as

scalar product of

(A-B)

This in reality does not differ essentially from scalar multi

The scalar in this case merely happens to be the scalar product of the two vectors A and B. Moreover inasmuch as two vectors cannot stand side by side in the plication (Art. 6).

form of a product as BC without either a dot or a cross to unite them, the parenthesis in (AB) C is superfluous. The expression

cannot be interpreted in any other of the vector C by the scalar AB. i

way

than as the product

Later (Chap. V.) the product BC, where no sign either dot or cross occurs, But it will be seen there that (A.B) C and A-(B C) are identical and consequently no ambiguity can arise from the omission of the parenthesis. will be defined.

VECTOR ANALYSIS

The second

37.]

two

vectors, of

triple

which one

the scalar product of itself a vector product, as is

product is

A-(BxC)

(AxB>C.

This sort of product has a scalar value and consequently is often called the scalar triple prod Its properties are

uct.

deduced from

easily

perhaps most

its

commonest

geometrical interpretation. Let A, B, and C be any three vectors drawn

from

same origin

the

Then BxC

B and C

allelogram of which scalar

BxC

the base and

This volume v side of the sides.

B C-plane

(14)

of the parallelepiped of

the slant height or edge.

positive ;

A and BxC

which

See Art. 28.

upon the same they lie on opposite

lie

but negative if if A, B, C form a right-handed or

In other words

positive system of three vectors the scalar A* (BxC) tive;

The

sides.

volume

(Fig. 20). the area of the par

two adjacent

are

will therefore be the

but

posi

they form a left-handed or negative system,

negative.

In case A, B, and C are coplanar this volume will be neither positive nor negative but zero. And conversely if the

volume

piped must

zero^the three edges A, B, C of the parallelolie in one plane. Hence the necessary and suffi

cient condition

for

of which vanishes

the coplanarity is

A-(BxC)

of three vectors A, B, C none As a corollary the scalar

= 0.

which two are equal or must vanish for any two vectors are coplanar. The two products A(BxC) and (AxB)-C are equal to the same volume v of the parallelepiped whose concurrent edges

triple product of three vectors of

collinear

are A, B, C. cases.

;

The sign

of the

Hence (AxB) . c

volume

(BxC)

the same in both

= ,.

(14)

DIRECT AND SKEW PRODUCTS OF VECTORS This equality

and

may

the cross in

be stated as a rule of operation.

a scalar

triple product may without altering the value of the product. It may also be seen that the vectors A, B, C

muted

The dot

be interchanged

may be

per

cyclicly without altering the product

= B-(CxA) = C-(AxB).

A-(BxC) For each

of the expressions gives the

parallelepiped and that

same

sign, because if

volume

A is

(15)

volume of the same

will have in each case the

upon the

positive side of the

B C-

B will be on the positive side of the C A-plane and C the The triple product upon positive side of the A B-plane. may therefore have any one of six equivalent forms

plane,

= B-(CxA) =: C.(AxB) = (AxB)-C = (BxC)-A = (CxA)-B A<BxC)

however the

product will

cyclic order of the letters is

= - A<CxBV

BxC

vectors, but it is reversed

CxB.

scalar triple product is not altered by interchanging

the dot or the cross or by

word

permuting

cyclicly the order

of the

in sign if the cyclic order be changed.

necessary upon the subject of parentheses this triple product. Can they be omitted without am

38.]

(16)

be seen from the figure or from the fact that

may

Hence

changed the

change sign.

A-(BxC) This

(35)

biguity

They

can.

The expression

A-BxC can have only the one interpretation

A<BxC). For the expression (A-B)xC is meaningless. It is impos form the skew product of a scalar AB and a vector

sible to

VECTOR ANALYSIS

only one way in which ABxC may be interpreted, no confusion can arise from omitting the

Hence

as there

Furthermore owing to the fact that there are six scalar triple products of A, B, and C which have the same parentheses.

value and are consequently generally not worth distinguish ing the one from another, it is often convenient to use the

symbol

[ABC] to denote

any one of the six equal products.

[A B C] then

= A.BxC = B*CxA = C AxB = AxB.C = BxC-A = CxA-B [A B C] = - [A B].

(16)

The all

scalar triple products of the three unit vectors i, j, k vanish except the two which contain the three different

vectors.

[ijk]

Hence

= _[ikj] = l.

(17)

three vectors A, B, C be expressed in terms of

i, j,

= ^ i + A, + 8 k, C = C i+C 2 + C 3 k,

then

This tions

may

[ABC]

C3 +

C2 A.+

be obtained by actually performing the multiplica which are indicated in the The result triple product.

may

be written in the form of a determinant. 1 -4i\

[A B C] 1

This

is the formula given in solid analytic geometry for the volume of a tetrahedron one of whose vertices is at the For a more general formula origin.

see exercises.

DIRECT AND SKEW PRODUCTS OF VECTORS If

more generally A,

non-coplanar vectors vectors,

B, a,

C are expressed in terms of any three b, c which are not necessarily unit

A=a

= &! C =c

where a^ # 2 stants,

#3,"

+ +

a2 b &2

+ +

J8 c

are certain con

ftp & 2 ,

then

[A B 0]

= (a

&2 C B

+ [a b c].

[A B C]

The

39.]

which one

(19)

the vector product itself a vector product. Such

third type of triple product

two vectors

be]

are

Ax(BxC) and (AxB)xC.

The vector Ax(BxC) is perpendicular to A and to (BxC). But (BxC) is perpendicular to the plane of B and C. Hence Ax (BxC), being perpendicular to (BxC) must lie in the plane of B and C and thus take the form

Ax(BxC) where x and y are two vector

= x B + y C,

scalars.

In like manner also the

(AxB)xC, being perpendicular to (AxB) must A and B. Hence it will be of the form

lie

in the plane of

(AxB)xC where

and n are two

= ra.A +

scalars.

From

this it is evident that

in general

(AxB)xC The

parentheses

changed.

It is

not equal

therefore essential to

Ax(BxC).

cannot be

removed or

know which

cross

inter

product

VECTOR ANALYSIS

formed

first

and which second.

This product

termed the

vector triple product in contrast to the scalar triple product.

The

vector triple product

ponent of a vector

B which

be used to express that com perpendicular to a given vector

may

This geometric use of the product is valuable not only in itself but for the light it sheds

AXB

upon the properties of the product.

Let

AXB*

(Fig. 21) be a given vector

and B another vector whose com ponents parallel and perpendicular to A are to be found. Let the

components of B

parallel

and per-

B and B" reDraw A and B from a spectively. common origin. The product AxB The product plane of A and B. pendicular to

(AXB) 2i

perpendicular to the Ax (AxB) lies in the plane of is

Hence

perpendicular to A.

A it

and is

It is furthermore

collinear with B".

examination of the figure will show that the direction of Ax (AxB) is opposite to that of B". Hence

Ax(AxB) where

c is

some

Now but if

cB",

scalar constant.

= - A* B sin (A, B) V - c B"-^= - c B sin (A, B) b",

Ax (AxB)

b" be a unit vector

in the direction of B".

Hence

A* A.

Ax(AxB)

The component of B perpendicular

(20) has been expressed

in terms of the vector triple product of A, A,

component B

parallel to

was found in Art 28

and

to be

The

DIRECT AND SKEW PRODUCTS OF VECTORS B B

=B +

=?A B

(21)

AA?A-^>. AA

(22)

The vector triple product Ax (BxC) may be expressed sum of two terms as

40.]

as the

Ax(BxC)=A-C B-A-B In the

first

place consider the product

vectors are the same.

when two

equation (22)

= A-B A - Ax(AxB) Ax(AxB) = A*B A - A- A B A-A B

of the

(22) (23)

This proves the formula in case two vectors are the same. To prove it in general express A in terms of the three non-coplanar vectors B,

and BxC.

A = bE + where

#, &, c are scalar constants.

Ax(BxC) The vector product

(23)

a (BxC),

(I)

Then

= SBx(BxC) + +

cCx(BxC)

(II)

a (BxC)x(BxC).

any vector by

itself is zero.

Hence

(BxC)x(BxC) = Ax(BxC) = 6Bx(BxC) + c Cx(BxC). Bx(BxC) = B-C B - B-B C

Cx(BxC) = - Cx(CxB) = - C-B C + C-C B. Hence Ax(BxC) = [(&B-C + cC-C)B- (6B-B + cCB)C]. But from (I) A-B = JBB + cC-B + a (BxC>B and

= b B-C + c C*C + a (BxC)-O. (BxC)-B = and (BxC)-C = 0. A-B = JB-B + cC-B, A-C = 5B-C + cC-C.

A-C Art. 37

Hence

(II)

(II)"

VECTOR ANALYSIS

Substituting these values in (II)",

Ax(BxC)

The It

relation

= A.C

therefore proved for

- A.B

(24)

any three vectors A, B, C.

Another method of giving the demonstration is as follows. was shown that the vector triple product Ax(BxC) was form

of the

Ax(BxC) Since it

Ax(AxC)

by A

is zero.

= #B +

yC.

perpendicular to A, the direct product of

Hence

A-[Ax(BxC)]

+ yAC =

A*B

= A*C AB. = n (A-0 B - A-B C), Ax(BxC) x :y

and

Hence where n

a scalar constant.

remains to show n

Multiply by B.

Ax(BxC>B = n (A-C B.B-A-B The

scalar triple product allows

cross.

C-B).

an interchange of dot and

Hence

Ax(BxC>B = A<BxC)xB = - A-[Bx(BxC)], if

the order of the factors (BxC) and

B be

inverted.

= -A-[B.C B-B.BC] = B-C AB + B-B AC. A-B C. Ax(BxC) = A.C B

-A-[Bx(BxC)] Hence n

and

(24)

From the three letters A, B, C by different arrangements, four allied products in each of which B and C are included in parentheses may be formed. These are Ax(BxC),

Ax(CxB),

a vector product changes

two

factors is

(CxB)xA,

(BxC)xA.

sign whenever the order of the above products evidently interchanged, its

satisfy the equations

Ax (BxC) = - Ax(CxB) = (CxB)xA = - (BxC)xA.

DIRECT AND SKEW PRODUCTS OF VECTORS The expansion

vector triple product in which the parenthesis comes first may therefore be obtained directly from that already found when the parenthesis comes last. for a

(AxB)xC

The

= - Cx(AxB) = - C-B A + CA

formulae then become

Ax(BxC) and

(AxB)xC

= A-C = A*C

- A.B C B - C-B A. B

(24)

These reduction formulae are of such constant occurrence and great importance that they should be committed, to memory. Their content may be stated in the following rule. To expand

vector triple product first multiply the exterior factor into the remoter term in the parenthesis to form a scalar coefficient for the nearer one, then multiply the exterior factor into the nearer

term in the parenthesis remoter one,

41.]

and

form a

scalar

coefficient

for

the

subtract this result from the first.

far as the practical applications of vector analysis

concerned, one can generally get along without any formulae more complicated than that for the vector triple are

product. But it is frequently more convenient to have at hand other reduction formulae of which all may be derived

simply by making use of the expansion for the triple product Ax(BxC) and of the rules of operation with the triple pro

duct

ABxC.

reduce a scalar product of two vectors each of which a vector product of two vectors, as

is itself

(AxB>(CxD). Let

this

be regarded as a scalar triple product of the three CxD thus

vectors A, B, and

AxB-(CxD). Interchange the dot and the cross.

VECTOR ANALYSIS

= A-Bx(CxD) Bx(CxD) = B-D C - B-C D. (AxB>(CxD) = A-C B-D - A-D AxB.(CxD)

Hence This

may

B.C.

(25)

be written in determinantal form.

(25)

and D be called the extremes B and C the means ; A and C the antecedents: B and D the consequents in this If

;

product according to the familiar usage in proportions, then the expansion may be stated in words. The scalar product of two vector products is equal to the (scalar) product of the antecedents times the (scalar) product of the consequents diminished by the (scalar) product of the means times the

product of the extremes.

(scalar)

To reduce is

a vector product of two vectors each of which itself a vector product of two vectors, as

(AxB)x(CxD). Let CxD = E.

The product becomes

(AxB)xE Substituting the value of

(AxB)x(CxD) Let F

= AxB.

= A-E

E back

- B-E A.

into the equation

(A-CxD)B

(B-CxD)

(26)

The product then becomes

Fx(CxD) = FD C F-C D (AxB)x(CxD) = (AxB-D)C - (AxB-C) all

(26)

equating these two equivalent results and transposing

the terms to one side of the equation,

[B C D]

This

A-

D A] B + [D A

B] C

[A B C] D

- 0.

(27)

an equation with scalar coefficients between the four vectors A, B, C, D. There is in general only one such equais

DIRECT AND SKEW PRODUCTS OF VECTORS

because any one of the vectors can be expressed in only one way in terms of the other three thus the scalar coeffi

tion,

cients of that equation

which

exists

between four vectors are

found to be nothing but the four scalar those vectors taken three at a time.

triple

products of

The equation may

also

be written in the form

[A B C]

D = [B C

More examples

A+

D] B

[A B D]

of reduction formulae, of

(27)

which some are

important, are given among the exercises at the end of the In view of these it becomes fairly obvious that chapter. the combination of any number of vectors connected in

any legitimate way by dots and crosses or the product of any number of such combinations can be ultimately reduced to

sum of terms each of which contains only one The proof of this theorem depends solely upon

cross at most.

analyzing the of and vectors combinations possible showing that they all fall under the reduction formulae in such a way that the crosses

may

be removed two at a time until not more than

one remains. *

The

formulae developed in the foregoing article have interesting geometric interpretations. They also afford a simple means of deducing the formulae of Spherical Trigo 42.]

nometry. These do not occur in the vector analysis proper. Their place is taken by the two quadruple products,

= A-C B-D - B-C A-D (AxB)x(CxD) = [ACD] B - [BCD] A = [ABD] C - [ABC] D, (AxB>(CxD)

and

(25)

(26)

which are now to be interpreted. Let a unit sphere (Fig. 22) be given. Let the vectors A, B, C, D be unit vectors drawn from a common origin, the centre of the sphere, and terminating in the surface of the The great circular arcs sphere at the points A,B, (7, D.

VECTOR ANALYSIS

A C)

AB,

and

etc.,

and B, give the angles between the vectors A a The points A, B, C, determine quadrilateral

C, etc.

upon the sphere.

and

BD are

B-D

- A-D

one

and C> the pair of opposite sides ; are the diagonals. and other.

(AxB).(CxD)

AxB = sin

CD = A-C

(A, B),

CxD

= sin (C, D).

The angle between AxB and CxD

the angle between the planes themselves.

the

Let

be denoted

Then

(AxB). (CxD)

The

angle between the normals to the ABand CD-planes. This is the same as

FIG. 22.

B-C

angles

circular arcs

= sin

(A,B) sin (C,D) cos a:.

D) may be replaced by the great which measure them. Then

(A, B), (C,

AB, CD

(AxB).(CxD) = sin A B sin CD cos#, A-C B-D- A.D B*C = cos AC cosBD - cos AD

cos

BC.

Hence sin

A B sin CD

cos x

= cos A C cos B D

cos

AD cos B C.

The product of the cosines of two opposite sides of a spherical quadrilateral less the product of the cosines of In words

the other two opposite sides sines

of the

equal to the product of the diagonals multiplied by the

cosine of the angle between them.

theorem

This

credited to Gauss.

C (Fig. 23) be a spherical tri the sides of which are arcs of great angle, circles. Let the sides be denoted by a, 6, c Let A, B,

FIG. 23.

respectively.

drawn from the center Furthermore

let

pa p b pe ,

Let A, B, C be the unit vectors

of the sphere to the points

-A,

B, C.

be the great circular arcs dropped

DIRECT AND SKEW PRODUCTS OF VECTORS perpendicularly from the vertices Interpret the formula

(AxB)-(CxA)

= sin

(AxB)

Then

(A, B)

(AxB) (CxA)

to the sides a,

- B.C A-A. (CxA) = sin (C, A) = sin

= A-C

= sin

-4, J9,

79 6,

B-A

= sin c sin

b cos #,

This angle between AxB and CxA. angle is equal to the angle between the plane of A, B and the which plane of C, A. It is, however, not the interior angle is one of the angles of the triangle but it is the exterior

where x

the

A, as an examination of the figure will show.

angle 180

Hence

= sin c sin b cos (180 A) = sin c sin 6 cos A BC A- A = cos & cos c cos a 1.

(AxB). (CxA)

AC BA By

equating the results and transposing,

= cos 6 cos c cos 6 = cos c cos a cos c = cos a cos 6 cos a

The

last

letters or

sin 6 sin c cos sin

sin a cos

A B

sin a sin 6 cos C.

two may be obtained by from the identities

cyclic permutation of the

= B-A C B - C-A, (CxAHBxC) = C-B A.C - B-C. (BxC).(AxB)

Next cases in

interpret the identity

(AxB)x(CxD)

which one of the vectors

in the special

repeated.

(AxB)x(AxC) = [A B C] A. Let the three vectors

a, b, c

be unit vectors in the direction of

AxB respectively. Then AxB = c sin AxC = cxb sin c sin & = A (AxB)x(AxC) =

BxC, CxA,

[A B C]

b sin sin

= cC sin c = cos (90 [ABC] A = sin c sin p A.

(AxB)-C

sin 6 sin

pc) sin c

VECTOR ANALYSIS

common

equating the results and cancelling the

factor,

= sin b sin A sin^? a = sin c sin B sin p = sin a sin C. sin^ c

The

last

letters.

triangle

two may be obtained by cyclic permutation of the The formulae give the sines of the altitudes of the in terms of the sines of the angle and sides. Again

write

= [ABC]A = [BCA]B (CxA)x(CxB) = [CAB]C.

(AxB)x(AxC) (BxC)x(BxA)

A= sin a sin c sin B = sin b sin a sin C =

Hence

sin

sin b sin

[A B C] [B C A] [C

A B].

The expressions [ABC], [BCA], [CAB] are equal. the results in pairs and the formulae

Equate

A = sin a sin B sin c sin B = sin b sin C sin a sin C = sin c sin A sin b sin

are obtained.

These may be written in a single sin

sin

sin a

sin

sin b

line.

sin c

The

formulae of Plane Trigonometry are even more easy to obtain. If B C be a triangle, the sum of the sides taken

as vectors

From

this

zero

for the

almost is

all

triangle

equation

b-f

a closed polygon.

the elementary formulae follow immediately.

to be noticed that the angles

from a to

from b to

from

DIRECT AND SKEW PRODUCTS OF VECTORS a are not the interior angles A, B, - B, 180 - C. angles 180 -A, 180 o to

=b+c + c) = b-b +

(7,

but the exterior

aa = If a,

c)*(b

be the length of the sides

J, c

The

last

=a +

area of the triangle

becomes

2 a 6 cos C. to the first

letters.

^axb = ^bxc = 2 cxa = a b sin C = % b c sin A = ^ c a sin

each of the last three equalities be divided by the product a b c, the fundamental relation sin

2 bc.

two are obtained in a manner similar

a, b, c, this

one or by cyclic permutation of the

The

c-c

obtained.

sin

Another formula for the area may be found from

the product

(bxc)(bxc) 2 Area (6

sin

2 Area

= (cxa)-(axb)

a sin B) (a sin -Z?sin

sin

The problem

of expressing

three non-coplanar vectors

Let

a, b, c

Reciprocal Systems of Three Vectors. 43.]

b sin (7)

Solution of Equations

any vector

may

r in

terms of

be solved as follows.

VECTOR ANALYSIS

where

three scalar constants to be determined

a, J, c are

Multiply by

= a abxc + 6 bbxc + [rbc] = a [a be].

r.bxc or

cc-bxo

In like manner by multiplying the equation by . a X b the coefficients b and c may be found.

x a and

= I [b c a] = c [c a b] [r a b] [r c a]

Hence

[be a]

The denominators

are

all

Hence

equal.

equation [a

[b c r] a

c] r

[c r a]

which must exist between the four vectors

The equation may r

= rb r -

- a

bxc _

a, ..

a b]

r, a, b, c.

cxa,b

x b e

axb o.

[abc]

three vectors which appear here multiplied

bxc

[a be]

[abc]

The

[abe] or

gives the

this

also be written

(28)

[c a b]

cxa _ b -

be]

by !, namely

axb

[a b c]

They are perpendicular respectively to the planes of b and c, c and a, a and b. They occur over and over again in a large number of important relations. For are very important.

this reason

they merit a distinctive

Definition

The system b x

cxa

axb

[abc]

name and

of three vectors

notation.

DIRECT AND SKEW PRODUCTS OF VECTORS

which are found by dividing the three vector products bxc, c

x b

of three non-coplanar vectors

[abc]

product

a, b, c

called the reciprocal system to

The word non-coplanar

bxc

axb

________

[a b c]

[a be]

were co

would vanish and

[a b c]

cxa

a, b, c.

If a, b, c

important.

planar the scalar triple product consequently the fractions

by the scalar

would all become meaningless. Three coplanar vectors have no reciprocal system. This must be carefully remembered.

when

Hereafter

the term reciprocal system

understood that the three vectors

The system

of three vectors

,_bxc

""[abc]

The

a ,

a x b

The

vector r

system a place a , b

it ,

and

c a,

may

r-b b

r.c

b, c

(29)

form (30)

be expressed in terms of the reciprocal

instead of in terms of

~[ac]

[abc]

will be

are not coplanar.

b, c

expression for r reduces then to the very simple r = r-a

used,

reciprocal to system

denoted by primes as a

will be

necessary to note that

a, b, c.

In the

first

are non-coplanar,

if a, b, c

which are the normals to the planes of b and c, a and b must also be non-coplanar. Hence r may

be expressed in terms of them by means of proper scalar coefficients #, y,

x a

Multiply successively by

?/b

-a, -b, -c.

z c

This gives

= x [b c a], x = r-a [abc]r-b = y [cab], y = r-b = z [a b c], z = r-c [a b c] r-c r = r-a a + r-b b + r-c c [a b c] r-a

Hence

(31)

VECTOR ANALYSIS

84 If a

44.]

be the system reciprocal to a, b, c the any vector of the reciprocal system into the

scalar product of

corresponding vector of the given the product of

system

unity

two non-corresponding vectors is zero.

a .a = bM>=:c .c = l = a .c = b -a = b *c = c -a = c a .b

Hence

(32)

non-coplanar the

are

Since a

= raa + r*bb + rcc = a -aa + a b V + a cc = b .aa + b -bb + = c aa + c -bb + c .cc

That

-b = 0.

This may be seen most easily by expressing a terms of themselves according to the formula (31) r

but

;

corresponding

coeffi

on the two sides of each of these three equations must be equal. Hence from the first cients

From

the second

From

the third

This

proves

the

=a =b =c

= a -b

l=b =c

relations.

= bxc

e. o.

also

proved

[abc]

= bxc-b = [abc]

=0 [abc]

forth.

Conversely

and

[abc]

and so

bxca = [be a] = 1

[abc] a

They may

directly from the definitions of a

b b

=a =b =c

a, b, c,

two

sets of three vectors each, say A, B, C,

satisfy the relations

Aa = Bb = Cc = 1 A-b = Ac = Ba = B-c = Ca = Cb =

DIRECT AND SKEW PRODUCTS OF VECTORS then the set A, B, C

the system reciprocal to reasoning similar to that before is

a, b, c.

+ Ab b + A-c c B = B-a a + B-b b + B-c c C = Caa + C-bb + C-c c A-a a

Substituting in these equations the given relations the re sult

Hence Theorem

The necessary and a

set of vectors

be the reciprocals of

sufficient conditions that the a, b, c is

that

they satisfy the equations

= b .b = c .c = l = b a = b .c = c -a = c

a .a

a -b

As to a

-c

(32) .b

these equations are perfectly symmetrical with respect and a, b, c it is evident that the system a, b, c may , c

be looked upon as the reciprocal of the system a , b c just as the system a , b , c may be regarded as the reciprocal of ,

a, b, c.

That

to say,

Theorem: If a then

V [a

be the reciprocal system of

be the reciprocal system of a

a, b, c will

x b

b=-c

]

x ab

x b

b c

a, b, c,

(29V v /

c ]

]

These relations may be demonstrated directly from the definitions of a

The demonstration

straightfor

ward, but rather long and tedious as it depends on compli cated reduction formulae. The proof given above is as short as could be desired. The relations between a b ,c and b c is the reciprocal a, b, c are symmetrical and hence if a ,

system of

a, b, c,

then

a, b, c

must be the

reciprocal system of

VECTOR ANALYSIS

86 45.]

Theorem

If a

and

be reciprocal systems and [a b c] are numerical

a, b, c

the scalar triple products [a b c ]

That

reciprocals.

bV] [abc]=l xc cxa axb"|

[a t. b .

]=[i[a "be]

~[abc]

[bxc cxa axb]

But Hence

[abc]

Hence

= (bxc)x(cxa>(axb).

[abc]c.

[abc] c-axb

bV] =

[abc] J

[bxc cxa axb].

(bxc) x (cxa) [bxc cxa axb]

[abc]

By means

[abc]

2 .

[abc]

(33)

[abc]

between [a b c ] and [a b an important reduction formula,

of this relation

possible to prove

(P.axE)(ABxC)

P-A

P.B

p.c

Q.A B*A

Q.B

a-c

c] it

(34)

which replaces the two scalar triple products by a sum of nine terms each of which is the product of three direct pro ducts.

Thus

the

two

crosses

products are removed. expressed as

But

give the proof let P,

= P-A A +

P.B B

B = B-A A +

B.B B

+ BC

Then

which occur in the two scalar

[POB]

P.A

P.B

P.C

a-A R-A

Q.B

a-c

R.B

R.C

B C

[ABC]

P.C C

B C

ft,

DIRECT AND SKEW PRODUCTS OF VECTORS P.A

Hence

[PQE] [ABC]

The system of

P-B a-B

Q.C

R.A

R.B

B*C

i, j,

system.

jxki this reason the

primes

a system of vectors reciprocal

is its

own

reciprocal

k==k

kxi

,, J

For

P-C

Q.A

three unit vectors

(35)

k are not needed to denote to i, j, k. The primes will

therefore be used in the future to denote another set of rect

angular axes i, j, k just as X* , F , Z* are used to denote a set of axes different from X, F, Z. The only systems of three vectors which are their own reciprocals ,

and left-handed systems of three unit the system i, j, k and the system i, j, k. Let A, B, C be a set of vectors which is its own reciprocal.

are the right-handed

That

vectors.

Then by (32) Hence the vectors

AA = B-B = CNC = 1. are all unit vectors.

A-B

Hence

perpendicular to

B-A

Hence B

= A-C = B and

= B-C = 0.

perpendicular to

C-A

and

=C.B

= O.

Hence C is perpendicular to A and B. Hence A, B, C must be a system like i, j, k * 46.]

A scalar equation

or like

i, j,

of the first degree in a vector r is

an equation in each term of which r occurs not more than once. The value of each term must be scalar. As an exam ple of such an equation the following

a a-bxr

6(oxd)(exr)

may

be given.

+ c fr +

= 0,

VECTOR ANALYSIS

where

known vectors

are

a, b, c, d, e, f

;

and

a, &, c, d,

known

Obviously any scalar equation of the first degree in an unknown vector r may be reduced to the form

scalars.

r-A

where

known

vector

;

and

known

scalar.

complish this result in the case of the given equation proceed as follows.

+ axb +

a axbor {a

(cxd)xe-r

(cxd)xe

In more complicated forms

+ c fr + d = + c f}r = d.

may

be necessary to

make use made

of various reduction formulae before the equation can be to take the desired form,

]>A

As a vector

= a.

has three degrees of freedom

it is

clear that one

Three

scalar equation is insufficient to determine a vector. scalar equations are necessary.

The geometric

interpretation of the equa

tion

r.A => a is

origin.

drawn from a

fixed origin.

Let

be a fixed vector drawn from the same

The equation then becomes r

Let r be a variable vector

interesting.

(Fig. 24)

(36)

cos (r,A)

T cos (r,A) r be the magnitude of r

;

and

r cos

A (r,

= a, a ,

that of A.

The

expression

is the projection of r upon A. The equation therefore states that the projection of r upon a certain fixed vector A must

DIRECT AND SKEW PRODUCTS OF VECTORS

always be constant and equal to a/A. Consequently the ter minus of r must trace out a plane perpendicular to the vector

equal to a/A from the origin. The projec of any radius vector drawn from the origin to a

at a distance

tion

upon

and equal

point of this plane is constant

to a/A.

This gives

the following theorem.

Theorem

scalar equation in

an unknown vector may be which is the locus of the

regarded as the equation of a plane,

unknown

vector if its origin be fixed. three scalar equations in an unknown why Each equation de vector determine the vector completely.

terminus of the It

easy to see

termines a plane in which the terminus of r must lie. The Hence one vec three planes intersect in one common point. tor r

The

determined.

equations

analytic solution of three scalar If the equations are

extremely easy.

rA = a

=b r-C = c r-B

(37)

it is

only necessary to call to mind the formula r

Hence

The

= r.A A + r-BB + r-C r = a A + 6 B + c C

solution

therefore accomplished.

(38) It is expressed in

the reciprocal system to A, B, C. One terms A B caution must however be observed. The vectors A, B, C will ,

C which

have no reciprocal system In this solution will fail.

they are coplanar.

case,

Hence the

however, the three planes de

termined by the three equations will be parallel to a line. They will therefore either not intersect (as in the case of the lateral faces of a triangular prism) or

common

line.

Hence there

they will intersect in a

will be either

there will be an infinite number.

no solution

for r or

VECTOR ANALYSIS

From

four scalar equations

=a r.B = 6 rC = c rD =d

r-A

the vector r

may be

To accomplish

entirely eliminated.

solve three of the equations fourth.

(39)

this

and substitute the value in the

= aA + 6B +

A D + &B .D + cC -D = d [BCD] + b [CAD] + c [ABD] = d [ABC]. a

or *

47.]

A vector equation

of the first degree in an

(40)

unknown

an equation each term of which is a vector quantity containing the unknown vector not more than once. Such

vector

an equation

(AxB)x(Cxr)

+ D ET + n

+ F =0,

where A, B, C, D, E, F are known vectors, n a known scalar, and r the unknown vector. One such equation may in gen eral be solved for

general sufficient to

contained in

The method

That

to say,

determine the

one vector equation is in vector which is

unknown

to the first degree.

of solving a vector equation is to multiply it

with a dot successively by three arbitrary vectors.

may first

Thus

known non-coplanar

be solved by the methods of the foregoing place let the equation be

A ar + B br + where A,

These

three scalar equations are obtained.

B, C, D, a, b, c are

known

c-r

In the

article.

= D,

vectors.

scalar coeffi

may be incorporated in the vectors. Multiply the equation successively by A B , C . It is understood of course that A, B, C are non-coplanar. cients are written in the terms, for they

DIRECT AND SKEW PRODUCTS OF VECTORS

= D-A b-r = D-B c-r = D-C a-r

Hence

The

= a a-r + b b-r + c c-r. r = D-A a + D-B b + D-C

But

solution

therefore accomplished in case A, B, C are nonThe special cases in a, b, c also non-coplanar.

coplanar and

which either of these

sets of three vectors is coplanar will not

be discussed here.

The most general vector equation of the first degree unknown vector r contains terms of the types

A That

is it

a-r,

Exr,

which consist of a known

contain terms

will

vector multiplied by the scalar product of another known vec and the unknown vector ; terms which are scalar multi

tor

ples

of the

product of

unknown vector; terms which are the vector known and the unknown vector and constant ;

The terms

terms.

of the type

a-r

may always

For the vectors

number.

to three in

be reduced

which are

be expressed in terms of three nonmay coplanar vectors. Hence all the products a-r, b-r, or, may be expressed in terms of three. The sum of all terms of multiplied into r

the type terms, as

all

a-r therefore reduces to

A a-r + B The terms

of the types

b-r

an expression of three c-r.

and Exr may

also be expressed

in this form.

= a a-r + n b b-r + n c c-r Exr = Exa a-r + Exb b-r+Exc c-r. n

Adding to the

all

these terms together the whole equation reduces

form

a-r

M b-r + N c-r = K.

VECTOR ANALYSIS

This has already been solved as r

The

solution

= K.L

+ K-M

+ XJT c

in terms of three non-coplanar vectors a

These form the system reciprocal to the products containing the *

a, b, c in

unknown

f .

terms of which

vector r were expressed.

SUNDRY APPLICATIONS OF PKODUCTS to

Applications 48.]

Mechanics

In the mechanics of a rigid body a force is not a See Art. 3.

vector in the sense understood in this book.

A force has magnitude Two

of application.

and

direction,

and direction

but which

lie

do not produce the same

;

but

has also a line

which are alike in magnitude

forces

different lines in the

upon

body

Nevertheless vectors are

effect.

sufficiently like forces to be useful in treating them. If a

number

same point

on a body at the added as vectors is called

of forces f x , f 2 , f 3 , ---act

0, the

sum

of the forces

the resultant R.

E = f1 + In the same

way

if f

the term resultant

added just as

...

do not act at the same point

f2, f 8

applied to the

is still

sum

of these forces

they were vectors.

B=f + 1

The

...

(41)

idea of the resultant therefore does not introduce the

line of action of a force.

far as the resultant

concerned

a force does not differ from a vector. Definition:

The moment

of a force f about the point

equal to the product of the force tance from

however

nitude

as defined above.

by the perpendicular

to the line of action of the force.

best looked

upon

dis

The moment

as a vector quantity.

Its direction is usually

Its

mag

taken to

DIRECT AND SKEW PRODUCTS OF VECTORS

be the normal on that side of the plane passed through the and the line f upon which the force appears to pro point in the positive duce a tendency to rotation about the point Another method direction. of defining the trigonometric = is as follows moment of a force t PQ about the point :

The moment

of the force f

= PQ

to twice the area of the triangle

about the point is equal PQ. This includes at once

both the magnitude and direction of the The point P is supposed to be the origin

moment ;

(Art. 25).

and the point

the terminus of the arrow which represents the force letter

M will be

The

A subscript will

used to denote the moment.

be attached to designate the point about which the

moment

taken.

The moment of a number of forces f x f 2 sum of the moments of the individual forces.

the (vector)

moment

of the forces

This *

known

as the total or resultant

v *& 49.]

If f

be a force acting on a body and

drawn from the point the force, the

moment

if e

the

Mo dxf

d be the vector

of the force about the point

vector product of d into f

For

any

point in the line of action of

W = dxf

= d f sin

(42)

(d, f) e,

be a unit vector in the direction of dxf.

dxf

Now d

sin

(d, f) is

The magnitude

= dsm

(d,

f)/e.

the perpendicular distance from

dxf

accordingly equal to this perpen

dicular distance multiplied by/, the magnitude of the force.

VECTOR ANALYSIS

The direction the magnitude of the moment MO {f} as same the of the direction moment. the Hence of dxf This

the relation

proved.

Mo The sum

of the

of the resultant

= dxf.

moments about

acting at the

f p f 2,

{f}

same point

of a

P is

of forces

moment

to the

equal

For

of the forces acting at that point.

d be the vector from

to P.

Mo Mo

{f x> {f a |

= dxf = dxf a l

moment about

let

Then

= dx( + f The

number

+ ... (43) + -..)=dxB

of any number of forces f x f 2 a on is acting rigid body equal to the total moment of those forces about increased by the moment about of the total

resultant

M<x Let

dj,

fr f , 2

considered as acting at 0.

{f i> f 2

>} = Mo

{f r f 2 ,

}

Mo< {Bo

be vectors drawn from

d 2,

Let d/, d 2

respectively.

any point

be the vectors drawn

from O f to the same points in f x , f 2 be the vector from to_0 Then

(44)

Let

respectively.

d^d/H-c, Mo Mo

{f i, f 2

{f!,f 2 ,

= d2 +

+ d 2 xf 2 + } J=d xf 1 + d a xf a + ... = (d - c)xf + (d 2 - c)xf a + = d x xf +d2 xf 2 + ---- cx(f j + f a + xf j x

But is

the

c is the

vector drawn from

moment about

parallel in direction to f 1

to 0.

of a force equal in

but situated at 0.

Hence

magnitude and

Hence

DIRECT AND SKEW PRODUCTS OF VECTORS fa

Hence MO/

{f x

...)

}

f2 ,

= - cxBo = Mo {f r

;,} +

{Bo}.

MCX {Bo

(44)

The theorem is therefore proved. The resultant is of course the same at all The points. is attached to at show what it is subscript merely point to when act the moment about O is taken. For supposed f

the point of application of

The is

the

E affects

the value of that moment.

moment and the resultant same no matter about what point the moment be taken. scalar product of the total

In other words the product of the total moment, the result ant,

and the cosine

between them

of the angle

invariant

for all points of space.

E MO where O and f

}= B MO

{f i, f 2

are

{f !

}

any two points in space. This -important from the equation

relation follows immediately

Mo For E.Mo But

the

{*i, f a

if!,f 2

moment

the point

= Mo {fj f 2 }=* M {f^, }

} }

perpendicular to

+ Mo {Eo}. + E- M {B

E no

matter what

Hence

of application be.

E-MO* IE O }

and the relation is proved. The variation in the total moment due to a variation of the point about which the moment is taken is always perpendicular to the resultant. A point O may be found such that the total moment 50.] about it is parallel to the resultant. The condition for r

parallelism

{f x

-}=<)

VECTOR ANALYSIS

where by

its

any point chosen at random. Replace value and for brevity omit to write the f v f 2

braces

Mo {Eo} in the

Then

{ }.

RxMcy The problem

= ExMo - Ex(cxE) = 0.

to solve this equation for

EE c + R.c E = 0.

ExMo Now R

known

o is also supposed to be quantity. Let c be chosen in the plane through known. perpen and the equation reduces to Then dicular to E.

Ec = ExM = EE c ExMo E-E

moment about

If c be chosen equal to this vector the total

the point O which is at a vector distance from equal to c, will be parallel to E. Moreover, since the scalar product of r

moment and

the total

resultant itself

the resultant

constant

it is

constant and since the

clear that in the case

they are parallel the numerical value of the total

moment

about which

For If c

it is

about O

O f

unchanged by displacing the point

taken in the direction of the resultant.

jf !, f 2

}

= Mo cxE

parallel to E,

equal to that about

find not merely one point is

where

moment

minimum.

will be a

The

total

parallel to the resultant

;

{f !

f2 ,

}

- cxE.

vanishes and the 0.

Hence

it is

possible to

about which the total

but the total

moment moment

moment about any

point in the line drawn through parallel to E is parallel to E. Furthermore the solution found in equation for c is the only one which exists in the plane perpendicular to E unless the resultant E vanishes. The results that have been

obtained

may

summed up

as follows

DIRECT AND SKEW PRODUCTS OF VECTORS If

any system of forces

whose resultant

f 19 f 2 ,

97 not

upon a rigid body, then there exists in space one and only one line such that the total moment about any point of it is parallel to the resultant. This line is itself zero act

The

parallel to the resultant.

of it

the

same and

total

moment about

all

points

numerically less than that about any

other point in space.

equivalent to the one which states that any system upon a rigid body is equivalent to a single force (the resultant) acting in a definite line and a couple of which the plane is perpendicular to the resultant

This theorem

of forces acting

which the moment

and

may

be reduced to a single force (the resultant) acting at any of space and a couple the moment of which

minimum.

system of forces

desired point

(regarded as a vector quantity) is equal to the total moment about of the forces acting on the body. But in general the of this couple will not be perpendicular to the result plane ant, nor will its

moment be a minimum.

Those who would pursue the study of systems of forces acting on a rigid body further and more thoroughly may consult the Traite de Mecanique Rationnelle l by P. APPELL. The first chapter of the first volume is entirely devoted to the discussion of systems of forces.

Appell defines a vector

and point of His vectors are consequently not the same as

as a quantity possessing magnitude, direction,

application.

those used in this book. carried through in the

however may be analysis.

The treatment

of his vectors

Cartesian coordinates.

Each step

easily converted into the notation of vector

number

of exercises

given at the close of

the chapter. 51.] Suppose a body be rotating about an axis with a con stant angular velocity a. The points in the body describe circles concentric with the axis in planes perpendicular to 1

Paris, Gauthier-Villars et Fils, 1893. 7

VECTOR ANALYSIS

98 the axis.

The

velocity of

any point in

its

circle is

equal

to the product of the angular velocity and the radius of the It is therefore equal to the product of the angular circle.

and the perpendicular dis tance from the point to the axis. velocity

The

direction

the velocity

to the

axis

and to

perpendicular the radius of the circle described

by the

point.

Let a (Fig. 25) be a vector drawn along the axis of rotation in that direction in which a right-handed

screw would advance

turned in

the direction in which the body is Let the magnitude of a rotating.

FIG. 25.

The vector a may be taken to the angular velocity. represent the rotation of the body. Let r be a radius vector drawn from any point of the axis of rotation to a point in the be

body.

The

vector product

axr

= a rsin(a,r)

equal in magnitude and direction to the velocity v of the terminus of r. For its direction is perpendicular to a and r and its magnitude is the product of a and the perpendicular

distance r sin

(a, r)

from the point to the

v If the

a i* a 2> a a case

line a.

= axr.

That

(45)

body be rotating simultaneously about several axes which pass through the same point as in the

of the

rotations are

gyroscope, the velocities due

v i -=a 1 xr 1 v8

= a 8 xr 8

the various

DIRECT AND SKEW PRODUCTS OF VECTORS where

drawn from points same point of the body. Let be drawn from the common point of

are the radii vec tores

r x, r 2, r 3,

on the axis a 19 a 2 a 3 ,

to the

the vectors r x , r 2 , r 8 ,

Then

intersection of the axes. TJ

= ra = r8 =

and v

= vt + v2 + v3 +

== a xr x

a 2 xr

This shows that the body moves as angular velocity which

velocities a 19 a 2 , a 8 ,

This theorem

the vector

a 8 xr

rotating with the of the angular

sum is

sometimes known

law of angular velocities. It will be shown later (Art.) 60 that the motion of any rigid body one point of which is fixed is at each instant of time a rotation about some axis drawn through that point. as the parallelogram

This axis axis

position.

fixed

called the instantaneous axis of rotation.

not the same for

therefore represented

v where a

The

but constantly changes its The motion of a rigid body one point of which is all time,

= axr

(45)

the instantaneous angular velocity; and r, the drawn from the fixed point to any point of the

radius vector

body.

The most general motion

body no point of which Choose an arbitrary may point 0. At any instant this point will have a velocity v the body will have a motion of rotation Relative to the point about some axis drawn through 0. Hence the velocity v of

fixed

of a rigid

be treated as follows.

any point of the body may be represented by the sum of V the velocity of and axr the velocity of that point relative to 0.

=v +

axr.

(46)

VECTOR ANALYSIS

100

In case v

body moves around a and precisely the motion of a

parallel to a, the

along a simultaneously. This is screw advancing along a. In case v

perpendicular to a, it the vector r, such that possible to find a point, given by

its

velocity

This

may

zero.

That

be done as follows.

Multiply by xa.

= a-r a =

v xa

(axr)xa

Let r be chosen perpendicular to

The ity a,

point

this

= -

v xa.

Then ar

zero

and

v x a v x a

thus determined, has the property that

its

veloc

drawn through this point parallel to the body is one of instantaneous rotation

If a line be

zero.

the motion of

about

new

In case v

axis.

neither parallel nor perpendicular to a

may

be resolved into two components

vv n

vn "

which are respectively parallel and perpendicular to v =v +

point

may now

+ axr

be found such that

axr.

Let the different points of the body referred to r Then the equation becomes

denoted by

this point

The motion here expressed

=v +

axr

(46)

consists of rotation about an axis

a and translation along that axis. It is therefore seen that the most general motion of a rigid body is at any instant

DIRECT AND SKEW PRODUCTS OF VECTORS

101

the motion of a screw advancing at a certain rate along a The axis of the screw and its rate

definite axis a in space.

of advancing per unit of rotation

(i. e. its

pitch) change from

instant to instant.

The

52.]

conditions for equilibrium as obtained by the virtual velocities may be treated by vector

principle

methods.

Suppose any system of forces

tance

D whether

The

dis

finite or infinitesimal

the

forces is

work done

total

this distance

work done by the

act on a

body be displaced through a vector

If the

rigid body.

f x , f 2,

therefore

W^^i

D.f 2

...

body be in equilibrium under the action of the forces the work done must be zero. If the

W= D-fj + D-f

The work done by

the forces

The equation D-E

holds for

all

= D.E = 0.

equal to the work done by

This must be zero for every displacement

their resultant.

= D-Cfj +

Hence

vectors D.

E = 0. The total resultant must be zero if the body be in equilibrium. The work done by a force f when the rigid body is dis placed by a rotation of angular velocity a for an infinitesimal time t is approximately a-dxf t,

where d

tion a to

a vector drawn from any point of the axis of rota

any point of

components

To prove

parallel

a-dxf

this

break up

f into

and perpendicular respectively

= a-dxf +

a-dxf ".

two to

VECTOR ANALYSIS

102

d f ] vanishes. parallel to a the scalar product [a a-dxf

the other hand the

done by

= a-dxf

work done by

For

during the displacement.

a is perpendicular to its line of action.

equal to the work

being parallel to

h be the common

vector perpendicular from the line a to the force f ", the work " done by f during a rotation of angular velocity a for time t is

approximately

The vector d drawn from any point of a to any point of f may be broken up into three components of which one is h, another In the scalar is parallel to a, and the third is parallel to f ". triple

product

only that component of d which " f has any effect. Hence

[adf]

and

perpendicular alike to a

W= a-hxf

= a-dxf

act be dis body upon which the forces f v f 2 an placed by angular velocity a for an infinitesimal time t and if d x d 2 be the vectors drawn from any point of If a rigid

a to any points of f v by the forces f v f 2 ,

f2 ,

respectively, then the

W= (a-djXfj + ad = a.(d = a.M If the

body be

Hence

work done

will be approximately

xf 1 {f 1

xf 2

d 2 xf 2

,f 2 ,...}

in equilibrium this

a*M

\tl9 f2 ,

)

.-.)*

work must be }

zero.

= 0.

The scalar product of the angular velocity a and the total moment of the forces tv f2 about any point must be As a may be any vector whatsoever the moment itself zero. ,

must vanish.

{f r f r

}

= 0.

DIRECT AND SKEW PRODUCTS OF VECTORS

103

The necessary

conditions that a rigid body be in equilib rium under the action of a system of forces is that the result

moment about any

ant of those forces and the total

point in

space shall vanish. Conversely if the resultant of a system of forces and the moment of those forces about any one particular point in space vanish simultaneously, the body will be in equilibrium. If

E = 0,

then for any displacement of translation

DE = JF=D-f1 + and the

total

work done

D.f 2

zero,

...

when

= the body suffers any

displacement of translation. Let Mo {fp f 2 be zero for a given point 0. } other O point any >

Then

for

Mo< {fx

f 2,

But by hypothesis

= Mo

flf f 2 ,

also zero.

}

{Bo}-

Hence

Hence where a

equal to the

any vector whatsoever. But this expression is work done by the forces when the body is rotated

with an angular velocity a about the line a This work is zero. passing through the point O of a Any displacement rigid body may be regarded as a for

a time

translation through a distance D combined with a rotation for a time t with angular velocity a about a suitable line a in space.

It has

been proved that the

total

work done by the

forces during this displacement is zero. Hence the forces must be in equilibrium. The theorem is proved.

VECTOR ANALYSIS

104

Applications

Geometry

Relations between two right-handed systems of three Let i, j, k and i j k mutually perpendicular unit vectors. 53.]

Hence

r = = r ri

and

From

The

i -i i

*^,t ^*v, rj j + rk k

-j

-k

i .j j

-i i

scalarsflj,

a2 a3 ,

direction cosines of

That

+r /+ I

(47) .

this

/ i

b lt

;

a3 k

cl i

c2 j

c3 k.

are respectively the with respect to i, j, k. b 3 ; cv

cos

<&]_

== COS 0j

)

= cos

O t^ == COS

= cos

= cos -

COS

= cos

k).

( 4o)

In the same manner

= i-i j j-i j ( k = k.i i

They form their own reciprocal systems.

be two such systems.

+ i-j j + i-k k = ^ + j-j j + j-k k = a a + k.j j + k-k k = a a 7

i i

!/!_/ !?

and

fcf

)

] I

j i_^

.k !

9i ^

+ \ y + GI k + 6 2 j + c2 k + J 8 J + C 3 k/

n &

(47)"

9 4

=1= + =1=a2+ 2

j.j

( k-k

and

i i

= =\ = U = Cj

J 22

6 32

+ +

c2 2 3

&2 c2

b B cs

-f-

(49)

-f-

ttj

(50)

DIRECT AND SKEW PRODUCTS OF VECTORS and

j-k j.k

= = #a* 2

a 39

62 +K

6 I*3

and

= (a

- a3

(50)

(51)

105

62 ) i

But

(a2 & 3

- a3

6 2 ),

Hence

(52)

and similar relations may be found a v a2 a3 ,

;

coefficients of a transformation

X F 1

The

for the other six quantities

All these scalar relations between the

b v & 2 , &3.

, orthogonal axes important and well

which expresses one

Z* in terms of another set

known

JT,

set of

Z are

to students of Cartesian methods.

ease with which they are obtained here

may

be note

worthy.

A number of vector relations, which are known, but nevertheless important,

may

perhaps not so well be found by multi

plying the equations i

a3 k

in vector multiplication. &!

Cj j

3 j

(53)

The quantity on either side of this equality is a vector. From its form upon the right it is seen to possess no component in

VECTOR ANALYSIS

106 the its

direction but to lie wholly in the jk-plane ; and from form upon the left it is seen to lie in the j k -plane. i

Hence Its

must be the

magnitude

line of intersection of those

V af + a

V b^ + c^.

two

planes.

This gives the

scalar relations

af + a*

=V+

= 1 - a*.

The magnitude 1 a^ is the square*of the sine of the angle Hence the vector between the vectors i and i .

^k -cj ^sj-aak

(53)

and jk-planes, and j k of the sine the between the planes. angle magnitude Eight other similar vectors may be found, each of which gives one of the nine lines of intersection of the two sets of mu -

the line of intersection of the

its

The magnitude of the vector is in tually orthogonal planes. each case the sine of the angle between the planes. Various examples in Plane and Solid Geometry may 54.] be solved by means of products.

The perpendiculars from the vertices of a trian meet in a point. Let A B be the Let the perpendiculars from A to BC and from B triangle. to CA meet in the point 0. To show is perpendicular Example 1

gle to the opposite sides

A B. =C.

Choose

as origin

and

let

OA =

OB = B,

and

Then

= C-B, By

hypothesis

- B) = B<A - C) = 0. C<B - A) = 0, A.(C

and

Add;

which proves the theorem. Example 2 : To find the vector equation of a through the point B parallel to a given vector A.

line

drawn

DIRECT AND SKEW PRODUCTS OF VECTORS be the origin and

Let

dius vector from

the vector

OS.

Let

any point of the required

be the ra

Then

line.

Hence the vector product

parallel to

107

vanishes.

Ax(B-B) = 0. This

It is a vector equation in the the desired equation. unknown vector B. The equation of a plane was seen (page 88) to be a scalar equation such as is

BC = c in the

unknown

The point

vector B.

of intersection of a line

found at once.

The equations (

Ax(B B-C

and a plane may be

are

AxB = AxB B - C-B A = (AxB)xC A-C B - c A = (AxB)xC (AxB)xC + c A

A-C

Hence

A-C

AC = 0.

solution evidently fails when ever the line is parallel to the plane

The

or, if it lies

in the plane, there are

In this case

how

and there

infinite

is no solution number of solu

;

tions.

Example 3: The introduction of vectors to represent planes. Heretofore vectors have been used to denote plane areas of The direction of the vector was normal to definite extent. the plane and the magnitude was equal to the area to be re presented. But it is possible to use vectors to denote not a

plane area but the entire plane itself, just as a vector represents a point. The result is analogous to the plane coordinates of analytic geometry. a plane in space.

Let

be be an assumed origin. Let is to be denoted b^ a vector

The plane

VECTOR ANALYSIS

108

whose direction

the direction of the perpendicular dropped

and whose magnitude is the upon the plane from the origin of that perpendicular. Thus the nearer reciprocal of the length a plane

represents If r be

to the origin the longer will be the vector

which

it.

any radius vector drawn from the origin to a point and if p be the vector which denotes the plane,

in the plane

then r-p is

the equation of the plane.

Now p,

=1 For

= r cos

(r,

p) p.

the length of p is the reciprocal of the perpendicular to the plane. On the other hand r cos (r, p)

distance from is

Hence rp must be i, j, k

that perpendicular distance. If r

and p be expressed

unity.

in terms of

= #i + yj + zk p = ui + vj + wit rp = xu + yv + zw = L. r

Hence

The quantities u, v, w are the reciprocals of the intercepts of the plane p upon the axes.

The

relation

tion of duality.

between r and p

symmetrical.

It is a rela

If in the equation

r-p

r be regarded as variable, the equation represents a plane

which

the locus of

all

however p be

points given by regarded as variable and r as constant, the equation repre r.

sents a point r through which all the planes p pass. The of the of idea will not be carried out. development duality It is familiar to all students of The use of vec geometry. tors to denote planes will scarcely be alluded to again until

Chapter VII.

DIRECT AND SKEW PRODUCTS OF VECTORS

SUMMARY OF CHAPTER The

scalar product

two vectors

of their lengths multiplied

them.

A-B

The necessary and of

109

equal to the product

by the cosine of the angle between

= A B cos (A, B) A-B = B.A A.A = ^.

(1) (2)

(3)

sufficient condition for the perpendicularity

two vectors neither of which vanishes

The

product vanishes.

that their scalar

scalar products of the vectors

are (4)

^=J!Uk!=o A.B

= A B + A,, B + AS Bz = A* = A* + A* + A*. 1

If the projection of a vector

B upon

AB -R

The

(7)

a vector

vector product of two vectors

(8)

(*\

equal in magnitude to

the product of their lengths multiplied by the sine of the an The direction of the vector product is the gle between them.

normal to the plane of the two vectors on that side on which a rotation of less than 180 from the first vector to the second appears positive.

AxB = A B

sin (A,

(9)

The vector product is equal in magnitude and direction to the vector which represents the parallelogram of which A and B are the two adjafcent sides. The necessary and sufficient con dition for the parallelism of

two vectors neither of which

VECTOR ANALYSIS

110 vanishes

that their vector product vanishes.

The com

mutative laws do not hold.

AxB = -BxA ixi ixj

jxk kxi

AxB =

(10)

= kxk = jxi = k

=jxj

= = = i

kxj

(12)

= + (A a B - A ixk

(13)

AxB =

(13)

scalar triple product of three vectors [A B C] is equal volume of the parallelepiped of which A, B, C are three edges which meet in a point.

The

to the

[AB C]

= A-BxC = B.CxA = C-AxB = AxB-C = BxCA = Cx A-B [ABC]

=- [A OB].

(15)

(16)

The dot and

the cross in a scalar triple product may be inter changed and the order of the letters may be permuted cyclicly

without altering the value of the product ; but a change of cyclic order changes the sign.

[ABC]

(18)

be]

(19)

DIRECT AND SKEW PRODUCTS OF VECTORS If the

B perpendicular

= _AX(AXB)

component

be B",

A*A

= A-C B - A-B C (AxB)xC = A-C B - C-B A (AxB>(CxD) = A.C B-D - A-D B-C (AxB)x(CxD) = [A CD] B- [BCD] A = [ABD] C-[ABC] D. Ax(BxC)

The equation which

111

subsists

(24)

(24) (25)

(26)

between four vectors A, B,

A-[CDA]B + [DAB] C- [ABC] D = 0.

[BCD]

(27)

Application of formulae of vector analysis to obtain the for mulae of Plane and Spherical Trigonometry. The system of vectors a V, c is said to be reciprocal to the ,

system of three non-coplanar vectors

when

= _bxc., .

A vector r may be its

reciprocal in

a, b, c

= =cxa -=>

= axb

[abc]

be]

(29)

[abc]

expressed in terms of a set of vectors and

two similar ways

= r.a a + r.V b + r-c c r = r-aa + r.bb + r.cc

The necessary and

(30)

(31)

sufficient conditions that the

non-coplanar vectors

b, c

and a b ,

two systems of

be reciprocals

that

a, b, c

will

= Vb = c c = 1 a .b = a -c = b .c = b .a = c -a = e -b = 0. a .a

If a

form a system reciprocal to

form a system reciprocal to a

a, b, c

;

then

VECTOR ANALYSIS

112

P.A a-A R.A

[PaK][ABC] = The system system be

its

i, j,

own

system of three

its

own

reciprocal

P.B

P.C

ft-B

a-c

B-B

R-C

(34)

reciprocal and if conversely a must be a right or left handed

mutually perpendicular unit vectors.

Appli

cation of the theory of reciprocal systems to the solution of scalar and vector equations of the first degree in an unknown vector.

The

vector equation of a plane

r-A

= a.

(36)

Applications of the methods developed in Chapter II., to the treatment of a system of forces acting on a rigid body and in of forces to a single particular to the reduction of any system force and a couple of which the plane is perpendicular to that

Application of the methods to the treatment of instantaneous motion of a rigid body obtaining

force.

v where v

=v +

(46)

the velocity of any point, v a translational veloc and a the vector angular velocity of ro Further application of the methods to obtain the

ity in the direction a, tation.

conditions for equilibrium

by making use

of the principle of

Applications of the method to obtain the relations which exist between the nine direction cosines virtual velocities.

of the angles

between two systems of mutually orthogonal

axes.

Application to special problems in geometry including the form under which plane coordinates make their appear

ance in vector analysis and the method by which planes (as distinguished from finite plane areas) may be represented

by vectors.

DIRECT AND SKEW PRODUCTS OF VECTORS

113

EXERCISES ON CHAPTER II Prove the following reduction formulae

= [ACD]B-A-BCxD = BD AxC B-C AxD.

Ax{Bx(CxD)}

[AxB CxD ExF] = [ABD] [CEF]- [ABC] [DBF]

= [ABE] = [CD A] [AxB

BxC

CxA]

[ABC]

2 .

Q.B

P Q

RA RB

P.A

4 [PQE] (AxB)

P.B

Ax(BxC) + Bx(CxA) + Cx(AxB)

[AxP

Bxtt

- [ABF] [BCD] - [CDB] [AEF].

[FCD] [BEF]

CxR] + [Axtt

= 0.

BxR CxP]

+ [AxR BxP Obtain formula

(34) in the text

Cxtt]

= 0.

by expanding

[(AxB)xP].[Cx(ttxR)] in

two

different

ways and equating the

results.

Demonstrate directly by the above formulae that if a V, c form a reciprocal system to a, b, c; then a, b, c form 8.

a system reciprocal to a

Show

tors

the connection between reciprocal systems of vec Obtain some of the triangles upon a sphere*

and polar

geometrical formulae connected with polar triangles by inter preting vector formulae such as (3) in the above list. 10.

The perpendicular

meet in a 11.

two

bisectors of the sides of a triangle

point.

Find an expression for the common perpendicular same plane.

lines not lying in the

VECTOR ANALYSIS

114

Show by

12.

ume

vector methods that the formulae for the vol

of a tetrahedron

whose four

vertices are

13.

Making use

of formula (34) of the text

[abo] where

a, &, c I

14.

which

= cos

m = cos

(b, c),

(c, a),

= cos

the perpendicular (as a

and where

(a, b).

vector quantity)

dropped from the origin upon a plane determined by

the termini of the vectors given in Art. 46. 15.

are the lengths of a, b, c respectively

Determine is

= a be

show that

Show

that the

a, b, c.

Use the method

volume of a tetrahedron

of solution

equal to one

two opposite edges by the perpendicu between them and the sine of the included angle.

sixth of the product of lar distance

drawn in each face plane of any triedral angle the vertex and perpendicular to the third edge, the through three lines thus obtained lie in a plane. 16.

If a line is

CHAPTER

III

THE DIFFERENTIAL CALCULUS OF VECTORS Differentiation of Functions of One Scalar

Variable

IF a vector varies and changes from r to r the incre ment of r will be the difference between r and r and will be 55.]

denoted as usual by

A r.

Ar = r -r, where

A r must

(1)

be a vector quantity. If the variable r be A r is of course also unrestricted

unrestricted the increment it

may have any magnitude and any

direction.

If,

however,

the vector r be regarded as a function (a vector function) of a single scalar variable t the value of A r will be completely

determined when the two values

two values

and

are

and

which give the

known.

To it

obtain a clearer conception of the quantities involved will be advantageous to think of the vector r as drawn

from a fixed origin

(Fig. 26).

When

the independent variable t changes its value the vector r will change, and as t possesses one degree of freedom r will

vary in such a way that its terminus describes a curve in space, r will be FIG. 26.

the radius vector of one point of the curve ; r , of a neighboring point chord of the curve. The ratio

Ar A*

f .

r will be the

VECTOR ANALYSIS

116

the ratio 1

When A t

chord PP

proach P, the

with the chord

will be a vector collinear

but magnified

approaches zero

will

will approach the tangent at P,

ap and

the vector

Ar which

(t t

a vector tangent to the curve at

sense in which the variable If r

rfr

...

will approach

be expressed in terms of r

i, j,

=r +

r2 j

directed in that

increases along the curve.

the components r v r 2 , r3 will be functions of the scalar r

+ Arj)i+

(r 1

Ar = r

= Ar

Ar _ A

+ Ar2 )j +

(^ 2

+ Ar3 )k

+ Ar2 j + Ar3 k

A r2

?*!

(r 3

J+

A r8

and

Hence the components spect to

of the first derivative of r with re

are the first derivatives with respect to

components derivatives.

The same

_ - -

j__

dt*~ dt* dn

dt n

non-coplanar vectors

a, b, c

dn r

. ,

of the

dt*

d d __ __ dt dt* n

In a similar manner

dt*

r,l

true for the second and higher

r* ?

n J

r be expressed in

terms of any three

= aa +

&b + cc

dn a

(2)

dn c

THE DIFFERENTIAL CALCULUS OF VECTORS Example :

The vector

Let

= a cos + t

b sin

117

then describe an ellipse of which a and b are two conjugate diameters. This may be seen by assum r will

ing a set of oblique Cartesian axes X,

and

Then

which

X = a cos

Y = 6 sin

coincident with a

the equation of an ellipse referred to a pair of con diameters of lengths a and b respectively. jugate is

dr -3

a sin

+ b cos

Hence

= a cos (t +

The tangent

to the curve is parallel to the radius vector

for

+ 90).

The second

derivative

90) + b

(a cos

sin (t

b sin

the negative of

90).

t).

Hence

evidently a differential equation satisfied by the ellipse.

Example

Let

cosh

b sinh

The

vector r will then describe an hyperbola of which a and b are two conjugate diameters.

= a sinh +

b cosh

= a cosh +

b sinh

and

Hence d is

a differential equation satisfied by the hyperbola.

VECTOR ANALYSIS

118

combination of vectors all of which depend on the same scalar variable t may be differentiated very much as in 56.]

ordinary calculus.

d For (a

+ Aa)

A(a-b)

= (a + Aa)

A* Hence

in the limit

Ab A*

when A

+ Ab) - a-b

Aa -A*

Aa-Ab -

= 0,

= a-b _(a.bxc) v dt

The

last three of these formulae

may

be demonstrated exactly

as the first was.

The formal process

differentiation in vector analysis

no way from that in scalar analysis except in this one point in which vector analysis always differs from scalar differs in

analysis,

namely

The order

of the factors in a vector product

THE DIFFERENTIAL CALCULUS OF VECTORS

119

cannot be changed without changing the sign of the product.

Hence

of the

two formulae

d and

first is evidently incorrect, but the second correct. In other words, scalar differentiation must take place without The altering the order of the factors of a vector product.

the

factors

must be

differentiated in situ.

This of course was to

be expected. In case the vectors depend upon more than one variable the results are practically the same. In place of total deriva

with respect to the scalar variables, partial derivatives occur. Suppose a and b are two vectors which depend on tives

three scalar variables

#, y, z.

The

scalar product

depend upon these three variables, and partial derivatives of the

The second

first

will

have three

order.

partial derivatives are

formed in the same way.

\3x5y

VECTOR ANALYSTS

120 Often

it is

more convenient This

the differentials. first differentials.

to use not the derivatives

particularly true

The formulas

(3), (4)

= da, b + a = ds, x b + a

d (a b) d (a X b) and so

forth.

example.

As an

when

but

dealing with

become db,

(3)

x db,

(4)

consider the following

illustration

If r be a unit vector

rr = 1. The

locus of the terminus of r

a spherical surface of unit

radius described about the origin, ables. Differentiate the equation.

depends upon two vari

= 0.

Hence

Hence the increment di

of a unit vector

the vector.

perpendicular to This can be seen geometrically. If r traces a is

sphere the variation d r must be at each point in the tangent

plane and hence perpendicular to r. Vector methods may be employed advantageously *57.j in the discussion of curvature and torsion of curves. Let r

denote the radius vector of a curve

where

some vector function of the scalar t. In most appli cations in physics and mechanics t represents the time. Let s be the of arc measured from some definite length point of f is

the curve as origin. curve. Hence A r / A

The increment A r

the chord of the

approximately equal in magnitude and approaches unity as its limit when A s becomes Hence d r / d s will be a unit vector tangent to infinitesimal. the curve and will be directed toward that portion of the to unity

s is

THE DIFFERENTIAL CALCULUS OF VECTORS curve along which unit tangent

The curvature

s is

increasing (Fig. 27).

Let

121

be the

UAt

of the curve

the

limit of the ratio of the angle through

which the tangent turns of the arc.

to the length

The tangent changes by

of unit length, the length of

the increment At.

approximately the angle has turned which the measured in circular tangent through measure. Hence the directed curvature C is is

LIM

As=0 The vector C to

collinear with

inasmuch as

t; for

ds*

and hence perpendicular to

a unit vector

perpendicular

The

tortuosity of a curve

the limit of the ratio of the

angle through which the osculating plane turns to the length of the arc. The osculating plane is the plane of the tangent vector t and the curvature vector C. The normal to this

N = txC.

planei8

If c be a unit vector collinear

with C

normal (Fig. 28) to the osculating plane and the three vectors t, c, n form an i, j, k system, will be a unit

a right-handed rectangular system. Then the angle through which the osculating plane turns will be given approximately by A n and hence the tortuosity is by definition that

is,

d n/d

From vectors

the fact that

t, c,

n form an

k system

of unit

122

VECTOR ANALYSIS

and

= c c = nn = 1 tc = cn = nt = 0. t

Differentiating the first set

= cdc = ndn = 0,

t-dt and the second

But d t

rft

=cdn + dcn = ndt + dnt==0. and consequently perpendicular

parallel to c

n- dt = 0. d n t = 0.

Hence

The increment of

perpendicular to

also perpendicular to n.

As the tortuosity is T = dn/ds, it to

to n.

But the increment

therefore parallel to

parallel to

dn and

hence

The

tortuosity

~ds^ T *

The

first

VCC/

d*r d*r j O v d s*

term of

this expression vanishes.

been seen to be parallel to C

= d 2 r/ds 2

T moreover has

Consequently the magnitude of T is the scalar product of T by the unit vec tor c in the direction of C. It is desirable however to have .

the tortuosity positive when the normal n appears to turn in the positive or counterclockwise direction if viewed from that side of the n c-plane upon which t or the positive part of the curve lies. With this convention d n appears to move in the direction

turns

away from

c c.

therefore be given

when The by

the tortuosity is positive, that is, n scalar value of the tortuosity will T.

THE DIFFERENTIAL CALCULUS OF VECTORS T

But

d 2 i/d s 2

c is parallel to the vector

123 1

Hence

dr ~~

ds 2

And

c is

T. -c-T = -

Hence

(12)

The is

Hence

a unit vector in the direction C.

tortuosity

may

somewhat shorter

(13)

be obtained by another method which not quite so straightforward.

tc = cn = nt = 0. dtc = dct dn c dc n =

Hence

dn*t

Now

dt n

= 0.

dtn.

hence perpendicular to n. Hence Hence dnt = 0. Butdnis perpendicular to n. Hence d n must be parallel to c. The tortuosity is the mag nitude of dn/ds taken however with the negative sign is

parallel to c

;

because d n appears clockwise from the positive direction of the curve. Hence the scalar tortuosity T may be given by

r=-

dn .c

= n. dc ds

(14)

r = txc-^-

(14)

VECTOR ANALYSIS

124

V cc

d C

dc ds

---VC.C ds

!; ds

C-C

But

= 0.

A/C

C-C _

dC T~

ITTc

(13)

ds*

In Cartesian coordinates

this

becomes

(13)

T= Those who would pursue the study of twisted curves and surfaces in space further from the standpoint of vectoi-s will find the book " Application de la Methode Vectorielle de Grass-

maun d

la Greometrie 1

Infinitesimale"

Paris, Carre et

by FEHB extremely

Naud, 1899.

THE DIFFERENTIAL CALCULUS OF VECTORS helpful. is

125

He works

with vectors constantly. The treatment The notation used is however slightly different

elegant.

from that used by the present writer.

The fundamental

points of difference are exhibited in this table

One used

~ Oi

x a3

[a x a 2 a 3 ]

[a x

method need have no

to either

a 2 aj. difficulty

with the

All the important elementary properties of curves and surfaces are there treated. They will not be taken other.

here. * Kinematics

Let r be a radius vector drawn from a fixed origin to a moving point or particle. Let t be the time. The equation 58.]

of the path

then

The

velocity of the particle is its rate of

This

the limit of the increment

V This velocity

LIM A *

r"|

change of position. increment A t.

r to the

a vector quantity. Its direction is the direction of the tangent of the curve described by the par is

The term

used frequently to denote merely This convention will be the scalar value of the velocity. ticle.

followed here.

speed

Then

.-,

(16)

be the length of the arc measured from some fixed point of the curve. It is found convenient in mechanics to denote

if s

differentiations with respect to the time by dots placed over the quantity differentiated. This is the oldfliixional notation

VECTOR ANALYSIS

126

introduced by Newton. It will also be convenient to denote The equations become the unit tangent to the curve by t.

--T. =v

The is

acceleration

<">

(17)

the rate of change

Let

a vector quantity.

of velocity.

be denoted by A.

Then by

definition

A v _ d v =_

LIM

-At=OA7-rf7 dv

and

Differentiate the expression A

dv __

d(vt) v

dt _ d t

where C

_ dt

dv .

dt dz

ds d

dt __ dt .

_=C

dt d

the (vector) curvature of the curve and v

the

speed in the curve. Substituting these values in the equation the result

A=st+ The

acceleration of a particle

v* C.

moving

in a curve has there

fore been lei

broken up into two components of which one is paral to the tangent t and of which the other is parallel to the

curvature

that

is,

perpendicular to the tangent.

resolution has been accomplished

That

this

would be unimportant were

THE DIFFERENTIAL CALCULUS OF VECTORS it

not for the remarkable fact which

brings to light.

127

The

component of the acceleration parallel to the tangent is equal It is entirely in magnitude to the rate of change of speed. sort of curve what the of particle is describing. independent

would be the same if the particle described a right line with the same speed as it describes the curve. On the other hand the component of the acceleration normal to the tangent It

equal in magnitude to the product of the square of the speed of the particle and the curvature of the curve. The is

sharper the curve, the greater this component. The greater the speed of the particle, the greater the component. But the of change of speed in path has no effect at all normal component of the acceleration. rate

If r be expressed in terms of

=# +

z k,

From

(16)

A = v=s =

(18)

+y * +z% x + y* + z 2

x x

this

= V ** + y* + * 2 A = v = r = ii + yj + * k, v

i/ i/

these formulae the difference between s t the rate of

change of speed, and A is apparent. Just when

= r, the

rate of

change of velocity, became clearly

this difference first

But certain mind when he stated

recognized would be hard to say.

Newton must have had it in law of motion. The rate of change to the impressed force

of velocity

his

that

second

proportional

but rate of change of speed is not. The 59.] hodograph was introduced by Hamilton as an aid to the study of the curvilinear motion of a particle. ;

With any assumed origin the vector velocity r is laid off. The locus of its terminus is the hodograph. In other words, the radius vector in the hodograph gives the velocity of the

VECTOR ANALYSIS

128 particle in

possible to

magnitude and direction at any instant. It is proceed one step further and construct the hodo

graph of the hodograph. vector acceleration

A=r

done by laying off the from an assumed origin. The This

radius vector in the hodograph of the hodograph therefore gives the acceleration at each instant.

Example 1

Let a particle revolve in a

circle (Fig. 29) of radius r with a uniform

^-r^

angular velocity

The

speed of the particle will then be equal to v

Let velocity v

perpendicular to f

The

vector v

be the radius vector

drawn to the particle. r and to a. It is

=v=a

The

always perpendicular and of constant magni is therefore a circle of radius v = a r.

radius vector r in this circle

advance of the radius vector

r in

just ninety degrees in

its

quently describes the circle with the

The

The hodograph

tude.

The

acceleration

which

circle,

and

conse

same angular velocity

the rate of change of y

always perpendicular to v and equal in magnitude to

A = a v = a 2 r. The

acceleration

But as a Hence

The

A may

be given by the formula

= A = axv = ax(axr) = ar a

perpendicular to the plane in which r r

= A = aa r =

a 29

a-a

lies,

a r

= 0.

acceleration due to the uniform motion of a particle in is directed toward the centre and is equal in magni

a circle

tude to the square of the angular velocity multiplied by the radius of the circle.

THE DIFFERENTIAL CALCULUS OF VECTORS Example 2:

Consider the motion of a projectile.

acceleration in this case

the

= A = g.

hodograph reduces to a constant

the

It is easy to find is merely a point. Let v be the velocity of the projectile

hodograph.

in path at

any given

=v +

Thus the hodograph ing through the

instant.

will be

particle

a later instant the velocity t

a straight line parallel to g and pass The hodograph of a extremity of v is

moving under the influence of gravity

The path

straight line.

Example 3

The

the acceleration g due to gravity.

The hodograph of The curve vector.

129

well

known

hence a

to be a parabola.

In case a particle move under any central

acceleration

= A = f(r).

The tangents to the hodograph of r are But these tangents are approximately chords between two successive values

That

vector in the hodograph.

the accelerations collinear

and

with the

of the radius

approximately

A* Multiply by rx.

=rx

Since r and r are parallel r

Hence

But J

r r

x x

- r ) = 0. r = r x r

f is the rate of description of area.

Hence the

equation states that when a particle moves under an ac celeration directed towards the centre, equal areas are swept

over in equal times by the radius vector. 9

VECTOR ANALYSIS

130 it

Perhaps

would be well

this question.

go a

little

more carefully into

If r be the radius vector of the

particle in

its path at one instant, the radius vector at the next instant The area of the vector of which r and r r. r are is r

the bounding radii is approximately equal to the area of the This triangle enclosed by r, r + A r, and the chord A r. area

The

rate

of description

of area

by the radius vector

consequently

irx(r+ Ar) Lm 1 A* ~A*-=02

LIM

A* = 02

A*~2

Let r and r be two values of the velocity at two points which are near together. The acceleration r at P

P and P is

the limit of r

A* A

Break up the vector parallel

A ^t

_" A r

= ?^IlI? A

into

two components one

and the other perpendicular

to the acceleration r

Ar. if

n be a normal to the vector

The quantity x ap The quantity y

proaches unity when A t approaches zero. approaches zero when A t approaches zero.

Hence

x x

Ar = r-r = #A*r + yA*n. - r ) = x A* r x r + y A* r x (r (f

- r ) = r x r - (r +

^A M x

THE DIFFERENTIAL CALCULUS OF VECTORS

131

Hence

Ar rxr-r xf = /A xr 6

But each

A* + zA*rxr + yA* rxn.

upon the right-hand side is an order. Hence the rates of descrip

of the three terms

infinitesimal of the second

tion of area at

and

differ

by an

second order with respect to the time.

Hence the

point of the curve.

rates

infinitesimal of the

This is true for any must be exactly equal

This proves the theorem. of a rigid body one point of which is at any instant a rotation about an instantaneous axis

at all points.

The motion

60.]

fixed

passing through the fixed point. Let i, j, k be three axes fixed in the body but moving in space. Let the radius vector r be drawn from the fixed point to any point of the body. Then

But

= (d r

i) i

(d r

Substituting the values of d r the second equation

= (xi

di+

di + yj

j) j 4-

(d r

+ *dj + d

k) k.

d r k obtained from

2 i

d k)

= 0. Hence dj +j di = Q or j-c?i = dj = = or j-dk j.dk + k.dj k.dj = = k.di + i.rfk k di. or idk Moreover i.i=j .j=kk = l. Hence d = d = k d k = 0.

But

VECTOR ANALYSIS

132

Substituting these values in the expression for d

This

dk +

(zi

yjdi)i+(jdi (y

-xi

d k)

a vector product.

= (Wj + idkj + jdik)x(>i +

dr Let

k).

k l+| -i J+J ii

-r

Then

d k

=axr

This shows that the instantaneous motion of the body is one of rotation with the angular velocity a about the line a. This angular velocity changes from instant to instant. The proof of this theorem fills the lacuna in the work in Art. 51.

Two

may be added

rotations

infinitesimal

d2 If r

be displaced by

T If it

a, it

Hence

becomes T

=T+

then be displaced by a 2 r 4-

xrdt.

becomes

= r + d r + % x [r + (a d r = aj x r d + a x r d + d

(a x

If the infinitesimals

like vectors.

The displacements

Let ax and a 2 be two angular velocities. due to them are d l r = a x x r d t,

r) (d

)

of order higher than the first be

neglected,

which proves the theorem. .

(a x

a 2)

both sides be divided by d

= dr = (a + 1

a 2)

THE DIFFERENTIAL CALCULUS OF VECTORS

133

the parallelogram law for angular velocities. was obtained before (Art. 51) in a different way.

This

In case the direction of stant, the

the instantaneous axis,

con

motion reduces to one of steady rotation about

The

acceleration r

= axr + axr = axr + ax

(axr).

a does not change its direction a must be collinear with That is, it is perpen a and hence a x r is parallel to a x r.

hand

dicular to

Inasmuch

as all points of the rotating

the other

r) is parallel to

body move in con

about a in planes perpendicular to a, it is unnecessary to consider more than one such plane. The part of the acceleration of a particle toward the centre centric circles

of the circle in

which

moves

a x (a x r).

This

equal in magnitude to the square of the angular It does not velocity multiplied by the radius of the circle. is

depend upon the angular acceleration a what is known as centrifugal force.

at all.

It corresponds

the other

the acceleration normal to the radius of the circle

hand

axr. This

equal in magnitude to the rate of change of angular It does not velocity multiplied by the radius of the circle. is

depend in any way upon the angular velocity itself but only upon its rate of change. 61.] The subject of integration of vector equations in which the differentials depend upon scalar variables needs but a

word. If

then

It is precisely like integration in ordinary calculus.

d r

=s+

VECTOR ANALYSIS

134

where C tion in

some constant

To accomplish

vector.

the integra

any particular case may be a matter of some

difficulty

of ordinary integration of scalars. just as it is in the case : 1 Integrate the equation of motion of Example

projectile.

The equation

of motion

simply

which expresses the fact that the acceleration tically downward and due to gravity.

always ver

= g + b,

where b

a constant of integration. 0. velocity at the time t

evidently the

r c is

ig*2 + b* +

another constant of integration.

of the point at time

= 0.

given by this so may be seen by

expressing

in terms of x

That this is and y and eliminating

last equation is a parabola. it

the position vector

The path which

Example % The rate of description of areas when a par ticle moves under a central acceleration is constant. :

Since the acceleration

But

For

x x

= f(r). parallel to the radius,

= a

r).

u/ t

Hence CL t

and

which proves the statement.

THE DIFFERENTIAL CALCULUS OF VECTORS

135

Example 3 : Integrate the equation of motion for a particle moving with an acceleration toward the centre and equal to a constant multiple of the inverse square of the distance from the centre. ^2

Given

Then

Hence

= C.

Multiply the equations together with x. r

xC = -1 ^-

rx (rxr)=

Then

= r r.

side of this equality

Then

Integrate.

x C c*

where

e I is

r 2.

o *

a perfect differential.

= + e I, r

x C

e r

I. *

e is its

magni

Multiply the equa

its direction.

But

- r-r r}.

the vector constant of integration,

tude and I a unit vector in tion

{r.r r

_ Lr

*2L o

Hence Each

(

-jjj-

Differentiate.

-1

VECTOR ANALYSIS

136 T

and cos u

Then

Or r

city.

r cos u.

cos

the equation of the ellipse of which e is the eccentri I is drawn in the direction of the major

axis.

(r, I).

= 1

This

= cos

The vector The length

of this axis

It is possible to cany the integration further and obtain the time. So far merely the path has been found.

Scalar Functions of Position in Space.

A function V (x,

62. ]

y, z)

which takes on a

value for each set of coordinates

definite scalar

#, y, z in space is called

scalar function of position in space.

ample,

The Operator

Such a function,

for ex

V O, y, z)

= r\

This function gives the square of the distance of the point from the origin. The function V will be supposed to be in general continuous and single-valued. In physics scalar (x, y, z)

functions

are of constant occurrence.

of position

In the

at any point of a body is a theory of heat the temperature scalar function of the position of that point. In mechanics

and theories of attraction the potential function.

This,

too, is

If a scalar function

is the all-important a scalar function of position. be set equal to a constant, the equa

tion

V(x,y,z)=c.

(20)

defines a surface in space such that at every point of it the has the same value c. In case be the tempera-

function

THE DIFFERENTIAL CALCULUS OF VECTORS ture, this is a surface of constant temperature.

In case

isothermal surface.

V be the

137

It is called

potential, this surface of

constant potential is known as an equipotential surface. As the potential is a typical scalar function of position in space,

and

most important of

as it is perhaps the

owing

all

such functions

to its manifold applications, the surface

V O, y,z)=c

V equal to a constant is frequently spoken an equipotential surface even in the case where V has no connection with the potential, but is any scalar function obtained by setting of as

of positions in space.

The tion

rate at

which the function

that

when x changes

constant

is,

V increases in the X direc

to x

x and y and

remain

LIM

Aa =

F" (a?

+ A a,

- T (x, y, z)

y, g)

the partial derivative of Fwith respect to x. Hence the rates at which increases in the directions of the three

This

axes X, Y)

Z are

respectively

3V Ty* Tz

3V ~Wx

Inasmuch as these are rates in a certain direction, they may Let i, j, k be a system be written appropriately as vectors. of unit vectors coincident with the rectangular system of are axes X, Y) Z. The rates of increase of

3V 1

JZ*

~3~z

The sum of these three vectors would therefore appear to be a vector which represents both in magnitude and direction the resultant or most rapid rate of increase of V. That this is

actually the case will be

shown

later (Art. 64).

VECTOR ANALYSIS

138

The vector sum which Fis denoted by

63.]

the resultant rate of increase

a directed represents a directed rate of change of or vector derivative of F^ so to speak. For this reason

will be called the derivative of

VF.

The terms

V F.

a vector

gradient and

customary to regard

V; and slope of

F, the primitive of

are also used for

V as an operator which obtains

V F from a scalar function F of position in space.

This symbolic operator

Hamilton and

now

was introduced by Sir

universal

There

employment.

l seems, however, to be no universally recognized name for it, although owing to the frequent occurrence of the symbol some name is a practical necessity. It has been found by

experience that the monosyllable del is so short and easy to occurs pronounce that even in complicated formulae in which a number of times no inconvenience to the speaker or hearer is read simply as " del F." arises from the repetition. has been defined as Although this operator

* v=i* + ji- +k dx 9z

Some

use the term Nabla owing to its fancied resemblance to an Assyrian harp. Others have noted its likeness to an inverted A and have consequently coined the none too euphonious name Ailed by inverting the order of the letters in the word Delta.

Foppl in

his Einfuhrung in die

Maxwell

sche Theorie der Elec-

avoids any special designation and refers to the symbol as "die Operation How this is to be read is not divulged. Indeed, for printing no particular

tricitdt

name

necessary, but for lecturing and purposes of instruction something is re too that does not confuse the speaker or hearer even when

quiredsomething often repeated.

THE DIFFERENTIAL CALCULUS OF VECTORS

appears to depend upon the choice of the axes, it This would be surmised in reality independent of them.

so that is

139

as the magnitude and direction from the interpretation of of To demonstrate the inde V. of the most rapid increase

pendence take another set of axes, variables # , y ^ z referred to them.

system

k and a new

Then

set of

referred to this

By making

?T7 ax

ao^7 y

k ^T7 dz

( 22 )

use of the formulae (47) and (47)", Art. 53, page

104, for transformation of axes

from

i, j,

k and by

actually carrying out the differentiations and finally by taking into account the identities (49) and (50), may

actually be transformed into V.

The

details of the proof are omitted here, because another

shorter

method

of demonstration

to be given.

Consider two surfaces (Fig. 30)

64]

y,z)=c

V (x,

and

upon which

V is

near together.

V=c. dius

Let

vector

y, z)

=c+

d c,

constant and which are moreover infinitely Let #, y, z be a given point upon the surface

denote the ra

drawn

point from any fixed

this

origin.

Then any the

point near by in neighboring surface

+ d c may be represented by the radius vector r + d r. c

The

actual increase of

Ffrom FIG. 30.

the is

first

surface to the second

a fixed quantity dc.

The

rate of increase

a variable

VECTOR ANALYSIS

140

quantity and depends upon the direction dr which is fol lowed when passing from one surface to the other. The rate of increase will be the quotient of the actual increase d

and

between the surfaces at the point in the direction d r. Let n be a unit normal to the

the distance x, y, z

Vdr

and d n the segment of that normal intercepted between the surfaces, n d n will then be the least value for d r. The quotient surfaces

\/d r will therefore be a

equal in

maximum when d r

magnitude of

d n.

therefore a vector of

most rapid increase of rate of that increase.

n and

which the direction

Fand

the direction of

which the magnitude

This vector

the

entirely independent of

be replaced by its equal d V which the increment of F^in passing from the first surface to the

the axes JT, Y, Z. is

parallel to

The expression

second.

Then

let

Let d

V V be defined again as

Vr=4^n. d n

(24)

From

is certainly the vector which this definition, and the rate the of most direction rapid increase of gives in that direction. VFis Moreover independent of the axes.

remains to show that this definition

first

given.

To do

this multiply

VF.dr =

equivalent to the one r.

-n.dr.

d n

(25)

is a unit normal. Hence n d r is the projection of d r on n and must be equal to the perpendicular distance d n between

the surfaces.

THE DIFFERENTIAL CALCULUS OF VECTORS

-dn

But

-z

=7rdx

where

(d x?

= dV

(d *)

= dr

takes on successively the values equation (25) takes on the values If

ids= dy

(25)

(d y)

141

d r. dx,

dy,

kdz the

~dx

sv dy

(26 )

If the factors

rf a;, rf

be cancelled these equations state

that the components

k of

in the

directions respectively are equal to

5_V

5y*

VF=(VF. i)i +

VF=

Henceby(26)

The second

|^ + j |T+

equivalent to

| reduced to the

definition of

y -

d x

V V.

satisfies

According

often taken as a

to ordinary calculus the deriv

the equation

dx dy dx ,

first

it.

The equation (25) found above

*65.]

(Vr-j)j + (VT. k)k.

definition (24) has been

and consequently

ative

VF* j, VF-

= dy

VECTOR ANALYSIS

142

equation defines dy / dx. In a similar manner is possible to lay down the following definition. of a scalar function of Definition: The derivative

Moreover it

this

position in space shall satisfy the equation

for all values of

d r.

This definition

certainly the

from theoretical considerations. either of the

are

They

definitions

tangible.

most natural and important

But

for practical purposes

before given seems to be better. The real significance of this last

definition cannot be appreciated until the subject of linear

vector functions has been treated.

The computation frequently

carried

of the derivative

See Chapter VII.

V of

a function

on by means of the ordinary

most

partial

differentiation.

Example 1 :

(ix

The This

derivative of r

+ jy +

kz)

a unit vector in the direction of

evidently the direction of most rapid increase of r and the rate of that increase. is

THE DIFFERENTIAL CALCULUS OF VECTORS Example % :

Let

V1F

-k _1

Hence

V ~ == 7 r

derivative of 1/r

-r -

square of the length

g 2 )t

V log

a vector whose direction

that

equal to the reciprocal of the

n rn ~ 2

= n r* i>r

Let

F(#,

y, z)

= log y# 2 +

V^Tp = TT 22 + i

y*.

+ f 22

denote the vector drawn from the origin to the point z) of space, the function V may be written as

If r y,

2 2 )*

left to the reader.

Example 4

V rn

Example 3: is

-r = ~T 3

and whose magnitude

The proof

and

The

143

= log Vr.r-(k.r) 2 k kr. ix + )y = T

2/1

and

Hence

V log

V^ + y T

- k kr

(r-kk.r).(r-kk.r)

VECTOR ANALYSIS

144 There

another method of computing

which

based

upon the identity

V = Vrr = r.

Example 1 : Let

=^ =

Hence Example 2 :

Hence Example 3:

Let

i>r

where a

V= r

Let

r is

a constant vector.

F=dr.a = dr Vr. V V = a.

F= (rxa)

(rxb), where a and b are

constant vectors.

V = rr ab dV =

2cZr*r a-b

drb

dr-a r-b

V F == 2 r a-b

Hence

r-a rb.

a r.b

Vr= (ra-b-ar-b) + Which

of these

two methods

for

= di V Fl

b r^a

(ra-b

= bx(rxa) -fax

r-a

-br.a)

(rxb).

computing

shall be

applied in a particular

case depends entirely upon their relative ease of execution in that case. The latter method is

independent of the coordinate axes and preferred.

may

therefore be

It is also shorter in case the function

Fcan

cannot be so expressed easily in terms of r. But when the former method has be to. to resorted expressed

The

in mathe great importance of the operator matical physics may be seen from a few illustrations. Sup *66.]

pose

T (#,

y, z)

be the temperature at the point

#, y, z of

THE DIFFERENTIAL CALCULUS OF VECTORS That direction

heated body.

145

which the temperature de

most rapidly gives the direction of the flow of heat. T, as has been seen, gives the direction of most rapid

creases

Hence the flow

increase of temperature. f

where k body.

of heat f is

= _& vr,

a constant depending upon the material of the be the gravitational potential Suppose again that is

due to a fixed body. the point (#, y, z) potential and

The

force acting upon a unit mass at in the direction of most rapid increase of

magnitude equal to the rate of increase Let F be the force per unit

per unit length in that direction. mass. Then

F As

= VF.

different writers use different conventions as regards the

sign of the gravitational potential, it

might be well to

state

that the potential Preferred to here has the opposite sign to If the potential energy. denoted the potential energy of a mass situated at #, y, z, the force acting upon that mass

would be

F In case

= - VfF.

represent the electric or magnetic potential due

to a definite electric charge or to a definite

spectively the force as the case might be

The

force

magnetic pole re

acting upon a unit charge or unit pole

= - VF.

the direction of most rapid decrease of

In dealing with electricity and magnetism poten potential. tial and potential energy have the same sign whereas in ;

problems they are generally considered to have opposite signs. The direction of the force in either case is in the direction of most rapid decrease of potential energy. The

attraction

difference between

potential

and potential energy 10

this.

VECTOR ANALYSIS

146

Potential in electricity or magnetism is the potential energy per unit charge or pole ; and potential in attraction problems is

mass taken, however, with the

potential energy per unit

negative sign. often convenient to treat an operator as a quantity provided it obeys the same formal laws as that Consider for example the partial differentiators quantity. It is

*67.]

!.. 3z

far as combinations of these are concerned, the formal laws

are precisely

what they would be

instead of differentiators

three true scalars a,

For instance

were given.

the commutative law

99 Sx3y

3ySx

the associative law

and the

distributive

law

3\ + ._ --+_-)=_--_ 3x\3y 3zJ 3x3y dxdz 3

f 3

(

a(b

hold for the differentiators just as for scalars.

Of course such

formulae as

where u

a function of x cannot hold on account of the

properties of differentiators.

scalar function

u cannot be

placed under the influence of the sign of differentiators. Such a patent error may be avoided by remembering that an

operand must be understood upon which 3/3 #

to operate.

THE DIFFERENTIAL CALCULUS OF VECTORS In the same

a great advantage

way

may be

147

obtained by

looking upon

+ V-if 3x +jf dz dy kf as a vector.

It is not a true vector, for the coefficients

.., JL,

P# are not true scalars.

A dz

a vector differentiator and of

It is

is always implied with it. As far as formal For concerned it behaves like a vector.

course an operand operations

are

instance

V (u + v) = V u + V v, V(ttfl) = (Vtt) v + ^(Vtf), c V u = V (c u),

w and v are any two scalar functions of the scalar variables y, 2 and if c be a scalar independent of the variables with

regard to which the differentiations are performed. 68.] If A represent any vector the formal combination

A. Vis

A.V = A

/-x+

A=A

provided

This operator

A V is

to a scalar function

+ As

+ A^ j + A% k.

a scalar differentiator.

V (x,

(27)

j-,

y, z) it

When

<^ r -A+^+^Suppose for convenience that

(a.V)F=a

applied

gives a scalar.

a unit vector

I + a r +a8 r 2

(28)

(29)

VECTOR ANALYSIS

148 where a v

are the direction cosines of the line a referred

a^a B

to the axes Jf, F, Z.

well-known This

Consequently

derivative of

directional

V appears

as the

in the direction

often written

3F,

T^^+^-^sTIt expresses the

magnitude of the

(29)

rate of increase of

V in

In the particular case where this direction is the normal n to a surface of constant value of F, this relation the direction

becomes the normal derivative.

n x n 2 n 3 be the direction cosines ,

of the normal.

The

operator a applied to a scalar function of position yields the same result as the direct product of a and the

vector

(a.V)F=a.(VF). For

this reason either operation

may

(30)

be denoted simply by

without parentheses and no ambiguity can result from the omission. The two different forms (a V) Fand a- (V F)

may however (a a.

interpreted in an important theorem. the directional derivative of in the direction

V) F is On the other hand

the direction

any direction

VFin

Hence

(

V V)

The

equal to the

that direction.

theorem becomes

The

the component of

V F in

directional derivative of

F in

of the derivative

component Fdenote gravitational potential the

directional derivative of the potential

any direction gives the component of the force per unit mass in that direction. In case Fbe electric or magnetic

potential a difference of sign

must be observed.

THE DIFFERENTIAL CALCULUS OF VECTORS

149

Vector Functions of Position in Space

69.]

vector function of position in space is a function

(x, y, z)

which associates with each point x, y, z in space a definite The function may be broken up into its three com

vector.

ponents

(x, y, z)

y,z)i+

(x,

(x, y, z) j

+ F3

(a?,

y, z) k.

of vector functions are very numerous in physics. the function has occurred. At each point of Already F has in general a definite vector value. In mechan space

Examples

each point of the body is a vector function of the position of the point. Fluxes of heat, ics of rigid bodies the velocity of

fluids, etc., are all

magnetic force,

electricity,

vector functions

of position in space.

The tion

V to

Let

scalar operator a

may

be applied to a vector func

yield another vector function.

and

(a.V)V

+ F2

Then

y,z)

(x,

(

(a.V) F! a

(as,

y, z) j

+ F3

(x, y, z)

a 3 k.

3^+-af3

+ (a.V) F2 j + (a.V)F, k

3V -

9V\

9V,

9V2

VECTOR ANALYSIS

150 This

may be

Hence

function

written in the form

the directional derivative of the vector

in the direction

V) V

without parentheses.

when

defined.

=a VV -

For the meaning of the vector symbol

applied to a vector function V has not yet been Hence from the present standpoint the expression

V V can

It is possible to write

have but the one interpretation given to

Although the operation V V has not been defined and cannot be at present, 1 two formal combinations of the vector operator V and a vector function V may be treated. These 70.]

and the (formal) vector prod

are the (formal) scalar product uct of into V. They are

and

VxV =

V V is read del dot V; The

+ ]-

and

differentiators

by the dot and the

cross.

< 82 >

+kxV

(88)

V, del cross V.

That

being scalar operators, pass

(32)

(88)

These may be expressed in terms of the components

F",

ofV. i

definition of

will

be given

Chapter VII.

PI.

THE DIFFERENTIAL CALCULUS OF VECTORS Now

8V_9V} 3*

151

~5y

9y h

5V, 3

(34)

3F Jl ~3

i.fl-

Then

Hence

Moreover

This

may

V V = rf + ^7 + 3Ti

(32)"

be written in the form of a determinant

VxV=

333 i

(33)"

VECTOR ANALYSIS

152 It is to be

understood that the operators^are to be applied to

the functions

F"2 ,

when expanding the

determinant.

From some

standpoints objections may be brought forward as a symbolic vector and introducing V against treating as the scalar and vector x V respectively and symbolic

products of

simply laying

may be looked upon

x, which

quite distinct

from V,

and

means advisable

all

for

71.]

This symbol obeys the

as a vector just in so far as the differentiators

333T~ T~~

(33)

remembering formulae

as a symbolic vector differentiator.

operators

to regard

same laws

new

+ jx-4-kx-.

for practical purposes

seems by

as entirely

shall be

V xV = ix

and

But

These objections may be avoided by the definition that the symbols and

into V.

down

^)e

^e same

aws

That the two functions

^ or(^ nary sca ar quantities. l

V V

and

x V have very

important physical meanings in connection with the vector function V may be easily recognized. By the straight

forward proof indicated in Art. 63

was seen that the

From

is independent of the choice of axes. operator fact the inference is immediate that V and x

intrinsic properties of

this

V V represent In order

invariant of choice of axes.

to perceive these properties it is convenient to attribute to the

function

some

definite physical

flow of a fluid substance.

meaning such

as flux or

Let therefore the vector

denote

THE DIFFERENTIAL CALCULUS OF VECTORS

153

and the magnitude

of the

at each point of space the direction

flow of some fluid.

This

may

be a material fluid as water

or gas, or a fictitious one as heat or electricity. To obtain as great clearness as possible let the fluid be material but not necessarily restricted to incompressibility like water.

= i~+j.

Then is

called the divergence of

and

V V= div The reason

for this

term

that

*I dz

often written V.

VV gives at each

volume per unit time

rate per unit

which

point the

fluid is leaving

the rate of diminution of density. To prove Let the edges this consider a small cube of matter (Fig. 31). Let of the cube be dx, dy, and dz respectively.

that point

(x, y, z)

(x9

y,z)i+ V^ (x

y, z) j

+ F3 (x, y, z)

Consider the amount of fluid which passes through those faces of the cube

which are

parallel to the F^-plane,

perpendicular to the axis. The normal to the face

whose x coordinate

the lesser, that

is,

the nor

mal to the left-hand face of the cube

The

flux

xy2

of substance through this

face

-i.V

(x,y,z)

face,

FIG. 31.

the face whose

x coordinate through

dz.

to the oppo- z

The normal site

greater by the therefore

it is

amount dx,

and the

flux

VECTOR ANALYSIS

164

dx,

y, z)

dy dz

= i V (x

V(#,

dy dz

y, z)

3V dx

i dy dz

dz.

outward from the cube through these two This therefore the algebraic sum of these quantities.

The

total

faces

flux

simply

3V i

= -^3^

dx dy dz

dx dy

dz.

In like manner the fluxes through the other pairs of faces of the cube are

3V,,, dx dy dz ^

The

total flux

j,,c)V and k - dx dy

out from the cube

/. i (

3V 9x

dz.

+ t

3V\

therefore

. )

9zJ

dx dy

dz.

the net quantity of fluid which leaves the cube per unit time. The quotient of this by the volume dx dy dz of the cube gives the rate of diminution of density. This is

This

V.T.I. 9x + , dy

Because V thus represents the diminution of density or the rate at which matter is leaving a point per unit volume per unit time, it is called the divergence. Maxwell employed the term convergence to denote the rate at which fluid ap proaches a point per unit volume per unit time. This is the negative of the divergence. In case the fluid is incompressible, as

much matter must

leave the cube as enters

change of contents must therefore be

zero.

it.

For

The

total

this reason

the characteristic differential equation which any incompres sible fluid

must

satisfy is

THE DIFFERENTIAL CALCULUS OF VECTORS where

known

as the

the flux of the

This equation

fluid.

hydrodynamic equation.

flow of water, since water

often

It is satisfied

155

by any

The

practically incompressible.

great importance of the equation for work in electricity is due to the fact that according to Maxwell s hypothesis electric dis

placement obeys the same laws as an incompressible then D be the electric displacement, div

the operator X Maxwell gave the This nomenclature has become widely accepted.

V The

x V

name

curl.

= curl V.

curl of a vector function

of position in space.

= V D = 0.

72.]

fluid.

the

itself

name

a vector function

indicates, it is closely

connected with the angular velocity or spin of the flux at each point. But the interpretation of the curl is neither so easily obtained nor so simple as that of the divergence.

Consider as before that

Take

represents the flux of a fluid.

at a definite instant an infinitesimal sphere about

any

the next instant what has become of the

point (#, y, z). sphere ? In the first place it may have moved off as a whole in a certain direction by an amount d r. In other words it

may have a translational velocity of dr/dt. In addition to may have undergone such a deformation that it is no a It may have been subjected to a strain by longer sphere. this it

virtue

Finally

of it

which

may

becomes slightly

ellipsoidal

shape.

have been rotated as a whole about some

axis through an angle

dw.

That

to say, it

may have an

An angular velocity the magnitude of which is dw/dt. infinitesimal sphere therefore may have any one of three distinct types of

motion or

all

them combined.

First, a

translation with definite velocity. Second, a strain with three definite rates of elongation along the axes of an ellipsoid.

VECTOR ANALYSIS

156

Third, an angular velocity about a definite axis. It is this In fact, third type of motion which is given by the curl.

a vector which has at each point of space the direction of the instantaneous axis of rotation at that point and a magnitude equal to twice the instantaneous the curl of the flux

angular velocity about that axis.

The more

analytic discussion of the motion of a fluid presents than it is necessary to introduce in treating

difficulties

The motion of a rigid body is sufficiently complex an adequate idea of the operation. It was seen (Art.

the curl. to give

51) that the velocity of the particles of a rigid body at any instant is given by the formula

=v + a x curl v = Vxv = Vxv + Vx(axr). a = a + a% j + a 3 k r = r + r 2 + r k=:;ri + 2/j-fzk v

Let

expand

product of

x v

+ (V

if it

r) a

were the vector

- (V

Hence the term

a constant vector.

V As

X (a x r) formally V, a, and r. Then

V v

a constant vector

3x it

of the differential operator,

^+3

may be

x v

vanishes.

= 3. z

placed upon the other side

/ 3 3 3\ = a Vr=( ai^ + a 2j-+ a s^ Jr Vxv = 3a a = 2a. Hence

triple

Therefore in the case of the motion of a rigid body the curl of the linear velocity at any point is equal to twice the angular velocity in magnitude and in direction.

THE DIFFERENTIAL CALCULUS OF VECTORS

157

= curl v = 2 a,

x v

= ^Vxv=| curl v. + ^ (V x v) x r = v + \ (curl v) a

(34)

The expansion of V x (a x r) formally may be avoided by multiplying a x r out and then applying the operator V X to the result. 73.]

tion

It frequently happens, as in the case of the applica

just cited, that the operators

V>V% V X,

have to be

applied to combinations of scalar functions, vector functions, or both. useful.

The following Let

be found

rules of operation will

be scalar functions and

u, v

v vector func

Then

tions of position in space.

+ t?) = Vw + Vfl V.(u + v) = V.u + V-v Vx(u + v) = Vxu + Vxv V (u v) = v V u + u V v V (w v) = V M v + M V v V(t6

(35)

(36)

(37) (38) (39)

(40) (41)

+ v x (V x

+ u x (V x v)

V.(uxv)=v.Vxu u-Vxv (42) Vx (uxv) = v.Vu~vV-u-u.Vv + uV.v. (43) A word necessary upon the matter of the interpretation 1

of such expressions as

V u v, The

rule followed in this

book

to the nearest term only. 1

By Art. 69 the expressions V) uand u V) v *

(

That v

that the operator

V applies

is,

V n and

V v me

to be interpreted as

VECTOR ANALYSIS

158

V uv = (V u) v V u v = ( V u) v V u x v = ( V u) x v.

is to be applied to more than the one term which follows terms to which it is applied are enclosed in a paren the it, thesis as upon the left-hand side of the above equations.

The

proofs of the formulae may be given most naturally by expanding the expressions in terms of three assumed unit The sign 2 of summation will be found con vectors i, j, k. venient.

By means

of it the operators

V> V*

take the

form

The summation extends over

To demonstrate

#, y, z.

(wv)

^ Vx

Hence

(wv)

= Vwxv + ^Vxv.

To demonstrate

=v Vu+

Vv + v

(

x u)

n x

(

v).

THE DIFFERENTIAL CALCULUS OF VECTORS

V(u.v)

159

2^-v+^^.-

Now ,3u

^ v ._ 2 3 u

= vx(V

IE v ** 9x

In like manner

;r-

(

x u) + v

(

9 u

2v.i-

x v)

V u. +

V v.

V(uv)=vVu +

Hence

+ The

xu) +

v x

(V x

+ u x (V x v).

other formulae are demonstrated in a similar manner.

71]

The notation l

V(u-v) u

(44)

will be used to denote that in applying the operator

product (u

That

is,

v),

partially

the quantity u is to be regarded as constant. is carried out only partially upon

the operation

the product (u

V to the

In general

v).

upon any number

of functions

to be carried out

which occur

after

a parenthesis, those functions which are constant for the differentiations are written after the parenthesis as subscripts.

it in

Let

This idea and notation of a partial so to speak may be avoided by means But a certain amount of compactness and simplicity is u v )u is surely no more complicated than u v or ( thereby. The idea of

of the formula 41. lost

X (V X

n).

VECTOR ANALYSIS

160

n-v

then

and

+ u^v 2 + u z

3^0

But

and

V(u.v) T

Hence

But

V(u-v)

rrr^j

= w Vi? + ^ a Vi? a + w 8 V? 8 V(uv) v = ^j V^ + 2 V^ 2 + ^ 3 V^ 3 V (u- v) = V (u- v) u + V (u- v) v

V(u-v) n

and

Vw +

Hence

(44)

(45)

This formula corresponds to the following one in the nota tion of differentials

d (u v)

d (u

The formulae

= d (u v) = u

v) u

+ d (u v) T + d n v. (Art. 73) may be obvious from analogy

(35)-(43) given above

written in the following manner, as is with the corresponding formulae in differentials

V (u + V.

+ v)

v*)

= V (u + v\ + V (u + v) = V- (u + v) u + V- (u + v) v

x (u

v) u

+Vx

v) y

(35)

(36) (37)

THE DIFFERENTIAL CALCULUS OF VECTORS

V (u v) = V (u v\ + V (u v\ V- (u v) = V- (u v). + V- O v) r V x (u v) = V x (i* v) a + V x (u v) v V (u- v) = V (u. v) u + V (u. v) v V (u x v) = V (u x v) a + V (u x v) y V x (u x v) = V x (u x v) a + V x (u x v)

(38) (39)

(40) (41)

(42) (43)

This notation scalar product

In almost

formally.

where

Take

x v) u

= (V

Expand

v) u is

v) u

constant for the differ

Then

which occur in V.

may

was introduced.

can be done away without loss of

for instance (43)

in the case of the

for this reason it

must be understood that u

entiations

factor u

other cases

all

simplicity.

particularly useful

u^v and

161

in the last

term the

Hence

be placed before the sign V.

V X (uX v) u = u V v u- Vv. manner V x (u x v) = v V u v V u. Vx(uxv)=vVu v V u u V v + u V *

In like

Hence

There are a number of important relations in which

75.]

the partial operation

u x

V(u-v) u

(u v) u figures. v)

= V (u

V v = V (u

j, k.

as follows.

v) a

= u. Vv + u v) u

proof of this relation

terms of is

(V x

The

method

Expand

x (V x

+ (V x

may

- u V v, v),

v) x

(46) (46)

(46)"

be given by expanding in

remembering the result

the product

u x (V x v) ll

easily

VECTOR ANALYSIS

162 formally as

v were

Then

all real vectors.

ux(Vxv)=u.vV

V v.

The second term is capable of interpretation as it stands. The first term, however, is not. The operator V has nothing upon which

to operate.

that

have u v as

shall

of the parenthesis in tiations.

must be transposed so an operand. But u being outside

It therefore

u x (V x v)

Hence

constant for the differen

V = V (u v) u u V v. v) = V (u v) u

u v u x (V x

and If

u be a unit vector, say

(46)

the formula

a-Vv = V(av) a + (V x

x a

(47)

v of a expresses the fact that the directional derivative a in vector function v the direction a is equal to the derivative of the projection of the vector v in that direction plus the vector product of the curl of v into the direction a. Consider the values of v at two neighboring points.

and

v (x

(x, y, z)

dx, y

dy> z

dz)

= v (x + dx, y + dy, z + dz) v (#, y, z). v = v{i + v 2 j + v 8 k Let dv = dv i + dv%j + dv 3 k. dv = dr* But by (25) d v% = dr dv% = dr d v s= d r (V v Hence + V v + V v z k). dv = dr Vv Hence d v = V (rfr v) dr + (V x v) x dr. By (46)" dv

2 j

(48)

THE DIFFERENTIAL CALCULUS OF VECTORS Or

V denote the value of v at the point

(#, y, z)

163

and v the

value at a neighboring point

= v + V (d r

v) dr

+ (V x

This expression of v in terms of the

dels,

and the displacement d

its

v) x dr.

(49)

value v at a given point,

r is analogous to the

expan

sion of a scalar functor of one variable by Taylor s theorem,

/<*>=/(*>+ .TOO ** The That

derivative of (r

when v

constant

V (r v) = v. = v V r - (V x

equal to v.

V (r

For

v) v

r> x

Vr = v

V (r

Hence In like manner

v%j

= v,

= 0. v) v = v.

instead of the finite vector

vector d r be substituted, the result

an infinitesimal

still is

V (d r v) v = v. v = V + V (d r v) dr + (V x v) x d r V (d r v) = V (d r v) d + V (d r vV V (d r v) dr = V (d r v) v.

V/fllO*^

(49)

;

Hence

Substituting :

= ^ vo + ^V(dr.v) + ^(Vxv)xdr.

This gives another form of (49) which

convenient

It is also slightly

(50)

sometimes more

more symmetrical.

VECTOR ANALYSIS

164 *

Consider a moving

76.]

Let v

fluid.

(#,

T/, 3,

velocity of the fluid at the point (#, y, z) at the time

round a point

(a;

z )

=v +

In the increment of time B

moved

The

= c2

Sur

point of this sphere the velocity

be the

with a small sphere.

dr dr

At each

V v.

the points of this sphere will have

the distance

point at the center will have

moved

the distance

The

distance between the center and the points that were upon the sphere of radius d r at the commencement of the interval $

has become at the end of that interval S

find the locus of the extremity of

it is

necessary to

eliminate d r from the equations

The

first

The

result will

= dr

be solved for d r by the method of Art. 47, page 90, and the solution substituted into the second. equation

may

show that the

infinitesimal sphere

has been transformed into an ellipsoid by the motion of the fluid during the time 8 1.

A may

definite account of the

change that has taken place

be obtained by making use of equation (50)

THE DIFFERENTIAL CALCULUS OF VECTORS v

= iv + |v(rfr.v) + 2-(Vx

165

xdr,

= v +i[V(dr.v)-v ] + |(Vxv)xdr; S

or of the equation (49)

= v + V(dr-v) dr +(Vx

=v + [V (dr The

first

v) dr

(V x

x d

v)xrfr,

r]+

(V x

x d r.

term v in these equations expresses the fact that is moving as a whole with an instan

the infinitesimal sphere

taneous velocity equal to V . This of the motion. The last term

^(Vxv)xdr =

the translational element

curl v

x dr

undergoing a rotation about an instantaneous axis in the direction of curl v and with an angu

shows that the sphere

magnitude to one half the magnitude of The middle term

lar velocity equal in

curl

v(dr.v) dr -

(Vx

x dr

expresses the fact that the sphere is undergoing a defor mation known as homogeneous strain by virtue of which it

becomes

ellipsoidal.

For

this

term

equal to

dx V^j + dy V# 2 + dz if

Vj,

tions

v 2 , v s be respectively the i,

components of v in the direc

fairly obvious

that at any given point

mutually perpendicular axes i, j, k be chosen such that at that point V^, V# 2 V# 3 are re-

(#o> 2A

may

zo) ^ set of three

VECTOR ANALYSIS

166

Then

them.

spectively parallel

the

expression

above

becomes simply

dx *i i+dyy dx 9y

^i + dz ^. 9^

The point whose

coordinates referred to the center of the

infinitesimal sphere are

dx,

dy,

will

endowed with this velocity. have moved to a new position

The

totality of the points

therefore

In the time S

t it

upon the sphere

the ellipsoid of goes over into the totality of points upon which the equation is 2

dx 2

dz*

The statements made before (Art. 72) concerning the three types of motion which an infinitesimal sphere of fluid may possess have therefore

now been

The symbolic operator

demonstrated.

V may be

applied several times This will correspond in a general way to forming derivatives of an order higher than the first. The will all be independ expressions found by thus repeating 77.]

succession.

ent of the axes because

V itself

is.

There are six of these

dels of the second order.

Let

The

(#, y, z)

derivative

be a scalar function of position in space. is a vector function and hence has a curl

and a divergence.

Therefore

V-VF,

VxVF

THE DIFFERENTIAL CALCULUS OF VECTORS two derivatives obtained from V.

are the

of the second order

which may be

V-VF=div VF

The second expression

(51)

VF=curl VF.

(52)

V x V V vanishes identically. V possesses no V V in terms of

scalar function

the derivative

167

That

curl.

is,

This

of any All x be seen by expanding i, j, k. the terms cancel out. Later (Art. 83) it will be shown con possesses no curl, i. e. if versely that if a vector function

may

W = curl W =

then

W = VF,

first

expression

V VF

when expanded

in terms of

k becomes

V V=

Symbolically,

The

the derivative of some scalar function F.

The i, j,

operator

Laplace.

V V is

Laplace

^ y2

-5

<i/2

+ -rO

therefore the well-known operator of

Equation

becomes in the notation here employed

V-VF=0.

(53)

V V

When applied to a scalar function F the operator yields a scalar function which is, moreover, the divergence of the derivative.

Let

flow f

be the temperature in a body. Let c be the con The ductivity, p the density, and k the specific heat.

VECTOR ANALYSIS

168

The

rate at

unit time

which heat

leaving a point per unit volume per

The increment

of temperature is

rfr=-^-V.f K

dt.

d -

= -^.VT. pK

at This

Fourier

equation for the rate of change of tempera

ture.

Let

ponents.

be a vector function, and

The

Vv Vv Vz

operator V V of Laplace

may

three

its

v.vv = v-vr + v-vr + v.vr i

If a vector function its

satisfies

2 j

order

Laplace

three scalar components does.

Other

Equation, each of second

dels of the

be obtained by considering the divergence and curl The divergence V has a derivative

VV-V = VdivV. curl

(55)

V X V has in turn a divergence and a curl, V V x V, VxVxV. V V x V = div curl V V x V x V = curl curl V.

and

(54)

may

of V.

The

these expressions

V VxV

(56) (57)

vanishes identically.

the divergence of the curl of any x V in terms of seen by expanding

vector is zero.

V V

will be

shown conversely that

83) vector function

i, j,

This

That

is,

may

Later (Art.

the divergence of a

W vanishes identically, V W = div W = then W = V x V = curl V, i. e. if

com

be applied to V.

the curl of

some vector function V.

THE DIFFERENTIAL CALCULUS OF VECTORS

If the expression

according to the

169

x (V x V) were expanded formally

law of the

triple vector product,

Vx(VxV) = V-VV-V.VV. The term

V VV

meaningless until

the beginning so that

be transposed to

operates upon V.

VxVxV = VV.V-V-VV, curl

This formula

(58)

curl V = V div V - V VV.

(58)

very important. It expresses the curl of the curl of a vector in terms of the derivative of the divergence is

and the operator of Laplace. satisfy Laplace

Should the vector function

Equation,

V VV = and curl curl V = V div V. Should the divergence of

curl curl

be zero,

Should the curl of the curl of

V VV. vanish,

V div V = V VV. To sum

There are six of the

up.

dels of the

second order.

V-VT, VxVF,

V-VV, V V Of

these,

two vanish

V Vx

identically.

VxVr=0, V-VxV = 0. A third may be

expressed in terms of two others.

VxVxV = VV.V-V.VV. The

operator

(58)

V V is equivalent to the operator of Laplace.

VECTOR ANALYSIS

170 *

The geometric

78.]

interpretation of

V Vw is interesting.

upon a geometric interpretation of the second u of the one scalar variable x. u at the Let it be required the value of Let u be point x to find the second derivative of u with respect to x at the Let xl and x2 be two points equidistant from # point x That is, let It depends

derivative of a scalar function

*""

~~~

Then

the ratio of the difference between the average of u at the points x l and #2 and the value of u at x to the square of the is

distance of the points x v #3 from x

d*u .

easily

That

LIM.

proved by Taylor s theorem.

Let u be a scalar function of position in space. three mutually orthogonal lines

Choose

k and evaluate the

expressions

Let

;

and a?! be two points on the line i at a distance a from 2 #4 and #3 two points on j at the same distance a from #e and #6 , two points on k at the same distance a from x o?

?^_.

LIM ._2

u~ :

THE DIFFERENTIAL CALCULUS OF VECTORS

171

Add:

-LIM r

~a=OL As

V and V-

are independent of the particular axes chosen,

this expression

then for

still

may

be evaluated for a different set of axes,

a different one,

etc.

adding together

all

these results

u\ + u% +

a==

Let n become

infinite

n terms

and

same time

at the

let the different

from #

sets of axes point in every direction issuing

The

fraction

n terms

then approaches the average value of u upon the surface of a Denote this sphere of radius a surrounding the point x .

by u a

V V u is equal to six times the of the excess of

u on the surface

limit approached

by the

ratio

of a sphere above the value

at the center to the square of the radius of the sphere. same reasoning holds in case u is a vector function.

The

u be the temperature of a body V-V u (except for a constant factor which depends upon the material of the If

VECTOR ANALYSIS

172 body) 77).

equal to the rate of increase of temperature (Art. If positive the average temperature upon a is

VV^is

small sphere is greater than the temperature at the center. The center of the sphere is growing warmer. In the case of a steady flow the temperature at the center

constant.

flow

must remain

Evidently therefore the condition for a steady

V V u = 0. That

the temperature is a solution of Laplace Maxwell gave the name concentration to is,

u be a

be called the dispersion of the function

The

Equation.

V V u whether Consequently V V u may

scalar or vector function.

or vector.

u whether

be scalar

proportional to the excess of

dispersion the average value of the function on an infinitesimal surface above the value at the center. In case u is a vector function the average

The

a vector average.

additions in

are

vector additions.

SUMMARY

OF CHAPTER III

If a vector r is a function of a scalar

with respect to

the derivative of

a vector quantity whose direction is that of the tangent to the curve described by the terminus of r and whose magnitude is equal to the rate of advance of r

that terminus along the curve per unit change of t. The derivatives of the components of a vector are the components of the derivatives.

dn r dt*

r,.

dt*

j J

dt*

combination of vectors or of vectors and scalars

(

may

differentiated just as in ordinary scalar analysis except that the differentiations must be performed in situ.

THE DIFFERENTIAL CALCULUS OF VECTORS

173

(3)

(4)

d (a

=da

(3)

d(axb) = daxb + axdb, and so

The

forth.

(4)

differential of a unit vector is

perpendicu

lar to that vector.

The

derivative of a vector r with respect to the arc s of the curve which the terminus of the vector describes is

the unit tangent to the curves directed toward that part of the

curve along which

$ is

supposed to increase.

r."The

derivative of t with respect to the arc *

direction

a vector whose

normal to the curve on the concave side and

whose magnitude

The

equal to the curvature of the curve.

tortuosity of a curve in space

the derivative of the

unit normal n to the osculating plane with respect to the arc

The magnitude

^n_^_/rfr X r ds~ ds\ds ds* <?

of the tortuosity

VCTC/

rdr d*r 3 r"| L^s ^T2 rf^J cZ

VECTOR ANALYSIS

174

denote the position of a moving particle, A the acceleration,

If r

the time,

v the velocity,

*-*---* The

be broken up into two components of parallel to the tangent and depends upon the

acceleration

which one

may

change of the scalar velocity v of the particle in its path, and of which the other is perpendicular to the tangent and depends upon the velocity of the particle and the curva rate of

ture of the path.

s t

(19)

Applications to the hodograph, in particular motion in a circle, parabola, or under a central acceleration. Application to the proof of the theorem that the motion of a rigid body

one point of which is fixed is an instantaneous rotation about an axis through the fixed point. Integration with respect to a scalar of differentiation.

merely the inverse due to

Application to finding the paths

given accelerations.

The

operator applied to a scalar function of position in a vector whose direction is that of most rapid space gives of that increase function and whose magnitude is equal to

the rate of that increase per unit change of position in that direction

THE DIFFERENTIAL CALCULUS OF VECTORS

The

is invariant of the axes operator denned by the equation

may be (24)

W-dT = dV.

175

(25)

by two methods depend Computation of the derivative V and Illustration of the oc (25) ing upon equations (21) .

currence of

V in

mathematical physics.

may be looked upon as a fictitious vector, a vector It obeys the formal laws of vectors just in differentiator. so far as the scalar differentiators of 51 5 x> "9 / d y, 9 1 3 z obey the formal laws of scalar quantities

A If a

=^ ^i^ l7 +

be a unit vector a

in the direction

< 28 >

the directional derivative of

a.VF = (a-V) F=a(VF). If

V is

a vector function a

(30)

the directional derivative

of that vector function in the direction a.

3 x

+ k.^-, ,-J~ 9 3 z

+ VxV=ix|^ 3x jx

V.V= ^

(32)

3 x

^+ 3y

+ kx

(33)

^ 3 z

(32)

VECTOR ANALYSIS

176 Proof that

V V is

of V.

the divergence of

V x V, the curl

and

V V = div V, V X V = curl V. V O + = V u + Vtf,

(35)

(36)

t;)

u + V v, V (u + v) Vx(u + v)=Vxn + Vxv, V (u v) = v V u + u V v, V (u v) = V u v + u V v, V x (u v) = V u x v + u V x v, V(nv)=vVu + U Vv + vx (V x

Vx (u x

(V x

n x

(37) (38) (39) (40)

n) v), (41)

V (n x v) = v Vxu-u-Vxv, u Vv + uV* v. v V u v) =v .Vu

Introduction of the partial ferentiations are

del,

V (u

performed upon

v) u

(42) (43)

which the

dif

the hypothesis that u

constant.

u x If a

(V x v)

= V (u

v) u

V v.

(46)

be a unit vector the directional derivative a

The expansion

Vv=V

v) a

+ (V x

(47)

any vector function v in the neighborhood of a point (x# y# z ) at which it takes on the value of v is

=v + V

= \ v + V (d r

(d r

v) dr .

+ (V x v) x

+ \ (V x

dr,

(49)

x d r.

(50)

Application to hydrodynamics. The dels of the second order are six in number.

THE DIFFERENTIAL CALCULUS OF VECTORS

V x VF= curl VF= 0, x)

(52)

wv &v

+ Vp V-VJ^vV^f^+^ d x d y 9 2

V V place s

Laplace s operator.

The

Equation.

(51)

VVF=0, V satisfies

operator

may

interpretation of

be applied to a vector.

VV. V = VdivV, V V x V = div curl V = 0, Vx VxV=curlcurlV = VV.V- V. VV. The geometric

177

V V as giving

(55) (56)

(58)

the disper

sion of a function.

EXERCISES ON CHAPTER III Given a

Let the particle moving in a plane curve. Obtain the formulae the for be the plane ij-plane. compo nents of the velocity parallel and perpendicular to the radius 1.

vector

These are

rp where is

kxr,

the angle the radius vector r

makes with

and k

the normal to the plane.

Obtain the accelerations of the particle parallel and perpendicular to the radius vector. These are 2.

Express these formulae in the usual manner in terms of x

and

VECTOR ANALYSIS

178

3. Obtain the accelerations of a moving particle parallel and perpendicular to the tangent to the path and reduce the

results to the usual form. If r,

be a system of polar coordinates in space,

</>,

where r is the distance of a point from the origin, </> the meridianal angle, and 6 the polar angle ; obtain the expressions for the components of the velocity and acceleration along the radius vector, a meridian, and a parallel of latitude.

Reduce

these expressions to the ordinary form in terms of #, y,

Show by

V is

the operator 6.

where 7.

method given

V of a triple product

a function of

#, y, z in

= (r

d, e, f

in Art. 63 that

independent of the axes.

the second

the derivative is

method suggested

the direct

case

r) r,

computing V find [a be] each term of which for

are constant vectors.

Compute

V V F when Fis r 2 r, -, or -r r r* V V V, VV V, and V x V x V when V ,

Compute equal to r and when

equal to

-j>

and show that in these

cases the formula (58) holds. 9.

Expand

V x V V and V V x V in

show that they vanish (Art. 10.

Show by expanding

Prove

in terms of

i, j,

k that

VV.

A.V(7-W) = VA.VW+ WA- VV,

and

(VxV)

i, j,

77).

Vx VxV=VV. V-V 11.

terms of

W=Vx

(Vx

k and

CHAPTER IV THE INTEGRAL CALCULUS OF VECTORS 79.]

Let

space.

Let

be a vector function of position in be any curve in space, and r the radius vector (#, y, z)

drawn from some fixed

origin to the points of the curve.

Divide the curve into infinitesimal elements dr.

sum

of the scalar product of these elements

of the function

thus

The in

From

the

d r and the value

W at some point of the element 2W d r.

limit of this

sum when

the elements

dr become

number, each approaching zero, is called the the curve C and is written

W along

infinite

line integral of

.dr.

and

= dx + j dy + i

dz,

r W dr = r [W+dx -\-W*dy +W%dz\. t/

(1)

The the

definition of the line integral therefore coincides with definition usually given. It is however necessary to

specify in which direction the radius vector r

supposed to

describe the curve during the For the elements integration. d r have opposite signs when the curve is described in oppo-

VECTOR ANALYSIS

180

C and

method

If one

site directions.

the other by

of description be denoted by

(7,

= --

JI c

-G

d r.

C is

a closed curve bounding a portion of surface the curve will always be regarded as described in

In case the curve

such

a direction that the enclosed area appears positive

(Art. 25).

which may be supposed to vary from (7, the work done by the force

If f denote the force

point to point along the curve

when r

its

point of application

of the curve

C to

J Theorem

The

scalar function

moved from

dr= fr

line integral of

V(x,y,

dr.

the derivative

That

F (#,

of a

along any curve from the point

r to the point r is equal to the difference

of the function

the initial point

point r is the line integral

its final

y, z) at

between the values

the point r and at the point r

is,

Vr.dr =

F(r)

- F(r ) =

V(x,y,z)

- V(xy*d.

definition

fdV= F(r) Theorem

The

F(r

)

V F" = d V =

Ffey,^)

line integral

of the

- V(xyz.). derivative

single valued scalar function of position closed

The is

(2)

V F" of

taken around a

curve vanishes. fact that the integral

denoted by writing a

taken around a closed curve

circle at the foot of the integral sign.

To show (3)

THE INTEGRAL CALCULUJS OF VECTORS The

initial

point r and the final point r coincide.

Conversely

V (x, y, z)

closed curve vanishes,

function

Hence

fvF.dr = 0.

Hence by (2) Theorem

181

the line integral of about every is the derivative of some scalar

of position in space.

Given

W = V V.

To show

be any fixed point in space and r a variable point.

Let

The

line integral

J di

independent of the path of integration C. For let any two paths C and C be drawn between r and r. The curve which is

consists of the path

C from

to r is a closed curve.

to r

W c

J/ c

-c

fw.dr = 0, /

/Wdr = J

C from

Hence by hypothesis

/W*cZr+ j

Hence

and the path

W*dr.

dr.

Hence the value of the integral is independent of the path and depends only upon the final point r.

of integration

VECTOR ANALYSIS

182

The value

of the integral is therefore a scalar function of

the position of the point r whose coordinates are x, y,

Let the integral be taken between two points

infinitely near

together. y,z).

But by

definition

= d V.

Hence

The theorem

therefore demonstrated.

be the force which acts upon a unit mass near 80.] the surface of the earth under the influence of gravity. Let

Let

a system of axes

k be chosen so that k

i, j,

The work done by

the force

moves from the position

f*dT

Hence

when

its

is vertical.

Then

point of application

r to the position r is

Jr (z

# k

z )

= g (z

J r gdz. I

z).

The force f is said to be derivable from a force-function V when there exists a scalar function of position V such that the force

Evidently

equal at each

V is one to V any

point of the derivative VF. may be obtained

force-function, another

arbitrary constant. the force-function is

by adding ample

is if

V=w = g(z

Or more simply The force is

-z).

V= g f = VF=-0k. z.

In the above ex

THE INTEGRAL CALCULUS OF VECTORS The necessary and

V (z,

y, z)

exist, is

183

sufficient condition that a force-function

work done by

that the

the force

when

its

point of application moves around a closed circuit be zero. The work done by the force is

w= If

when taken around every

integral vanishes

this

closed

contour

And

conversely

= VV

the integral vanishes. The force-function and the differ only by a constant.

V= w +

In case there

work done by closed circuit

The

friction

a particle

For the

exist.

moved around

in a

never zero.

by a fixed mass

force of attraction exerted

a unit mass

directed toward the fixed mass and

masses. f

M upon

propor the distance between the

to the inverse square of

tional

const

no force-function can

when

work done

M = -c-r. 6 r

This It is

the law of universal gravitation as stated by Newton. easy to see that this force is derivable from a force-

function V.

Choose the origin of coordinates at the center M. Then the work done is

of the attracting mass

M ?

But

= r d r,

= -c$r

r dr I

d r.

=-cM

M---1) }

r 3

VECTOR ANALYSIS

184

a proper choice of units the constant c may be made The force-function equal to unity. may therefore be

chosen as

had been several attracting bodies

If there

the force-function would have been

where r r r 2 r 8 are the distances of the attracted unit mass from the attracting masses v M% B ,

The law of the conservation of mechanical energy requires that the work done by the forces when a point is moved around a closed curve shall be zero.

on the assump tion that none of the mechanical energy has been converted into other forms of energy during the motion. The law of This

conservation of energy therefore requires the forces to be from a force-function. Conversely if a force-

derivable

function exists the work done by the forces when a point is carried around a closed curve is zero and consequently there is

loss of energy.

function exists

A mechanical

system for which a forcecalled a conservative system. From the

just cited above

it is clear that bodies example moving under the law of universal gravitation form a conservative system at least so long as they do not collide.

Let

be any vector function of position in Let S be any surface. Divide this surface into in space. finitesimal elements. These elements may be regarded as 81.]

(x, y, z)

plane and may be represented by infinitesimal vectors of which the direction is at each point the direction of the normal to the surface at that point and of which the magni

tude

equal to the magnitude of the area of the infinitesimal

THE INTEGRAL CALCULUS OF VECTORS Let

element.

this infinitesimal vector

185

which represents the

element of surface in magnitude and direction be denoted by

d a.

Form

which

the

sum

of the scalar products of the value of

surface.

The

limit of this

face approach zero

the surface $, and

sum when

the elements of sur

called the surface integral of

and the (vector) element of

at each element of surface

W over

written

(4)

The value

pressed in terms of their

and da be ex three components parallel to i, j, k

of the integral

= dy dz

scalar.

i -f

dz dx

+ dx dy k, (5)

The

surface integral therefore has been defined as is cus tomary in ordinary analysis. It is however necessary to

determine with the greatest care which normal to the surface

is.

integral

That is

is,

which

taken over.

side of the surface (so to

are the negatives of each other.

taken over the two sides will

Hence the

sides

surface integrals

differ in sign.

In case the

upon as bounding a portion of space d a considered to be the exterior normal. always If f denote the flux of any substance the surface integral

surface be looked is

speak) the

For the normals upon the two

f.rfa s

VECTOR ANALYSIS

186

gives the amount of that substance which is passing through the surface per unit time. It was seen before (Art. 71) that the rate at which matter was leaving a point per unit

volume per unit time was V f The total amount of mat ter which leaves a closed space bounded by a surface S per .

unit time

the ordinary triple integral

(6)

Hence the very important

relation connecting a surface in

tegral of a flux taken over a closed surface

and the volume

integral of the divergence of the flux taken over the space enclosed by the surface

///

CO Written out in the notation of the ordinary calculus

this

becomes I

\Xdy dz + Ydzdx + Zdxdy~\

3Y, The where X, F, Z are the three components of the flux f familiar when each of the three theorem is perhaps still more .

components

treated separately.

(8)

This

known

as Gauss s Theorem.

It states that the surface

integral (taken over a closed surface) of the product of a

X and

the cosine of the angle which the exterior normal to that surface makes with the X-axis is equal to the volume integral of the partial derivative of that function

function

THE INTEGRAL CALCULUS OF VECTORS

187

with respect to x taken throughout the volume enclosed by that surface. If the surface

be the surface bounding an infinitesimal

sphere or cube

ff f-da = V-f dv where d v

the volume of that sphere or cube.

V.f =

fff-da.

dv J J

Hence

(9)

be taken as a definition of the divergence The divergence of a vector function f is equal to the f. limit approached by the surface integral of f taken over a sur

This equation

may

face

bounding an infinitesimal body divided by that volume the volume approaches zero as its limit. That is

when

V.f= dvQA -f-da. dvJJs ,

From

(10)

which is evidently independent of the the properties of the divergence may be deduced. In order to make use of this definition it is necessary to develop at least the elements of the integral calculus of vectors before axes

this definition

all

the differentiating operators can be treated. This definition of f consequently is interesting more from a theoretical

than from a practical standpoint. Theorem : The surface integral of the curl of a vector 82.] function

equal to the line integral of that vector function taken around the closed curve bounding that surface. is

f fV x JJ8 This

w-dr.

the celebrated theorem of Stokes.

great importance in

number

W-da=

all

(11)

account of

its

branches of mathematical physics a

of different proofs will be given.

VECTOR ANALYSIS

188 First Proof

(Fig. 32).

Consider a small triangle 1 23 upon the surface at the vertex 1 be

Then by (50), Chap.

W = ~{ W

III.,

Let the value of

the value at any neighboring point

+ V (W* 8 r) + (V x W) x

8 r

where the symbol 8 r has been introduced for the sake of dis tinguishing it from d r which is to be used as the element of integration.

The

integral of

W taken

around the triangle

FIG. 32.

Cw-dr=l fwo -dr + + The

first

term

I 2

f (V x W) x A

fv(W-Sr).<Zr Sr-dr.

fw JA

.dr

= iw

Cdr

vanishes because the integral of d r around a closed figure, in this case a small triangle, is zero. The second term

Jfv(W-Sr).dr A

vanishes by virtue of (3) page 180.

Hence

THE INTEGRAL CALCULUS OF VECTORS

Cw*di = JA

189

fvxWxSr-dr.

Interchange the dot and the cross in this triple product.

=| JV xW-Sr When dr

equal to the side

x dr.

of the triangle,

also

Hence the product

equal to this side.

Sr x di vanishes because 8 r and d r are collinear.

when dr

In like manner

is the same side 13, but taken Hence the vector product vanishes. side #5, Sr is a line drawn from the vertex to this side S3. Hence the product 8 r x d r

the side 31, 8r

in the opposite direction.

When dr

1 at which

W= W

the

twice the area of the triangle. Hence positive area 1 % 3.

|r x

This area, moreover,

the

where d a denotes the positive area of the triangular element For the infinitesimal triangle therefore the of surface. relation

holds.

Let the surface 8 be divided into elementary triangles. For convenience let the curve which bounds the surface be made up of the sides of these triangles.

Perform the

integration

fw-dr

around each of these triangles and add the results together.

2/ JA

VECTOR ANALYSIS

190

The second member

]

3 is

the surface integral of the curl of

2 V x W-rfa=JJv x W In adding together the line integrals which occur in the

member

first

the sides of the ele

that all

necessary to notice mentary triangles except those which lie along the bounding curve of the surface are traced twice in opposite directions.

Hence

it is

all

the terms in the

sum

from those sides of the triangles lying within the cancel out, leaving in the sum only the terms which arise from those sides which make up the bounding

which

arise

surface

curve of the surface. tegral of

Hence the sum reduces

to the line in

W along the curve which bounds the surface S. = fw Jo

Hence

= fW

d r.

Second Proof

Let

C be any closed

contour drawn upon the surface It will be assumed that (Fig. 33). is

continuous and does not cut

Let to C.

S C

itself.

be another such contour near

Consider the variation S which

FIG. 33.

takes place in the line integral of in passing from the contour C to the

contour

C".

THE INTEGRAL CALCULUS OF VECTORS

191

/V.dr = t/f

fwdr =

But

d(W-

and

Hence

J*W

CdW *ST.

form a perfect

differential.

&dT= Cw*dST=f d(W*Sr)

The expression d The value of the

8 r)

its

integral of that expression will therefore be

the difference between the values of

at the

integral

end and at

In this case the

the beginning of the path of integration.

taken around the closed contour C.

Hence

Jc Hence

and

But

fsw.dr- f

fw-rfr=

W = 9W d J

-K

W = PW &x -^

+ ^d

and vx

-7T

dJ y

3W k -^

d r,

VECTOR ANALYSIS

192

Substituting these values

i.Br-~ ox

similar terms in y

and

-8r i-dr

z. [

But by (25) page 111

Hence

sfwdr=/

+ or

Srxdr

similar terms in y

= fV

and

x d r.

In Fig. 33 it will be seen that d r is the element of arc along the curve C and 8 r is the distance from the curve C to the curve

r .

Hence

X dr

equal to the area of an ele

mentary parallelogram included between surface S. That is

Let the curve

fw-dr= fv

C and C upon f

the

da.

in expand until starting at a point coincides with the contour bounding S. The line integral

will vary

from the value

at the point

/ t/O

to the value

THE INTEGRAL CALCULUS OF VECTORS

193

taken around the contour which bounds the surface S.

This

sum

of the

total variation of the integral will be

equal to the

variations 8

f Wdr=

ff Vx W-da.

(11)

theorem that the surface integral of the curl of a vector function is equal to the line integral of the func Stokes

83.]

which bounds the surface

tion taken along the closed curve

The converse

has been proved. integral of a

vector function

W taken around

also true.

equal

the surface

to the line integral

of the

and

this relation holds for all surfaces in space, then TT is the curl

function

That

Form V x W.

the curve

bounding

the surface

= f Wdr, thenU=Vx

fll. da

S over which

Let the surface infinitesimal.

The

factor vanishes.

the integration

any element

Hence

V x W = 0. IT = V x W.

IT-

performed be

this equation holds for

The converse

and

integral reduces to merely a single term

(U_V

Hence

f W-dr = 0,

f f (TI- V x W)-da =

first

(12)

the surface integral df the difference between

// (tf~ Vx W)*da=f W*dr -

therefore demonstrated. 13

of surface

the

VECTOR ANALYSIS

194

which is independent of the axes definition of V x k may be obtained by applying Stokes s theorem to an in

A i, j,

Consider a point P. Pass a plane through P and draw in it, concentric with P, a small circle of finitesimal plane area.

area d

Vx W.da=f W*dT. When d a

has the same direction as

line integral will be a

between

W and

maximum,

(13)

W the

value of the

for the cosine of the angle

d a will be equal

For

to unity.

this

value of da,

IM =rfa rfa=:0

Hence

the curl

VxW

F/V

f W-rfrl J

(13)

Lda.dajo

of a vector function

W has

at each

normal to that plane in taken about a small circle con

point of space the direction of the

which the

line integral of

centric with the point in question is a

nitude of the curl at the point

maximum.

The mag

equal to the magnitude of

that line integral of maximum value divided by the area of the circle about which it is taken. This definition like the

one given in Art. 81 for the divergence is interesting more from theoretical than from practical considerations. Stokes

theorem or rather

duce Maxwell

its

converse

may

be used to de

equations of the electro-magnetic field in a

simple manner. Let E be the electric force, B the magnetic the magnetic force, and C the flux of electricity induction, per unit area per unit time (i. e. the current density).

It is a fact learned from experiment that the total electro motive force around a closed circuit is equal to the negative of the rate of change of total magnetic induction through

the circuit.

The

total electromotive force is the line integral

of the electric force taken

around the

Edr.

circuit.

That

THE INTEGRAL CALCULUS OF VECTORS The

195

magnetic induction through the circuit is the sur B taken over a surface

total

face integral of the magnetic induction

bounded by the

circuit.

That

B d*. i

Experiment therefore shows that

B d a.

JJa J/E-dr=/l o

Hence by the converse

x E

of Stokes s theorem

= - B,

curl

= - B.

experiment that the work done in carry unit positive magnetic pole around a closed circuit is ing a equal to 4?r times the total electric flux through the circuit. It is also a fact of

The work done

in carrying a unit pole around a circuit is around the circuit. That is the line integral of

The

through the circuit is the surface integral of C taken over a surface bounded by the total

circuit.

flux

That

electricity

///* Experiment therefore teaches that

= 47r C f

JJs

VECTOR ANALYSIS

196

the converse of Stokes s theorem

V With a proper

H=4

TT C.

interpretation of the current C, as the dis

placement current in addition to the conduction current, an interpretation depending upon one of Maxwell s primary hypotheses, this relation and the preceding one are the funda mental equations of Maxwell Heaviside and Hertz.

The theorems

of Stokes

s theory, in the

form used by

and Gauss may be used

demon

strate the identities.

V V

div curl

Vx VF=0, According to Gauss

curl

VF=0.

theorem

VX Wdv= According to Stokes

this to

the sphere point

;

theorem

fvxW-da = CW

fffv-VxWdtf=

Hence

Apply

Cw*dr. The

an infinitesimal sphere. closed.

Hence

its

and the integral around

dr.

surface bounding

bounding curve reduces

it,

to zero.

V-VxWdv = Jfw-dr = o

V V

to a

THE INTEGRAL CALCULUS OF VECTORS Again according

to Stokes s

197

theorem

ffvxvr.<2a = fvr-dr. to

Apply this bounding

any

infinitesimal portion of surface.

line integral of

VF" vanishes.

the derivative

V As

Hence the

this surface is closed.

The curve

this equation holds for

any d a,

follows that

Vx VF=0. In

similar

manner the converse theorems may be

TT of a vector function divergence TJ is everywhere zero, then TT is the curl of some vector function W.

demonstrated.

If the

If the curl

then

V x II of

between the dels/ viz.

x W*

a vector function

TT is

everywhere zero,

the derivative of some scalar function

By making

84.]

F",

use of the three fundamental relations

line, surface,

and volume

integrals,

and the

JYv x W-rfa=

f W.rfr,

(2)

(11)

(7)

it is

possible to obtain a large

transformation of integrals.

number

of formulae for the

These formulae correspond to

VECTOR ANALYSIS

198

" in ordinary connected with u integration by parts are obtained both sides of the calculus. They by integrating those

formulae, page 161, for differentiating.

First

(u v)

= u V v + v V u.

~Jc

Hence

C V JG

(

[uv]

vV u* dx.

(14)

The expression

[u v]

represents the difference between the value of (u v) at r, the end of the path, and the value at r the beginning of the path. If the path be closed ,

f^Vvdr = -

C V Jo

Jo Second f*

J J

x (u v)

x (wv)*^ a=

=uV

J J

x v +

u*dr.

x (*

^Vxvrfa+/

f* I

J J

(14)

Vwxv-da.

Hence f*

J J

V^xvda=l

uv dr

f* f* I

wVxvda,

J J

(15)

&Vxvda= Third

(wV

But Hence

uv dr

V?txvrfa,

J J

y)^^VxV^ + V^xV

V x Vv = (u V v) = V u

V v,

(15)

199

THE INTEGRAL CALCULUS OF VECTORS f*

J J

Hence f*

IIVO AVt/ f*

J J

S-4

^-J

Fourth

,-y

(u v)

///

=uV

/>V

+ C C C

r r r

V-vd JJJV.(T)^=JJJVT^ JJJ *

Hence

C C C ^7uv dv=

v(V^xv) = VXV^*v

Fifth

V (V M x Hence

rfv^xvrfa =

all

surface

rrr^V

Mvda

V^ V

V^i;,

;

(17)

v^-VXv. x

fffv^vxvdi;.

(18)

these formulae which contain a triple integral the is the closed surface bounding the body throughout

which the integration

Examples

performed.

of integration

given here.

known

the most important of functions of position,

by parts

like those

above can be

Only one more will be as Greens Theorem and is perhaps

multiplied almost without

all.

limit.

u and v

are

any two scalar

VECTOR ANALYSIS

200

JJJ^

fl)

VOV

= V^ V + V V^ = V u V v + v V V u> 24

J J J V- (uvv)dv C f Cu^

vclv==

Hence

^ /

/^VvcU

/V^-Vtfdfl=/

/ /

r/^V-Vvdi?,

I v^*V udv.

(19)

subtracting these equalities the formula

/ / / (^

V V^

V V w) ^ ^ =

(20)

/ (^

V ^)

obtained. By expanding the expression in terms of i, j, k the ordinary form of Green s theorem may be obtained. further generalization due to Thomson (Lord Kelvin) is the is

following

lw^/u*Vvdv=l

= where

w is

vwVU"d*

uwVv*d& I

v\?

[w^ u^

dv,

(21)

a third scalar function of position.

The element of volume dv has nothing

do with the scalar

function v in these equations or in those that go before. The use of v in these two different senses can hardly give rise to

any misunderstanding. * 85.]

tions

In the preceding

articles the scalar

which have been subject

and vector func

to treatment have been sup-

THE INTEGRAL CALCULUS OF VECTORS

201

posed to be continuous, single-valued, possessing derivatives of the first two orders at every point of space under consider

When

ation.

valued, or

discontinuous or multiplefirst two orders

the functions are

possess derivatives of the

fail to

in certain regions of space,

some caution must be exercised

applying the results obtained.

Suppose for instance

VFThe

line integral

Introducing polar coordinates

= r cos 6, y = r sin 0, x

Form

along two

circle lying

path

(

+ 1,0)

to the point

Let one path be a semi and the other, a semicircle

different paths.

above the JT-axis

lying below that

axis.

;

The value

of the integral along the

along the second path,

From

d r

the line integral from the point

(1, 0)

first

/-* d6

TT.

this it appears that the integral does not depend merely the limits of integration, but upon the path chosen, upon

VECTOR ANALYSIS

202

of the value the value along one path being the negative which is a circle the around The integral along the other. 2 TT. to closed curve does not vanish, but is equal were false Art. 79 of results the therefore seem It

might and that consequently the entire bottom of the work which This however is not so. The difficulty is follows fell out. that the function

F=tan

not single-valued. At the point (1,1), for instance, the function V takes on not only the value

-i

tan

TT -r>

but a whole series of values 7T

-+&7T, where k

the origin,

any positive or negative integer. Furthermore at which was included between the two semicircular

becomes wholly inde paths of integration, the function and fails to It will be seen terminate possess a derivative. therefore that the origin is a peculiar or singular point of the function V. If the two paths of integration from (+ 1, 0) to (1,0) had not included the origin the values of the integral

would not have

differed.

In other words the value of the

integral around a closed curve

which does not include

the

origin vanishes as it should.

Inasmuch

appears to be the point which vitiates the results obtained, let it be considered as marked as

the

by an impassable

origin

barrier.

Any

closed curve

which does

not contain the origin may be shrunk up or expanded at will ; but a closed curve which surrounds the origin cannot be so distorted as no longer to enclose that point without break ing its continuity. The curve C not surrounding the origin

THE INTEGRAL CALCULUS OF VECTORS may but

203

shrink up to nothing without a break in its continuity ; can only shrink down and fit closer and closer about

It cannot be

the origin.

shrunk down to nothing.

must

always remain encircling the origin. The curve C is said to be reducible ; (7, irreducible. In case of the function F, then, true that the integral taken around any reducible circuit vanishes; but the integral around any irreducible circuit C does not vanish.

it is

Suppose next that tinuous

first

such point reducible;

any function whatsoever.

circuit

may be shrunk up to nothing and is said to be but a circuit which contains one or more such

stated: The line integral of the derivative

around any

reducible circuit C.

vanish around an irreducible circuit circuit

all

be marked as impassable

which contains within

points cannot be so shrunk up without breaking and it is said to be irreducible. The theorem

V vanishes

Let

to be continuous or to have con

partial derivatives

Then any

barriers.

V is

V fails

the points at which

C may be

irreducible circuit

its

continuity

may then be VF" of any function It may or may not

In case one irreducible

distorted so as to coincide with another

without passing through any of the and without breaking its continuity,

singular points of the two circuits are said to be reconcilable and the values of

the line integral of about them are the same. region such that any closed curve C within

may

shrunk up to nothing without passing through any singular point of V and without breaking its continuity, that is, a region every closed curve in which is reducible*, is said to be All other regions are cyclic. acyclic. By means of a simple device any cyclic region may be ren

dered acyclic.

Consider, for instance, the region (Fig. 34) en closed between the surface of a cylinder and the surface of a

cube which contains the cylinder and whose bases coincide with those of the cylinder. Such a region is realized in a room

VECTOR ANALYSIS

204 in

which a column reaches from the

floor to the ceiling.

circuit which passes evident that this region is cyclic. is It the column irreducible. cannot be contracted to around is

~^x

nothing without breaking its continuity. If now a diaphragm be inserted reaching from

the surface of the cylinder or column to the surface of the cube the region thus formed

bounded by the surface

of the cylinder, the

surface of the cube, and the two sides of the diaphragm is acyclic. Owing to the inser tion of the diaphragm to

draw a

circuit

which

it is

no longer possible around the cyl

shall pass completely

the diaphragm prevents it. Hence every closed cir which may be drawn in the region is reducible and the

inder cuit

region

is acyclic.

In like manner any region inserting a sufficient

surfaces of the

number

new region

may

be rendered acyclic by

of diaphragms.

consist of the

The bounding

bounding surfaces of

the given cyclic region and the two faces of each diaphragm. In acyclic regions or regions rendered acyclic by the fore going device all the results contained in Arts. 79 et seq.

hold true. true.

For cyclic regions they may or may not hold enter further into these questions at this point is

unnecessary.

Indeed, even as

much

discussion as has been

given them already may be superfluous. For they are ques tions which do not concern vector methods any more than the corresponding Cartesian ones.

They belong properly

to the

subject of integration itself, rather than to the particular notation which may be employed in connection with it and

which

the

primary object of exposition here.

In this

respect these questions are similar to questions of rigor.

THE INTEGRAL CALCULUS OF VECTORS The Potential

The Integrating Operators. 86.]

205

Hitherto there have been considered

line,

surface,

and volume integrals of functions both scalar and vector. There exist, however, certain special volume integrals which,

owing

with the differentiating

to their intimate connection

operators V,

Vx, and owing

to their especially frequent

occurrence and great importance in physics, merit especial consideration. Suppose that

^0** Vv is

*a)

a scalar function of the position in space of the point

For the sake of definiteness

may

density of matter at the point (# 2 , yv 22 ).

body

V is

be regarded as the In a homogeneous

In those portions of space in which no identically zero. In non-homogeneous dis

constant.

matter exists

V is

tributions of matter

varies

from point to point; but at

each point it has a definite value. The vector z 2 i + y 2 j + * 2 k, r2

drawn from any assumed the point (# 2 , y 2 , z 2 ).

origin,

may

be used to designate

Let

yi.

*i)

be any other fixed point of space, represented by the vector

drawn from the same r2 is

-r = x

the vector

(#2> IJy 2 2)-

O 2

origin.

!>!

Then (y2

- yi ) j +

(z 2

- *j)

drawn from the point (x v y v Zj) to the point A S s vec ^or occurs a large number of times

in the sections immediately following, r i2

~~ r 2 i-

will be denoted

VECTOR ANALYSIS

206

The length

of r 12

then r 12 and will be assumed to be

positive. -i2

= V r 12

r 12

= V

(* 2

- x^ +

(y2

- ^) + 2

- ^) 2

Consider the triple integral

The

integration that ^2> ^2> ^ 2

is is,

performed with respect to the variables with respect to the body of which V represents the density (Fig. 35). During the integration the point (x v y v z^ re

mains

fixed.

The

integral

/ has

a definite

value at each definite point (x v y v zj. The inIt is a function of that point.

FIG. 3o.

terpretation of this integral

easy, if

the function

be regarded as the density of matter in space.

The element

mass

V (# 2

at (# 2 , y 2 , z 2 ) is ,

dx^ dyz dz%

Vdv.

The integral / is therefore the sum of the elements of mass in a body, each divided by its distance from a fixed point

r dm

J what

termed the potential at the point (x v y v due to the body whose density is This

The in

limits of integration in the integral

either

two ways.

In the

first

/ may be

looked at

place they

may

body of which the most natural

regarded as coincident with the limits of the

V is

the density.

set of limits.

This indeed might seem the other

hand the

integral

/ may

THE INTEGRAL CALCULUS OF VECTORS

207

regarded as taken over all space. The value of the integral the same in both cases. For when the limits are infinite

the function

V vanishes

identically at every point (# 2 ,

situated outside of the body

2 2)

and hence does not augment

It is found most convenient the value of the integral at all. to consider the limits as infinite and the integral as extended

This saves the trouble of writing in special The function Vot itself then limits for each particular case. over

all space.

practically determines the limits tically at all points unoccupied

owing to its vanishing iden by matter.

The operation

of finding the potential is of such that a special symbol, Pot, is used for it. occurrence frequent 87.]

Pot

The symbol Pot V,

r=fff

* 2>

rf* 2

dy^ dz y

read "the potential of V." a function not of the variables is

The #.

(22)

potential,

with

which the integration is performed but of the point (x v y^ Zj) which is fixed during the integration. These

regard to

enter in the expression for r 12

variables

and Pot

V therefore

have

The function

different sets of variables.

has hitherto may be necessary to note that although in of matter been regarded as the density space, such an It

interpretation for

Whenever

V is

entirely too restricted for convenience.

becomes necessary to form the integral

i of

any scalar function V, no matter what

integral

< 22 >

represents, that

The reason for calling even in cases in which it has

called the potential of V.

such an integral the potential np connection with physical potential according to the

same formal law

that

formed

as the true potential

and

VECTOR ANALYSIS

208

by virtue of that formation has certain simple rules tion which other types of integrals do not possess. Pursuant

to this idea the potential of a vector function

WO may

of opera

z 2)

be written down.

Pot

W (*

* 2)

dx,

In this case the integral

the

sum

rfy. rf, r

of vector

(23)

quantities

consequently itself a vector. Thus the potential of a vector function is a vector function, just as the potential

and

of a scalar function tion in space.

2/2 ,

V was

seen to be a scalar function of posi

W be resolved into three components = X O yv ) + T <> yv ) ) its

Pot

X+

Pot

+ kZ <> 2

z 2)

Y+ k Pot Z.

(24)

The potential of a vector function W is equal to the vector sum of the potentials of its three components X, Y, Z. The potential of a scalar function V exists at a point (x v y v z p ) when and only when the integral

taken

over

all

space converges to a definite value.

If,

were everywhere constant in space the in tegral would become greater and greater without limit as the limits of integration were extended farther and farther for instance,

out into space.

Evidently therefore

Jhe potential

to exist

must approach zero as its limit as the point (#2 , yv 32 ) few important sufficient conditions recedes indefinitely. for the convergence of the potential may be obtained by

transforming to polar coordinates.

Let

THE INTEGRAL CALCULUS OF VECTORS x

= r sin

6 cos fa

sin 6 sin fa

z = r cos

209

= r sm0 2

dr dO

Let the point (xv y v ^) which be chosen at the origin. Then

d<f>.

fixed for the integration

r i2

and the integral becomes

PotF= CCCVrsmff dr d0

or simply

the

function

dfa

V decrease so rapidly that the product Vr*

remains finite as r increases indefinitely, then the integral con verges as

For

far as

the

distant

regions of space are concerned.

let

= 00

dr d0d<f>

dr d0

d<f>

= QO

Hence the

triple integral

taken over

space outside of a supposed to be a large quan all

sphere of radius R (where R is jR, and consequently converges as far tity) is less than %TT* as regions distant from the origin are concerned.

VECTOR ANALYSIS

210

the function

V remain

finite or if it become infinite so

weakly that the product

Vr remains finite when r approaches zero, then the integral converges as far as regions near to the origin are concerned. For let

Vr<K f CCrrsmddr d0 r

triple integral

R (where than 2 Tr2

of radius

regions near to concerned.

R is now

the point

d<f>.

space inside a sphere supposed to be a small quantity) all

and consequently converges as the origin which is the point (x v y v

If at any point (x 2 y2 i.

d<t>

taken over

is less

< C C fadr d0

CCC Hence the

d<f>

(x x , y v

far as

z 2 ) not coincident with the origin,

z x ), the function

weakly that the product of the value

0/V

becomes infinite so at

a point near

( X 2> J2> Z 2) ty the square of the distance of that point

y2 z 2 ) ,

remains

The

from

finite as that distance approaches zero, then

the integral converges as far as regions

are concerned.

are

Zj)

near

to the

proof of this statement

point (x 2

z2)

like those given

These three conditions for the convergence of the integral Pot V are sufficient. They are by no means neces The sary. integral may converge when they do not hold.

before.

It is

however indispensable

know whether or

not an integral

under discussion converges. Unless the tests given above show the convergence, more stringent ones must be resorted to.

Such, however, will not be discussed here.

They belong

to the theory of integration in general rather than to the

THE INTEGRAL CALCULUS OF VECTORS

211

theory of the integrating operator Pot. The discussion of re the convergence of the potential of a vector function three which are scalar duces at once to that of its components

functions and

may

be treated as above.

a function of the variables x v y v z l which are constant with respect to the integration. Let the

The

88.]

potential

value of the potential at the point (x v y v z^ be denoted by

The is

first partial

derivative of the potential with respect to x l

therefore

LIM

The value

of this limit

may

be determined by a simple

Consider device (Fig. 36). the potential at the point

due to a certain body T. This is the same as the potential at the point

FlG 36 -

due to the same body T displaced in the negative direction by the amount A x r For in finding the potential at a point P

due to a body T the absolute positions in space of the body T and the point P are immaterial. It is only their positions

which determines the value of the poten both body and point be translated by the same amount in the same direction the value of the potential is un

relative to each other tial.

But now if T be displaced in the negative direction amount A#, the value of Fat each point of space is

changed.

by the

changed from

where

v C*2> y* A# 2 = A x r

**) to

v 0*2

VECTOR ANALYSIS

212

Hence [Pot

V(x v yv

z^ +

[ Pot

LlM

Hence

A #! =

It will be

AX,, yt , *,

= [Pot F<> 2 + A

HX. + A

..,,, .t

2 ,y2 ,

- [Pot

j /

found convenient to introduce the

limits

Let the portion of space originally filled by the denoted be ; and let the portion filled by the by body in the negative direction through translation its after body

integration.

the distance

x l be denoted by

The

regions

Let the region common to both be m remainder of be m; the remainder of overlap.

Pot

V (a, + A *

y r * 2) f

M and M

and

;

let the

Then

1 .

rrr

JJJ m

Pot

///

J J JM /

"\F ( r I O/ft

(II

t/n

I ^j 2

/*/*/* ill

* ^O/, *y *\

1^ f V O1 tl 9 ^ *V* V91 " J /

rft>,+ / / / J J Jm

Hence (25) becomes, when

A ^j

T YL

replaced *by

t As all the following potentials are for the point indices have been dropped.

^_Mrf r2

yi,

its

equal Aic 2

the bracket and

THE INTEGRAL CALCULUS OF VECTORS

213

C r J Jf J

my(* A

Or,

==0

LIM

v yy

r12

= rrr

jJJ

LIM

(

r 12

9 X

when A

^! approaches zero as its limit the regions mand which are at no point thicker than A #, approach zero ;

and Jf both approach t

-Jf

There are cases in which

; ,

as a limit.

this reversal of the order in

which the two

limits

are taken gives incorrect results. This is a question of double limits and leads to the mazes of modern mathematical rigor.

Fis to exist at the surface bounding the values of the diminish continuously to zero upon the surface. If Fchanged suddenly from a finite value within the surface to a zero value outside the de rivative QVlS^i would not exist and the triple integral would be meaningless. J

If the derivative of

function

V must

For the same reason within the region T.

V is

supposed to be

finite

and continuous at every point

VECTOR ANALYSIS

214

Then

be assumed that the region vanishes upon the surface bounding T if it

T/nvr i <t\ jvi

rrr

A^ojjj

V(Y V ^o* \

<>/

T is

z \ ^9y

riaAa;2

finite

and that

Consequently the expression for the derivative of the poten tial

reduces to merely

3 Pot

F= r r r

J1/1 J JM

d xl

i r 12

3# 2

dv*

Pot

3^2

^%^ partial derivative of the potential of a scalar function equal to the potential of the partial derivative of V.

V of the potential ofVis equal to the potential of the derivative V V. VPotF=PotVF (27) The derivative

This statement follows immediately from the former. the

upon the left-hand

ables x v

Zj,

side applies to the

be written

may

set

of vari

In like manner the

V upon

the right-hand side may be written atten 2 to call tion to the fact that it applies to the variables #2 y 2 , z2 of F. ,

Then

Pot

F= Pot V F

demonstrate this identity

.3 PotF

V may be expanded

.3 PotF

^J I

(27)

. I

T-

+jPot 3-l-

lr *

PotF ^

in terms of

215

THE INTEGRAL CALCULUS OF VECTORS As

i, j,

are constant vectors they

may

be placed under

be integration and the terms may

the sign of by means of (26)

The curl function

and

W are equal

divergence

collected.

Then

of the potential of a vector

respectively to the

potential of the curl

and

divergence of that function.

W = Pot V x W, curl Pot W = Pot curl W Vj Pot W = Pot V W, div Pot W = Pot div W.

V, x Pot or

and

(28)

be proved in a manner analogous to the even possible to go further and form the dels

These relations above.

It is

may

of higher order

v v Pot r= Pot v

VF;

(30)

Uf>iac<3*

V- V Pot W = Pot V V W, V V Pot W = Pot VV W, V x V x Pot W= Pot V x V x W.

(31)

(32) (33)

The dels upon the left might have a subscript 1 attached to show that the differentiations are performed with respect to the variables x v y v z v and for a similar reason the dels upon the right might have been written with a subscript 2. results of this article may be summed up as follows:

Theorem: The

differentiating operator

The

V and the integrating

operator Pot are commutative. In the foregoing work it has been assumed that the *89.] T was finite and that the function Fwas everywhere region finite and continuous inside of the region T and moreover

decreased so as to approach zero continuously at the surface bounding that region. These restrictions are inconvenient

VECTOR ANALYSIS

216

and may be removed by making use of a surface integral. The derivative of the potential was obtained (page 213) in essentially the form

otr_ r r r J J J ,f

r 12 2x 2 J.

-LJ-LJjl

\^o

a V2

LIM

rrr

T r(g a>

y ff

region

The element

therefore equal to dt? 2

L -

Hence

volume dv z

=A#

the r

da.

r !2

volume d v 2 ^ n *^ e di, 2

Hence

$ bounding

in the region

,^2^-^2^2112)

f JfJfJm-

= TfV JJ The element

fgr

Let d a be a directed element of the surface J!f.

g a)

re gi

equal to

= -Aa; 2 i.da.

/Tf J\X^J J Jm

Consequently

i-da. ^ r !2

(34)

THE INTEGRAL CALCULUS OF VECTORS

217

M with

The volume

integral is taken throughout the region the understanding that the value of the derivative of

the surface

derivative of

V at

shall be equal to the limit of the value of that

when

the surface

approached from the interior

This convention avoids the difficulty that arises in

connection with the existence of the derivative at the surface

S where V becomes

discontinuous.

taken over the surface

The

S which bounds

M becomes

Suppose that the region the conditions imposed the potential

upon

V to

surface integral

the region. infinite.

virtue of

insure the convergence of

Vr* < K.

S be

Let the bounding surface tity which is large. i

a sphere of radius 2

a quan

6 d<f>.

<//*-. s

The surface integral becomes smaller and smaller and ap becomes infinite. proaches zero as its limit when the region Moreover the volume integral

JLJT^ remains

V is

finite as

M becomes

infinite.

such a function that Pot

Consequently provided

V exists as

far as the infinite

regions of space are concerned, then the equation

= holds as far as those regions of space are concerned. ceases to be continuous or becomes infinite Suppose that

at a single point (x^ y v z^) within the region T.

Surround

VECTOR ANALYSIS

218 this point

surface of this sphere and

all

the region

denote the

T not

included

Then

within the sphere.

r r r

=JJj^^ By

Let

with a small sphere of radius R.

the conditions imposed upon

dv * +

r r

JJ*-^

Vr<K V

//>"

<//.*

Consequently when the sphere of radius

d^ R

becomes smaller

and smaller the surface integral may or may not become Moreover the volume integral 1

may

when

a limit

may not approach

and smaller.

zero.

becomes smaller

Hence the equation

SPotF

has not always a definite meaning at a point of the region T at which becomes infinite in such a manner that the

product

remains

finite.

remains

the point in question so that the product Vr approaches zero, the constant is zero and the surface integral becomes smaller and smaller as If,

however,

finite at

approaches zero.

Moreover the volume integral

THE INTEGRAL CALCULUS OF VECTORS approaches a definite limit as

becomes

infinitesimal.

219

Con

sequently the equation

5 Pot 7\

V _ Ap Ob dV

holds in the neighborhood of all isolated pointe at which remains finite even though it be discontinuous.

Suppose that V becomes infinite at some single point 22 ) not coincident with (x^ y v z^). (iC2 y 2 According to the ,

conditions laid upon

V VI*

where

I is

near to

it.

the distance of the point (z2 y2 z2 ) from a point Then the surface integral ,

r !2

need not become zero and consequently the equation

5PotF

= Pot SV TT

need not hold for any point if V becomes infinite at # 2 y2 ,

r yv z^) of the region. z2 in such a manner that

(a? ,

But

<K,

then the surface integral will approach zero as

its

limit

and

the equation will hold.

remains finite upon the Finally suppose the function surface S bounding the region jT, but does not vanish there.

In this case there exists a surface of discontinuities of V.

Within

this surface

V is

finite

;

surface integral

without,

it is

zero.

The

VECTOR ANALYSTS

220

does not vanish in general.

Hence the equation

SPotF =--- = dX

Pot

9V -^r

v%i

cannot hold. Similar reasoning may be applied to each of the three x By combining partial derivatives with respect to v y v z r the results

it is

seen that in general

PotF= Pot V2 F+ f f

da.

(35)

F be any function in F exists. Surround

space, and let it be granted that each point of space at which V Let the surface of the ceases to be finite by a small sphere. Draw in all those surfaces denoted be S. by space sphere

Let

Pot

which are surfaces of discontinuity of

Let these sur

by S. Then the formula (35) holds where the surface integral is taken over all the surfaces which have been designated by S. If the integral taken faces also be denoted

over

all

these surfaces vanishes

when

the radii of the spheres

above mentioned become infinitesimal, then

(27)

This formula

V PotF=PotV 1

a point (x x yv Zj) if remains always or becomes infinite at a point (x 2 y2 z 2 ) so that the

will surely hold at finite

product finite, if V possesses no surfaces of and the discontinuity, if furthermore product V r remains finite

and

remains

as r becomes infinite.

In other cases special

tests

must be

whether the formula (27) can be used or the more complicated one (35) must be resorted to. applied to ascertain

For extensions and modifications of

this theorem, see exercises.

THE INTEGRAL CALCULUS OF VECTORS The

relation (27) is so simple

formation that

and so amenable

will in general be

assumed

In cases in which

function that (27) holds.

S of

to trans

to be

221

such a

possesses a

discontinuity frequently found convenient to consider V as replaced by another function V which has in general the same values as Fbut which instead of possess surface

it is

ing a discontinuity at S merely changes very rapidly from one value to another as the point (#2 , y2 2 2 ) passes from one Such a device renders the potential side of S to the other. ,

actual

simpler to treat analytically and probably conforms to physical states more closely than the more exact

conception of a surface of discontinuity. This device prac tically amounts to including the surface integral in the

symbol Pot VF: In fact from the standpoint of pure mathematics it is better to state that where there exist surfaces at which the function

V becomes

discontinuous, the full value of Pot

should always be understood as including the surface integral

//. in addition to the

volume integral

>VF U SSSr*J

10 12

and other Pot V X W, New V met in the future must be regarded as consisting not only of a volume integral but of a surface possesses integral in addition, whenever the vector function In like manner Pot

similar expressions to be

a surface of discontinuities. It is precisely this convention in the interpretation of formulae which permits such simple formulae as (27) to hold in general, and which gives to the treatment of the integrat ing operators an elegance of treatment otherwise unobtainable.

VECTOR ANALYSIS

222

The

irregularities

which may

thrown into the inter

arise are

pretation, not into the analytic appearance of the formulse. This is the essence of Professor Gibbs s method of treatment.

The

90.]

first partial

derivatives of the potential

also

may

be obtained by differentiating under the sign of integration. 1

= CCC Jj J

*>-

S Pot

3/21*2)

manner

/r/-

.. /..

( 3 7)

for a vector function

/* /* /*

""I

rrr (*,-*!> r^y,,*,) "^^^ V[(*a-* )H(y ^ )2+(v-^)T

~ ..

~ \2^ia

/^.

^!W=

and

But 1

*"-,-

" " d, r

(38)

an attempt were made to obtain the second partial derivatives in the same it would be seen that the volume integrals no longer converged.

manner,

THE INTEGRAL CALCULUS OF VECTORS

V Pot F =

Hence

/// ^f- d

223

(39)

In like manner

^,,

and

Pot

W = fff

(40)

These three integrals obtained from the potential by the differentiating operators are of great importance in mathe matical physics.

Each has

its

own

Conse

interpretation.

quently although obtained so simply from the potential each Moreover inasmuch as these given a separate name.

integrals

may

even when the potential

exist

divergent,

they must be considered independent of it. They are to be looked upon as three new integrating operators defined each upon its own merits as the potential was defined. Let, therefore,

(42) 12

.3 r

the potential exists,

= Max W.

then

V Pot F= New F VxPotW = LapW V-PotW = MaxW. The

first is

written

(44)

New

V and read

The Newtonian

(45)

VECTOR ANALYSIS

224

The reason

for calling this integral the

V represent the

Newtonian

that

density of a

body the integral gives the force, of attraction at the point (x^ y v Zj) due to the body. This The second is written Lap and will be proved later.

read "the Laplacian of W." This integral was used to a considerable extent by Laplace. It is of frequent occurrence

in electricity and magnetism. If represent the current C in space the Laplacian of C gives the magnetic force at the point (x v y v zj due to the current. The third is written Max and read " the Maxwellian of W." This integral was used by Maxwell. It, too, occurs frequently in electricity

and magnetism. of magnetization

For instance I,

represent the intensity the Maxwellian of I gives the magnetic

potential at the point (x^ y v z^)

due

to the magnetization.

To show

that the Newtonian gives the force of attraction to the law of the inverse square of the distance. according Let dm<i be any element of mass situated at the point f rce at

in magnitude

(

x v Vv

z i)

due to

equal to

and has the direction of the vector r12 from the

point (xv y v zj to the point (#2 y2 , z 2 ). ,

Hence the

force is

Integrating over the entire body, or over all space according to the convention here adopted, the total force is

where

V denotes the

density of matter.

THE INTEGRAL CALCULUS OF VECTORS The

integral

may

be expanded in terms of

225

The

three components

may

be expressed in terms of the po

tential (if it exists) as

(42)

It is in this

form that the Newtonian

generally found in

books.

To show that the Laplacian gives the magnetic force per unit positive pole at the point (x v yv z^) due to a distribution

W (#

f electric flux. z The magnetic force at (x v yv 2 y<p 2 ) due to an element of current d C 2 is equal in magnitude ,

)

the magnitude d C% of that element of current divided by the square of the distance r 12 ; that is

dC* T*2

The The

perpendicular both to the vector

dC 2 and

to the line r 12 joining the points.

direction of the force

element of current

direction of the force

vector product of r 12 and

dC 2

therefore the direction of the

The

r3 T 12 15

force

therefore

VECTOR ANALYSIS

226

Integrating over all space, the total magnetic force acting at the point (x^ y v z^) upon a unit positive pole is

c r r rJi2 x d CJ2

///

This integral

may

W (x v y v *

r r r*

-J J J

be expanded in terms of

= X(x v i

The

-^i)

i+(y-yi)J+

k components of Lap

i, j,

k^O

Let

z^ + j Y (x^ y v 4-

r i2=(^ 2

".-

z^) z%).

(a-*i)k-

W are respectively

C.-^^ (43)

In terms of the potential

Lap

(if

one exists) this

W = 3 gPot Z

S Pot

may

r=lP^_a|2t^

To show

be written

(43)

I be the intensity of magnetization at the that is, if I be a vector whose magnitude is point (x v %>*2)> equal to the magnetic moment per unit volume and whose

that

THE INTEGRAL CALCULUS OF VECTORS direction

227

the direction of magnetization of the element d v%

from south pole to north pole, then the Maxwellian of I is the magnetic potential due to the distribution of magnetization.

The magnetic moment of the element of volume d 2 is I d v%. The potential at (x v y v 24) due to this element is equal to its t>

magnetic moment divided by the square of the distance r 12 and multiplied by the cosine of the angle between the direc The potential is tion of magnetization I and the vector r 12 .

therefore r 12

dv%

Integrating, the total magnetic potential

seen to be

This integral

may

also be written out in terms of x, y,

Let-

*ia

xv yv

A+

(y a

~ Vi) B +

(*2

- *i) &

the variables x^ y, z; and instead of e variables %, ?;, f be used 1 the expression takes

If instead of

O -x

x v y& z z oq the form given by Maxwell.

According to the notation employed for the Laplacian

Max

w -fff (*.-i (44) 1

Maxwell

Electricity and Magnetism, Vol. II. p.

VECTOR ANALYSIS

228

The Maxwellian It

may

of a vector function

a scalar quantity.

be written in terms of the potential

Max

W = -=dx-- + SPotF + l

The Newtonian,

3y l

9z l

much used

This form of expression is upon mathematical physics.

(if it exists) as

(44)"

in ordinary treatises

Laplacian, and Maxwellian, however, should

not be associated indissolubly with the particular physical interpretations given to them above. They should be looked

which may be applied, as the of functions position in space. The New potential is, any tonian is applied to a scalar function and yields a vector as integrating operators

upon

The Laplacian

function.

and is

applied to a vector function

The Maxwellian and yields a scalar function.

a function of the same sort.

yields

applied to a vector function

Moreover, these integrals should not be looked upon as the are

its

the potential.

derivatives

derivatives.

the potential exists they exist when the

But they frequently

potential fails to converge.

Let

91.]

exist

and

and have

(29)

be such functions that their potentials

in general definite values.

V V PotF= V

and

(45)

and by

(45)

and

VF.

V PotF= V. NewF= Max VF = PotV.VF

(46)

W = V Pot V. W= Pot V V Pot W = Max W, V Pot V W = New V. W.

By (27) and (29) But by

VF = Pot V

(27)

V Pot V = New F, V.Pot VF=Max VF.

But by (45)

Hence

Pot

Then by

VV V

Pot

THE INTEGRAL CALCULUS OF VECTORS

Hence

Pot

229

W = V Max W = New V W = Pot VV.W

(28)

But by (45) and

Hence

And by or

Hence

And by

Hence or

x Pot

W=V x

Pot

V V

= Pot V x x W. V x Pot W = Lap W, V x Pot V x W = Lap V x W. V x Pot W = V x Lap W = Lap V x W = Pot V x V x W. (48)

V V x Pot W = 0, V Pot V x W = 0. V Lap W = Max V x W = 0. (49) III. V V PotF= x 0, (52), Chap. VxPotVF=0. V x New V = Lap V V = 0. (50) V x V x W = VV W - V V W, (58), Chap. III. V.VW = VV-W VxVxW.

(56),

(47)

Chap.

III.

V V Pot W New V W V V Pot W = V Max W

V X W, V X Lap W.

Lap

(51)

These formulae may be written out in terms of curl and div

desired.

Thus

New V = Max

V F,

(46)

V Max W = New div W

(47)

div

W = Lap curl W div Lap W = Max curl W = curl

curl

V V

Pot

Lap

New V = Lap

W = New div W

(48)

(49)

VF= Lap

(50) curl

(51)

VECTOR ANALYSIS

230

Poisson s Equation

Let

92.]

any function in space such

"be

that the potential

PotF has in general a

definite value.

Then

V V PotF= - 4 TrF, c>

PotF

This equation

The

PotF

known

Pot r= Vj . New

The

PotF

as Poisson s Equation.

integral which has been defined

solution of Poisson s Equation.

V V

subscripts 1

clearly

(52)

as the potential is a

The proof

as follows.

* F= Max V r=T f C 2

and ^ have been attached

what are variables with respect

V ^*

which the

dv v

designate differen

tiations are performed.

V .V PotF=V .NewF=ff TVJ--. V2 Fdv a 1

But

and

V (v V 2

r vt

= - V2

= V2

V F+ V Va Va 2

THE INTEGRAL CALCULUS OF VECTORS Hence

- V2

.v.r=rv. v2

V V 2

Hence

r !2

- V2

+ v..(V !2

\ r !2/

But

That

V V = V V2 V

y 13

Integrate

231

to say

satisfies

= 0.

Laplace

Pot V =

Equation.

f f fV

=rr J J

-x

V2 Vd v 2

(53)

.rfa.

And by (8)

7*12

The

surface integral is taken over the surface which bounds the region of integration of the volume integral. This is taken " over all space." Hence the surface integral must be taken over a sphere of radius R, a large quantity, and must

At the point (x r y^z^)^ of the surface the however, integrand integral becomes in finite owing to the presence of the term

be allowed to increase without limit.

VECTOR ANALYSIS

232

Hence the surface S must include not only the surface

of the

sphere of radius J2, but also the surface of a sphere of radius the point (x^y^z^) and B , a small quantity, surrounding must be allowed to approach zero as its limit.

has been assumed that the potential of exists, it is assumed that the conditions given (Art. 87) for the existence it

That

of the potential hold.

< Vr <

when

K, when r

and

large

small.

= - Va

x l

= -^

for the large sphere of radius

da =

r r2

*3 4*3

Then

polar coordinates with the origin at the point Then r12 becomes simply r

Introduce (#i> #i

-fiT,

sm0 d0

dd>.

Hence the surface For

integral over that sphere approaches zero

as its limit.

Hence when

-R

becomes

infinite the surface integral

large sphere approaches zero as

its

limit.

For the small sphere

1 t

da r !2 <r

Hence the

= --r5 r 2 sin

7*v

integral over that sphere becomes

d 6.

over the

THE INTEGRAL CALCULUS OF VECTORS Let

233

V be

supposed to be finite and continuous at the point Then for the ( xvl/v z i) which has been selected as origin. surface integral

and equal

practically constant

to its

value

V (*ii Vv *i) at the point in question.

- f /Vsi sintf d8

Hence

when

fssintf d

the radius

zero as

its limit.

of the sphere of integration approaches

Hence

ff, rv

=- 4 * F

V- VPotF=-47rF.

and In like manner

< 68)1

(52)

a vector function which has in

general a definite potential, then that potential satisfies Poisson s Equation.

V V Pot W = - 4

The proof

of this consists in resolving

ponents.

For each component

(52)

W into

its

the equation holds.

three

Let

vV. VPotF=-47r F, V V Pot Z = 4 TT Z. Consequently

V V Pot (JTi +

Fj + Zk)

= - 4 TT

(JTi

+ Fj +

com

VECTOR ANALYSIS

234

Theorem

If V and

W are such functions of position in space

that their potentials exist in general, then for all points at which

and

Poisson

and continuous

are finite

those potentials

Equation,

V- VPot r=-4irF; V V Pot W = - 4 TT W. The

satisfy

(52) (52)

modifications in this theorem which are to be

points at

V and W become

which

made

taken up here. It

93.]

Hence

was seen

(46) Art. 91 that

V VPotF = V- NewT=Max V New V = - 4 V

VF1

(53)

Max VF=-47rF.

In a similar manner

Hence or

was seen (51) Art. 91 that

V V Pot W = V Max W V x Lap W = New V W Lap V x W. V Max W - V x Lap W = - 4 W, New V W - Lap V x W = - 4-rr W. TT

virtue of this equality

Lap V x

-7

4-7T

where

and

W = t

W =

-r

4-rr

Lap

4-7T -

New V- W.

4?T

(54)

(55)

W = -4?r Lap curl W

NewV- W =

(54)

divided into two parts.

W = W! + W

Let

discontinuous will not be

Newdiv W.

4-7T -

(56)

(57)

THE INTEGRAL CALCULUS OF VECTORS

235

multiplied Equation (55) states that any vector function 4 TT is equal to the difference of the Laplacian of its curl by

and the Newtonian

of its divergence.

Lap

4?r

Furthermore

V xW= V- V x Lap W r 4-7T -7

But the divergence

of the curl of a vector function

Hence

V.W = divW =

W = - j Vx New V W = 2

VxV Max W

curl of the derivative of a scalar function

Hence

zero.

(58)

But the

is zero.

= curl W = 0.

(59)

which has a potential Consequently any vector function may be divided into two parts of which one has no divergence

and

which the other has no

curl.

This division of

W into

two such parts

is unique. In case a vector function has no potential but both

and divergence possess

potentials, the vector function

divided into three parts of which the

first

its

curl

may be

has no divergence

;

the second, no curl; the third, neither divergence nor curl.

W=4

Let

Lap V x

W-4

New V

W + W.

(55)

As before

4?r

and

V 1

Lap V x

V x New V W

4-7T

The divergence part of

V V

4-7T

VxV Pot V W =

of the first part

W are therefore zero.

x Pot

and the curl of the second

VECTOR ANALYSIS

236

4?r

x Lap

VxW =

Pot

= 4

-4?r

~ V V Pot V X W. W - 4?T

W = -- V P o V V x W =

P ot V x

Hence

^ 4-7T

Hence 4?r curl of

V VPot V x W = V x W.

V x Lap VxW = VxW = VxW

equal to the curl of the

Lap

-r

47T into

V V x W = 0.

for

The

VxVxPotVxW 4-7T

which

T: 7T

Thus

the

can have none.

New

divergence of

as the second part has

Moreover

W= V W

part

VxW

Hence

divided.

curl, the third part

first

equal to the divergence of

the second part -

4?r

New

is divided. Hence as the first part has no the third have can none. Consequently the third divergence This proves the part 3 has neither curl nor divergence. statement.

into

which

By means

be seen that any function 3 which possesses neither curl nor divergence, must either of Art. 96

may

THE INTEGRAL CALCULUS OF VECTORS

237

vanish throughout all space or must not become zero at In physics functions generally vanish at infinity. infinity. Hence functions which represent actual phenomena may be divided into two parts, of which one has no divergence and the other no curl. 94.]

Definition

A vector function the

divergence of which said to be solenoidal.

vanishes at every point of space is vector function the curl of which vanishes at every point of space is said to be irrotational.

In general a vector function is neither solenoidal nor irrota But it has been shown that any vector function which possesses a potential may be divided in one and only one tional.

way

Wv W

two parts

into

They have With

solenoidal

and

stated.

been proved in the foregoing sections. v the operators respect to a solenoidal function

4?r

Lap V x Wi

Applied

gives zero.

That

respect to

or curl

-j

4-rr

Lap Wi

= Wr either

W = an irrotational function W

New

are inverse operators.

_ _L New V

(60) of

these opera

and

4?r

irrotational function

Lap

Lap and

That

are inverse operators.

With

which one

all

4?r

tors

The following theorems may be

the other irrotational.

That

(61) the operators

div

=-V

-i-

New

W =W 2

(62)

VECTOR ANALYSIS

238

With

a scalar function

respect to

div and

the operators

New,

4-7T

and

also

Max and

4?r

That

are inverse operators.

-V.-i NewF= V

(63)

4 7T

~--Max VF=

and

4?r

TFttA respect to a solenoidal function -

Pot and

the operators

or curl curl

47T

That

are inverse operators.

Pot

4?r

With

respect to

irrotational function

Pot and That

are inverse operators.

With

respect

VV W .

Pot Wi 4?r

4?r

_ _L Pot

= Wr

the operators

= - VV -L Pot W = W .

any scalar or vector function V,

operators

Pot and 4-7T

are inverse operator*.

That

(64)

V V

(65)

the

THE INTEGRAL CALCULUS OF VECTORS

_ JL

Pot

v v v= - v v

-i- Pot

-^V V W = - V V 4?r

Pot

4?r

With

a solenoidal function

respect to

F= v

4?r

and

239

Pot

W = W. (66)

the differentiating

operators of the second order

V V are equivalent

and

respect to

Wr irrotational function W

V VW =V x

With

(67) the differentiat

ing operators of the second order

V V are equivalent

That

VBy

and

= V V-W2

(68)

integrating the equations

and

by means

4^^=- V.NewF W = V x Lap W - V Max W

of the potential integral Pot

V New F= - Max New F (69) Pot W = Pot V x Lap W - Pot V Max W 4 Pot W = Lap Lap W - New Max W. (70)

4<7rPotF=:-Pot 4 TT

Hence for scalar functions and irrotational vector functions

New Max

47T is

an operator which

equivalent to Pot.

functions the operator

Lap Lap

For solenoidal

vector

VECTOR ANALYSIS

240

For any

gives the potential. the potential

gives

of the

vector function the first operator

irrotational

part;

the second^ the

potential of the solenoidal part. *95.] There are a number of double

volume integrals which in are of such frequent occurrence mathematical physics as to merit a passing mention, although the theory of

them

will

These double

not be developed to any considerable extent.

They are not scalar func integrals are all scalar quantities. tions of position in space. They have but a single value. The integrations in the expressions may be considered for convenience as extended over

The

functions by vanishing identically outside of certain finite limits deter mine for all practical purposes the limits of integration in all space.

case they are finite. Given two scalar functions

The mutual potential of the

two functions

Z7,

of position in

or potential product, as is

may

space.

be called,

the sextuple integral

Pot (71)

One

of the integrations

may

be performed

(

* 2 y*

yi ,^)

* 2>

Pot

PotVdv,

Ud **

(T2)

In a similar manner the mutual potential or potential product of

two vector functions

W, W" is

(71)

This

ried out

also a scalar quantity.

One

integration

may

be car

THE INTEGRAL CALCULUS OF VECTORS Pot (W, W")

The

mutual

=w or

Laplacian

vector functions

(x v

Laplacian

241

Pot W" dv t

product

of position in space

two

the sextuple

integral

Lap(W ,W")

=ffffffw

(*! yi

*i)

x ;nr

w" (** y* *) <*i <*"

(73)

One

integration

may

Lap (W, W") =

be performed.

f ( fW"

(^ 2 , ya , * 2 )

Lap

va (T4)

v yi

The Newtonian product function

LaP

*i)

d r

of a scalar function F,

W of position in space

and a vector

the sextuple integral

rf* 2

(75)

performing one integration

New ( F, W)

=///W (*

* 2)

New Frf

(76)

In like manner the Maxwellian product of a vector function and a scalar function F of position in space is the

integral

Max (W,F)

=/////JV(*i^*i) J- W0r2

,2/ 2 ,*2 )rf

W (77)

VECTOR ANALYSIS

242

One

integration yields

Max (W, F)

=fff V(xv y v zj Max W d v

= - New

(

F, W).

(78)

(53) Art. 93.

UPotr = - (V New CO PotF. [New U Pot F] = (V New V) Pot F + (New IT) V Pot F. 4?r

-(V.NewOPotF=-V.[NewPTotF]+NewtT.NewF Integrate

47r

|VpotFdi>=-

f f fv.

[Ne

+ C f CtfewU- NewFdv. 4-Tr

Pot

IT,,

F)= f f fNewT. NewFdv -

Pot

F New Z7

(79)

surface integral is to be taken over the entire surface S bounding the region of integration of the volume integral.

The

this region of integration is " all space," the surface

S may

be looked upon as the surface of a large sphere of radius R. If the functions and vanish identically for all points out

side of certain finite limits, the surface integral

must

vanish.

Hence 4

Pot ( U, F)

= f f fNew U New Vd v.

(54) Art. 93,

47rW". PotW

= V x Lap W" Pot W - V Max W" Pot W .

(79)

THE INTEGRAL CALCULUS OF VECTORS

But

[Lap

W" x

Pot

W = Pot W V x

Lap

W" V x

Pot

]

and

Lap

W] = Pot W V Max W" + Max W" V Pot W. Lap W" Pot W = V [Lap W" x Pot W]

+ Lap W" Lap

V Max W"

and

Pot

W=V -

Hence 4

[Max W" Pot

Hence

243

[Max W" Pot Max W" Max -

substituting:

Pot

W = Lap W

Lap

+ Max

W -V [Max W" Pot W

+ V

[Lap

W" X

Pot

Max W"

]

Integrating

Pot (W, W")

ff fLap W

r c r Max W Max / J J J

PotW x Lap W" da

Lap W" dv

(80)

Max W"PotWWa.

and W" exist only in finite space these surface taken over a large sphere of radius B must vanish integrals and then If

now

Pot (W, W")

= f f fLap W + 11 fMax J J J

* 96.]

Lap

Max W" d v.

There are a number of useful theorems

theoretic nature

(80)

of a function-

which may perhaps be mentioned here owing

VECTOR ANALYSIS

244

to their intimate connection with the integral calculus of The proofs of them will in some instances be given vectors.

and in some

The theorems

not.

are often useful in practical

applications of vector analysis to physics as well as in purely

mathematical work.

Theorem : If (#, y, z) be a scalar function of position V in space which possesses in general a definite derivative and if in any portion of space, finite or infinite but necessarily is continuous, that derivative vanishes, then the function

constant throughout that portion of space.

VF=0.

Given

F= const.

To show

Choose a fixed point (#15 y v zj in the region.

(2) page

180 y> *

V F. d r = V(x, y,z)-V (xv y v zj.

ft* *i

But

fvr.dr =f<)

Hence

F(#,

Theorem in space

y, z)

F" (#, y,

which possesses

= 0.

= V (xv y v zj = const. 2;)

be a scalar function of position

in general a definite derivative

;

the divergence of that derivative exists and is zero through out any region of space, 1 finite or infinite but necessarily

continuous

;

and

furthermore the derivative

V V vanishes

any finite volume or of any finite portion of surface in that region or bounding it, then the derivative vanishes throughout all that region and the function re at every point of

duces to a constant by the preceding theorem. The term throughout any region of space boundaries of the region as well as the region 1

must be regarded as including the itself.

THE INTEGRAL CALCULUS OF VECTORS

V V V=

Given

V F=

and

for a region T,

for a finite portion of surface S.

J^=

To show Since

245

const.

V Evanishes for the portion of surface

Vis

certainly

Suppose that, upon one side of S and in the V upon The derivative were not constant.

constant in S.

region T,

this side of

has in the main the direction of the normal to

Consider a sphere which lies for the most part upon the outer side of S but which projects a little The surface integral of over through the surface S. the surface S.

the small portion of the sphere which projects through the S cannot be zero. For, as is in the main normal

surface to

must be nearly

parallel to the

normal to the portion

of spherical surface under consideration.

VT-

Hence the terms

da,

same sign and cannot surface integral of over the intercepted by spherical sur

in the surface integral all have the

cancel each other out. that portion of

S which

The is

V V is zero. Consequently the surface V V taken over the entire surface of the spherical

face vanishes because integral of

segment which projects through S

not zero.

f r vr-da= f r fv. vrd*=o.

But

f /Vr da =

Hence

therefore appears that the supposition that is not constant upon one side of S leads to results which contradict the given relation V 0. The supposition must there fore have been incorrect and V must be constant not only in It

V V

but in

all

portions of space near to

in the region T.

VECTOR ANALYSIS

246

an extension of the reasoning

seen to be constant

throughout the entire region T. Theorem : If (x, y, z) be a scalar function of position in and if through space possessing in general a derivative

out a certain region

T of

or discontinuous, the divergence exists

and

zero,

and

a constant value

and

V (x, y, z)

vanishes.

continuous

space, finite or infinite,

V VV

of that derivative

V possesses

furthermore the function

in all the surfaces as a limit

bounding the region

when

the point (x, y, z) approaches recedes to infinity, then throughout the entire region T the function has the same constant value c and the derivative c

The proof does not

differ essentially

in the case of the last theorem.

eralized as follows

from the one given

The theorem may be gen

Theorem: If V(x,y,

be any scalar function of position

in space possessing in general a derivative

U (x,

y, z)

be any other scalar function of position which is either posi tive or negative throughout and upon the boundaries of a region T, finite or infinite, continuous or discontinuous; of and the divergence [ V~\ of the product

exists

and

and at to c

function is

and

;

all the

equal to Theorem :

and upon the boundaries of T V be constant and equal

furthermore

boundaries of

and

at infinity

;

then the

constant throughout the entire region

zero throughout

infinity

upon

T and

V (#,

be any scalar function of position

y, z)

in space possessing in general a derivative

out any region

V V;

through

of space, finite or infinite, continuous or

V VV

of this derivative exists discontinuous, the divergence and is zero ; and if in all the bounding surfaces of the region

the normal

component

at infinite distances in 1

of the derivative

The region

(if

VF" vanishes and

such there be) the product

includes

its

boundaries.

THE INTEGRAL CALCULUS OF VECTORS

247

9 Vj 3 r vanishes, where r denotes the distance measured from any fixed origin then throughout the entire region T r2

;

V Evanishes

the derivative

T V

and

in each continuous portion

constant, although for different continuous portions

this constant

not be the same.

may

This theorem

may

be generalized as the preceding one

U V F) = r^SV/Sr = 0.

was by the substitution of the relation

V-VF=Oand Ur*3V/3r = As

for

(

the foregoing theorems the following made. The language is not so precise be may in as the theorems themselves, but will perhaps be under corollaries of

statements

when they

stood

are borne in mind.

V U = V V,

then

and

differ

most by a

constant.

V-V7=V.VF

VZ7 = VF = V U V V at all

and

portion of surface S, then from only by a constant at most. If

V V.VJ7= V- VF

V V 7 = V V F and

any

points

finite

and

differs

and if V in all the bounding surfaces of the region and at infinity (if the region extend thereto), then at all points 7 and Fare equal. in all the

bounding surfaces and VFare

the region the normal components of VZ7 2 equal and if at infinite distances r (3 U/Sr of

F/5r)

Fare equal at all points of the region and U differs from F only by a constant. and W" are two vector functions of position Theorem If in space which in general possess curls and divergences if zero,

then

?7and

;

any region I

or infinite but necessarily continuous, the curl of is equal to the curl of and the divergence of is equal to the of W"; and if moreover divergence for

finite

the

two functions

every point of in

any

Tor bounding

of the region T.

and

finite

it;

volume

then

are equal to each other at in

T or of

equal to

any

finite

W" at

surface

every point

VECTOR ANALYSIS

248

Since

W = V x W", V x

whose curl vanishes

tor function

of a scalar function

(W - W") =

vec

equal to the derivative

Let

(page 197).

VF=W

W".

Then V V V= The theorem Theorem tion if

owing to the equality of the divergences. therefore becomes a corollary of a preceding one. If and are two vector functions of posi

which in general possess

definite curls

and divergences

throughout any aperiphractic* region T,

necessarily continuous, the curl of

W" and W";

the divergence of and if furthermore in

region then

;

equal to

Theorem: If

equal to the curl of

equal to the divergence of all the bounding surfaces of the

and

are equal;

throughout the aperiphractic region T.

and

W" are

two vector functions

which in general possess

tion in space

;

but not

the tangential components

finite

definite

of posi

curls

and

acyclic region T, finite but not curl of the is equal to the curl necessarily continuous, and the divergence of is equal to the divergence of

divergences

;

throughout any

bounding surfaces of the region T the and W" are equal then the func normal components of and W" are equal throughout the region acyclic T. tions The proofs of these two theorems are carried out by means

W"; and

in all the

;

of the device suggested before. and are Theorem: If

V V

W" and V V W"

two vector functions such

have in general definite values in a certain region T, finite or infinite, continuous or discon tinuous ; and if in all the bounding surfaces of the region that

at infinity the functions

ponents of

tic.

and are equal ; then entire the region T. throughout equal to The proof is given by treating separately the three com

and

and W".

T may

The

A region which

region

If it encloses

have to be made acyclic by the insertion of diaphragms. encloses within itself another region is said to be periphrac-

no region

it is

aperiphractic.

THE INTEGRAL CALCULUS OF VECTORS

249

SUMMARY OF CHAPTER IV The

line integral of a vector function

W along a curve

C is

defined as

Jfc The

Wdr=f J

[Widx + W^dy + W.dz].

line integral of the derivative

V V of

(1)

a scalar function

along a curve C from r to r is equal to the difference at the points r and r and hence the between the values of

line integral taken

versely

around a closed curve

around any closed curve vanishes, then V V of some scalar function V.

zero

integral of a vector function

the line

taken

(2)

(3)

C W dr = Jo

and con

the derivative

/ TO

and

;

then

W = VF.

theorem by application to mechanics. surface integral of a vector function over a surface defined as

Illustration of the

The

= ff Theorem| The surface integral of a vector functiorTtaken over a closed surface is equal to the volume integral of the divergence of that function taken throughout

Gauss

the volume enclosed by that surface

VECTOR ANALYSIS

250

= f f {Xdydz+ Ydz if

Z be

7 X, I

Stokes

the three components Theorem: The surface

dx + Zdxdy],

(8)

of the vector function

integral of the curl of a

vector function taken over any surface is equal to the line integral of the function taken around the line bounding the

And

surface.

function

conversely if the surface integral of a vector taken over any surface is equal to the line integral

W taken

of a function

curl of

around the boundary, then

//,VxW.*.=/o W.*r, and

the

ffjj da =f W

rfr,

then

(11)

x W.

(12)

Application of the theorem of Stokes to deducing the equations of the electro-magnetic field from two experimental facts

due to Faraday.

and Gauss

Application of the theorems of Stokes

to the proof that the divergence of the

a vector function a scalar function

curl of

zero and the curl of the derivative of zero.

Formulae analogous to integration by parts

JJ8

t/O

v~]

rf r,

(14)

rr JJS

cc ^vwXv yaa=ic i6v yar = o 8

*/ */

r vVu

t/O

a r,

(16)

THE INTEGRAL CALCULUS OF VECTORS I

C CuV *vdv =

/Y vu Green /

da=- r r

V v dv =

.Vi;-i;V.Vw)rfi;= __

x * dv.

(i8>

f T TwV -V v

f ttV-y -da

= if v V ^

Kelvin

A/!*

(17)

Theorerii:

x v

ii fV u*vdv,

uv d&-

251

vV *V udv,

f C (uVv

(19)

t?Vw).rfa. (20)

generalization:

i I Tw^7u^vdv=

w^Vv-rfa

T T TV V

The integrating operator known

V w]

rf v.

as the potential

(21)

defined

by the equation Pot

Pot

V(xyv Z^

=*?

yy *

dxt dy 2 dz y

^^2 ^y 2 ^^-

VPot T=PotVF;

V x Pot W = Pot V x W, V Pot W = Pot V W, V V Pot F= Pot V VF,

(22)

(23)

(27) (28)

(29)

(30)

VECTOR ANALYSIS

252

V V Pot W = Pot V V W,

(31)

(32)

W = Pot VV W, V x V x Pot W = Pot V x V x W. The

Pot

(33)

integrating operator Pot and the differentiating operator

V are commutative. The

three additional integrating operators

known

as the

Newtonian, the Laplacian, and the Maxwellian. __

New T=

Lap

Max

W =j

*19 -^

fO? ^9 /

\ Ay *

^-^-

\XT ^7*

J J

i ***.

(42)

2 ^ 2

^^2

^2/2

dzv

( 43 )

2> I

dy 2 dz2

rf^2

rf^2 rfy

d^2

If the potential exists these integrals are related to it as fol

lows:

V Pot F= New V, V x Pot W = Lap W,

The interpretation of the

(45)

physical meaning of the Newtonian

on the assumption that V is the density of an attracting is electric body, of the Laplacian on the assumption that is the flux, of the Maxwellian on the assumption that

intensity of magnetization. The expression of these integrals or their components in terms of a?, y, % ; formulae (42) , (43) ,

(44) and

(42)", (43)", (44)".

V New F= Max V F, V Max W = New V W, V x Lap W = Lap V x W,

(46) (47)

(48)

THE INTEGRAL CALCULUS OF VECTORS

Lap

V V V

The

potential

W = New V W Lap V X W = V Max W - V x Lap W.

V. VPotF=-47rW.

(52)

Lap

divided

That (52)

4-7T

Hence

(51)

V. VPotF = -47rF;

(50)

a solution of Poisson s Equation.

and

(49)

New V = Lap V V= 0,

Pot

W = Max V x W =

253

V.NewF,

(53)

(55)

into

New V W.

two

parts

which one

is,

and the other

irrotational, provided the potential In case the potential does not exist a third term 3

solenoidal

exists.

must be added vanish.

A list

which both the divergence and the curl of theorems which follow immediately from of

equations (52), (52)

(53),

(55)

and which

state that certain

integrating operators are inverse to certain differentiating Let V be a scalar function, x a solenoidal vector

operators.

function, and

47T

Lap

an irrotational vector function.

4 7T Lap

-47T

Wj = V x

New V. W,

= 0, V

=-V

Lap

4?T

Then

(60)

W =

(61)

New W2 = W2

(62)

VECTOR ANALYSIS

254

f_V-

Pot

-NewF = V (63)

47T

Max

VF=

-- - Pot VV W = - V V 4

x Pot

[--4?r

Pot

- Pot L- -r 4?r

- -

Pot

= W,

(64)

(65)

vr=- v-v 4?r-

V V W = - V V ! Pot W = W. 4?r

(66)

-V-VWj^VxVxWj V V W = VV W

(67)

(68)

V = - Max New V 4 TT Pot W = Lap Lap W - New Max W. 4

Pot

(69) (70)

Mutual potentials Newtonians, Laplacians, and Maxwellians may be formed. They are sextuple integrals. The integra tions cannot all be

may

be.

grals.

performed immediately but the first three Formulae (71) to (80) inclusive deal with these inte The chapter closes with the enunciation of a number

of theorems of

;

a function-theoretic nature.

By means

these theorems certain facts

concerning functions may be inferred from the conditions that they satisfy Laplace s equa

tion

and have certain boundary conditions.

Among tion.

the exercises

The work done in

number the text

worthy of especial atten has for the most part assumed 6

But many of the formulce connecting Newtonians, Laplacians, and Maxwellians hold when the poten tial does not exist. These are taken up in Exercise 6 referred to. that the potential exists.

THE INTEGRAL CALCULUS OF VECTORS

255

EXERCISES ON CHAPTER IV I.

a scalar function of position in space the line

integral

a vector quantity.

That

Show

that

the line integral of a scalar function around a is equal to the skew surface integral of the deriv

;

closed curve

ative of the function taken over

the contour of the curve.

Show

any surface spanned into

further that

V is

constant

the integral around any closed curve is zero and conversely if the integral around any closed curve is zero the function V is

constant.

Hint

Instead of treating the integral as it stands multiply (with a dot) by an arbitrary constant unit vector and thus reduce it to the line integral of a vector function. :

a vector function the line integral

x dr =/w J c

a vector quantity.

of the function

be called the skew line integral any constant vector, show that if

may

If c is

the integral be taken around a closed curve

H 1

The

first

(cVW cVW)

= c/ Wxdr,

four exercises are taken from Foppl s Einfiihrung in die

well sche Theorie der Electricitat where they are

worked

out.

Max-

VECTOR ANALYSIS

256

H-c = c.

and

]JJ V Wda- J J V (W 8

Show when

taken over a plane curve and the the portion of plane included by the curve

In case the integral surface

that the integral taken over a plane curve vanishes is constant and conversely if the integral over any must be constant. plane curve vanishes 3. The surface integral of a scalar function is

This

a vector quantity.

Show

that the surface integral

taken over any closed surface is equal to the volume taken throughout the volume bounded by integral of of

that surface.

That

Hence conclude that the surface integral over a closed sur if V be constant and conversely if the surface integral over any closed surface vanishes the function V must face vanishes

be constant. 4.

W be a vector function, the surface integral T= f C d&x W

may

be called the skew surface integral.

quantity.

Show

that the

skew

a vector

surface integral of a vector

THE INTEGRAL CALCULUS OF VECTORS function taken over a closed surface

equal to the volume

integral of the vector function taken throughout the

bounded by

the surface.

That

257

volume

Hence conclude that the skew surface integral taken over is an any surface in space vanishes when and only when That is, when and only when the line irrotational function.

integral of 5.

W for every closed circuit vanishes.

Obtain some formulae for these integrals which are

analogous to integrating by parts. 6.

The work

potentials of

most part that the

in the text assumes for the

Fand

W exist.

Many

of the relations, however,

be demonstrated without that assumption. Assume that the Newtonian, the Laplacian, the Maxwellian exist. For

may

simplicity in writing let

Then

New V =

V Pn V(x v y y * t

=i^

Lap

Max

W =fffv

W (x v y v

W (z

d t> 2 ,

dv v

z^dv v

(81)

(82)

(83) (84)

c c r

-JJJr^rdvr 17

VECTOR ANALYSIS

258

exercise ( 3

It can be

)/// V

shown that

(Pit

V is

d v*

such a function that

New V

radius

then this surface integral taken over a large sphere of R and a small sphere of radius R* approaches zero

when

exists,

small.

becomes indefinitely great; and

indefinitely

Hence

NewF=PotVF.

(85)

Prove in a similar manner that

W = Pot V x W, Max W = Pot V W.

Lap

By means

of (85), (86), (87)

x Lap

it is

(86) (87)

possible to prove that

W = Lap V x W,

V-New F=Max VF, V Max W = New V W. Then prove /*/*/*

VxLapW=i

I^ 12 VV-W

Hence

x Lap

V-VWdi l J *Jf^ 12

V Max W = f/JJPii V V W

and

di? 2

%} <J *J

dvv

W - V Max W = -ffffv V V W d v

x Lap

integral used

W - V Max W = 4 by Helmholtz

(88)

THE INTEGRAL CALCULUS OF VECTORS or

W be a vector function H (W) =///

Show

that the integral converges

that

259

"2"

V diminishes

)

so rapidly

when

becomes indefinitely great. 2 Vtf(F) = #(VF) = New(r

(91)

F),

# (W) = # (V W) = Max (r W), V x H (W) = 5" (V x W) = Lap (r W), V

(92)

=Jff (V.

VP) = Max(r 2 VF) = 2

Pot

(93)

(94)

(95)

Pot Pot PI H ( F) = - -L J

(96)

^ (W) = - -?Pot Pot W. 2

(97)

~2W = VxVx^T(W) + VV.^r (W). Give a proof of Gauss

(98)

Theorem which does not depend

upon the physical interpretation of a function as the flux of a The reasoning is similar to that employed in Art. 51 fluid. and

in the first proof of Stokes s

Show

that the division of

Theorem.

W into

two

parts,

page 235,

unique. 10.

Treat, in a

manner analogous

the case in which

to that

upon page

has curves of discontinuities.

220,

CHAPTER V LINEAR VECTOR FUNCTIONS 97.]

AFTER

down One was

the definitions of products had been laid

and applied, two paths of advance were open. differential and integral calculus the other, higher algebra ;

in the sense of the theory of linear

The treatment of and new symbols and vector

the

first

homogeneous substitutions.

of these topics led to

new

ideas

to the derivative, divergence, curl, scalar

potential, that

is,

V, Vx,

and Pot with the

auxiliaries, the Newtonian, the Laplacian, and the Maxwellian.

The treatment

of the second topic will likewise introduce

the linear vector novelty both in concept and in notation function, the dyad, and the dyadic with their appropriate

symbolization.

The

simplest example of a linear vector function is the The vector r product of a scalar constant and a vector.

= CT

(1)

r. more general linear function be obtained may by considering the components of r individ Let i, j, k be a system of axes. The components of ually.

a linear function of

r are i

Let each of these be multiplied by a scalar constant which

may

be different for the different components. cl i

c2 j

LINEAR VECTOR FUNCTIONS Take these r

The

components of a new vector

as the

(Cji-^ + j

vector r

j-r)

(c a

(c 8

then a linear function of

261

k-r).

(2)

Its

components components of r each

are always equal to the corresponding

multiplied by a definite scalar constant. Such a linear function has numerous applications in geom If, for instance, i, j, k be the axes of a etry and physics.

and c v

strain

homogeneous axes, a point

= ix + j

becomes

i c

the elongations along these

c 2, c , 3

-f j c 2

+ bz y

+ r

2 j

3 z,

This sort of linear function occurs in the theory of elasticity and in hydrodynamics. In the theory of electricity and

magnetism, the electric force

a linear function of the

For isotropic bodies displacement the function becomes merely a constant electric

But

in case the

in a dielectric.

body be non-isotropic, the components of the

force along the different axes will be multiplied

constants k v & 2 , & 3

E The the

= i%

i*D +

linear vector function

phenomena

different

Thus

.D + k&gk-D.

2 j

indispensable in dealing with

of electricity,

magnetism, and optics in non-

isotropic bodies. 98.]

possible to define a linear vector function, as has

been done above, by means of the components of a vector. The most general definition would be

VECTOR ANALYSIS

262 Definition

vector r

said to be a linear vector func

when

tion of another vector r

the components of r

along

three non-coplanar vectors are expressible linearly with scalar coefficients in terms of the components of r along those same vectors.

= XB, + yb + zc, where [abc] r = # a + y b + z x = a x + b y + c z y = a^x + 6 2 y + c z, z = az x + lz y + c

and and

(3)

3 z,

then r

a linear function of

(The constants a^ l v c v etc., have no connection with the components of a, b, c par allel to i, j, k.) Another definition however is found to be more convenient and from it the foregoing may be deduced. is

Definition

continuous vector function of a vector

said to be a linear vector function

sum

vectors.

when

the function of the

any two vectors is the sum of the functions of those That is, the function /is linear if

/(r 1 +

)=/(r )+/(r a ). 1

(4)

a be any positive or negative scalar and if / be a linear function, then the function of a times r is a times

Theorem

the function of

/0r) = a/(r), And

(5)

hence

/(a

= The obvious

f<Ji)

+ .-) + <**f (r a )+ 8 /(*8) +

a 2 r2

a3 r3

(5)

proof of this theorem which appears more or less is a trifle long. It depends upon making repeated

use of relation (4).

LINEAR VECTOR FUNCTIONS Hence

/(2r)

In like manner

/ (n r)

where n Let

any positive

= 2/(r). = nf (r)

integer.

be any other positive integer. obtained just

is,

Then by

the relation

i r )=-?./ / (.i) m (,). \ m \ w / =/( /

Hence That

263

equation (5) has been proved in case the constant a

a rational positive number.

To show

the relation for negative numbers note that

/(0)=/(0 + Hence But

/(0) /(O) =/(r-r)

= 2/(0).

= 0.

=/( r

+(-r)) =/(r)

Hence

To prove

(5) for incommensurable values of the constant

becomes necessary to make use of the continuity of the function /. That is a, it

Let x approach the incommensurable number a by passing through a suite of commensurable values. Then

Hence

*****.

J (v xi}

=a ~

VECTOR ANALYSIS

264

LlM

v (ar)=ar.

/(") = / 00

Hence

which proves the theorem. Theorem:

mined when

linear vector f unction /(r) is entirely deter its values for three non-coplanar vectors a, b, c are

known. Let

l=/(a),

m=/(b), n=/(c). Since r

any vector whatsoever,

be expressed as

may

= #a + yb + 3C. / (r) = x + y m + z n. r

Hence 99.]

In Art. 97 a particular case of a linear function was

expressed as

i c

2 j

For the sake of brevity and to save repeating the vector which occurs in each of these terms in the same way may be written in the symbolic form

In like manner b3

be any given vectors, and b p b 2

ap a 2 , a 8

another set equal in number, the expression r

this

= a! b

a2 b2

a linear vector function of r

;

for

a3 b3

owing

to the distributive

(6)

character of the scalar product this function of r satisfies For the sake of brevity r may be written sym relation (4). bolically in the

form

( ai

aa b2

a3 b3

(6)

LINEAR VECTOR FUNCTIONS

265

particular physical or geometrical significance attributed at present to the expression

a^ + agbg + .)

(a^ +

to be

(7)

should be regarded as an operator or symbol which -con the vector r into the vector r and which merely

verts

affords a convenient

and quick way

of writing the relation

(6).

expression a b formed by the juxtaposition of two vectors without the intervention of a dot or a cross is Definition

The symbolic sum

called a dyad.

two dyads

called a

dyadic binomial of three, a dyadic trinomial ; of any num For the sake of brevity dyadic ber, a dyadic polynomial. binomials, trinomials, and polynomials will be called simply ;

The

vector in a dyad is called the antecedent ; and the second vector, the consequent. The antecedents of a dyadics.

first

dyadic are the vectors which are the antecedents of the individual dyads of which the dyadic is composed. In like manner the consequents of a dyadic are the consequents of the individual dyads. Thus in the dyadic (7) a p a2 a3 the antecedents and b r b 2 , b 3 the consequents.

are

Dyadics will be represented symbolically by the capital Greek letters. When only one dyadic is present the letter will generally be used. In case several are under consid eration other

Greek

this notation (7)

capitals will be

and (6) may now be written

= aj b

read

briefly in the

definition

employed

also.

With

becomes

form

(8)

a2 b2

a3 b 3

The symbol

<P-r

product of

into r because the consequents bj, b 2 , b 3

dot

It is called the direct -

are

VECTOR ANALYSIS

266

The multiplied into r by direct or scalar multiplication. order of the factors and r is important. The direct product of r

r into

=r =r

^+a

(a a

a x bj

Evidently the vectors Definition is

When

and

a3 b 3

a3 b3

)

(9)

multiplied into r as

When

are in general different.

the dyadic

said to be a prefactor to

<#,

a2 b2

r r

said to be a post/actor to

r is multiplied in

used either as a prefactor or as a postfactor to a dyadic vector r determines a linear vector function of r. The two linear

vector functions thus obtained are in general different from one another. They are called conjugate linear vector func tions.

The two dyadics

^ajbj + ajbg + =b

and each of which

may

b2 a2

agbg

b3 a3

...

be obtained from the other by inter

changing the antecedents and consequents, are called conjitr The fact that one dyadic is the conjugate of gate dyadics. another is denoted by affixing a subscript C to either.

Thus Theorem:

dyadic used as a postfactor gives the same

result as its conjugate used as a prefactor.

r 100.]

Definition

Any two

That

dyadics

(9)

and

are said to

be equal

when or

when

= W r =r W W r =B

r r B

for all values of

and

(10) r.

LINEAR VECTOR FUNCTIONS The

third relation

scalar

equivalent to the

first.

For,

the

r are equal, the scalar

products of any them must be equal. And conversely if the r product of any and every vector s into the vectors s

into

*T are equal, then those vectors must be equal. In it may be shown that the third relation is equiva

and like

and

vectors

vector

267

manner

lent to the second.

Theorem

Hence

all

dyadic

three are equivalent.

completely determined

when

the

values

0.a,

0.b,

0.c,

any three non-coplanar vectors, are known. This follows immediately from the fact that a dyadic defines

where

a, b, c

are

a linear vector function.

= 0.(#a +

2/b

+ zc)==#

?/*b-Mc,

and W are equal provided equa Consequently two dyadics tions (10) hold for three non-coplanar vectors r and three non-coplanar vectors s. Theorem : Any linear vector function /

may

be represented

to be used as a prefactor and by a dyadic , by a dyadic which is the conjugate of 0, to be used as a postfactor. The linear vector function is completely determined when its

values for three non-coplanar vectors (say

known (page

Then

to be

264).

the linear function

k) are

equivalent to the dyadic

used as a postfactor; and to the dyadic

= to be

i, j,

Let

<P (7

used as a prefactor.

= ia +

kc,

VECTOR ANALYSIS

268

The study of linear vector functions therefore is identical with the study of dyadics. dyad a b is said to be multiplied by a scalar Definition :

a when the antecedent or the consequent is multiplied by when a is distributed in any manner between

that scalar, or

the antecedent and the consequent.

a (ab)

(a a) b

= a (a b) =

=aa

(a a) (a" b).

is said to be multiplied by the scalar a when dyadic each of its dyads is multiplied by that scalar. The product is written

The dyadic a $

<Pa.

applied to a vector r either as a prefactor or

as a postfactor yields a vector equal to a times the vector

obtained by applying

(a 0)

Theorem utive.

that

to r

The combination

That

= a (0

r).

of vectors in a

dyad

distrib

(a + b)

and

a (b

+ c)

= ac + bc = ab + ac.

...

This follows immediately from the definition of equality of

For

dyadics (10). [(a

b) c]

= (a +

b) c

= ac

+ c)]

= a (b + c)

= ab -r +

ac-

= (ab + ac)

(a c

b c)

and [a(b

Hence

follows that a dyad which consists of two factors, is the sum of a number of vectors, may be

each of which

multiplied out according to the law of ordinary algebra except that the order of the factors in the dyads must be

maintained.

LINEAR VECTOR FUNCTIONS

269

bn+

...

(11)

+ cl-f cm-f cn+ The dyad

therefore appears as a product of the

two vectors

it is composed, inasmuch as it obeys the characteris the distributive law. law of products This is a justifi a cation for writing dyad with the antecedent and conse

which tic

quent in juxtaposition as

customary in the case of products

in ordinary algebra.

Nonion Form of a Dyadic

The 10L]

From

the three unit vectors

i, j,

k nine dyads may These are

be obtained by combining two at a time. ii,

ij,

ik,

ji,

jj,

jk,

ki,

kj,

If all the antecedents

pressed in terms of simplified

i, j,

(12)

kk.

and consequents in a dyadic be ex and if the k, resulting expression be

by performing the multiplications according to the law (11) and if the terms be collected, the dyadic

distributive

may

be reduced to the

sum

of nine dyads each of

which

a scalar multiple of one of the nine fundamental dyads given above.

= a n ii + a 12 ij + + iJi +a 22 jj +

a 13 ik

a 33 kk.

+ This

called the nonion

Theorem dyadics

a 31 ki

a 32 kj

(13)

form of 0.

The necessary and

and

a23

be equal

sufficient condition that

is that,

when expressed

two

in nonion

VECTOR ANALYSIS

270

form, the scalar coefficients of the corresponding dyads be equal.

be equal, then obviously

If the coefficients

r= W

<P. for

and the dyadics by (10) must be equal. the dyadics and W are equal, then by (10)

any value

Conversely,

of r

for all values of

and

Let

and

each take on the values (14)

0.i=j.

W.i,

j. </.j

=j.

= k.

?F.i,

k- 0-

= k-

Then

i,j,k.

<P-i

W .

?F.j,

k = i <P.k = j. 0- k = k

?T.k r.k.

But

these quantities are precisely the nine coefficients in the and W. Hence the corresponding expansion of the dyadics 1 This equal and the theorem is proved. statement of the of two analytic equality dyadics can some times be used to greater advantage than the more fundamental

are

coefficients

definition (10) based

upon the conception

of the dyadic as

defining a linear vector function.

dyadic may be expressed as the sum of nine dyads of which the antecedents are any three given non-

Theorem

coplanar vectors,

a, b, c

non-coplanar vectors Every antecedent

and the consequents any three given n.

be expressed in terms of a, in terms of 1, m, n. The dyadic

may

and every consequent,

b, c

may

then be reduced to the form

= an + -f1

As a

& 12

a 21 bl

am + bin +

fflai

a 13

a 23

m + ^33

(15)

c n.

corollary of the theorem it is evident that the nine dyads (12) are in None of them may be expressed linearly in terms of the others.

dependent.

;

LINEAR VECTOR FUNCTIONS

271

This expression of <P is more general than that given in It reduces to that expression when each set of vectors (13). with i, j, k. a, b, c and 1, m, n coincides Theorem

Any

dyadic

may

be reduced to the

sum

three dyads of which either the antecedents or the consequents,

but not both,

be arbitrarily chosen provided they be non-

may

coplanar.

Let of

which

be required to express 4> as the sum of three dyads Let 1, m, n be any other a, b, c are the antecedents.

three non-coplanar vectors.

then be expressed as in

may

Hence

(15).

= a (a n + 1

a 13 n)

+ or

In like manner

+ Osi 1 +

b (a 21 c

= aA + bB +

m + 23 n) m + a 32 n), 32 22

cC.

(16)

be required to express

as the

three dyads of which the three non-coplanar vectors the consequents

= Ll + Mm +

= an a + M = a 12 a + N = a lB a + L

where

Nn,

sum

m, n are

(16)

a 2l b

a 31

a 32

a 23 b

a ZB

are unique. Two equal expressions (15), (16), (16) for which have the same three non-coplanar ante dyadics

The

cedents,

a, b, c,

have the same consequents A,

- -

these

however need not be non-coplanar. And two equal dyadics which have the same three non-coplanar consequents 1, m, n, have the same three antecedents. 102. ]

Definition:

position of or a cross

a and

The symbolic product formed by

two vectors

the juxta b without the intervention of a dot

called the indeterminate product of the

two vectors

VECTOR ANALYSIS

272

The reason

for the

term indeterminate

The two

is this.

products a b and a x b have definite meanings. One is a certain scalar, the other a certain vector. On the other hand the product

neither vector nor scalar

it is

purely

symbolic and acquires a determinate physical meaning only when used as an operator. The product a b does not obey the commutative law.

law is

(11)

It does

however obey the

and the associative law

distributive

as far as scalar multiplication

concerned (Art 100). TJieorem

The indeterminate product a b

two vectors

the most general product in which scalar multiplication

is is

associative.

The most general product conceivable ought to have property that when the product is known the two factors also

known.

Inasmuch

are

Certainly no product could be more general.

as scalar multiplication is to be associative, that

a (ab) it

the

= (a a) b = a (a b) = (a* a)

(a"b),

will be impossible to completely determine the vectors a

and b when

may

their product a b is given. Any scalar factor be transferred from one vector to the other. Apart from

this possible transference of a scalar factor, the vectors

posing the product are other words

known when

the product

com

known. In

two indeterminate products a b and a b are equal, the vectors a and a b and b must be collinear and the product of the lengths of a and b (taking into account the Theorem

If the

and b have respec a and b ) is equal to the

positive or negative sign according as a tively equal or opposite directions to product of the lengths of a and b .

Let

6 3 k,

LINEAR VECTOR FUNCTIONS

v = Vi + yj + &, Then

and

Since

a2 &3

a 2 6 3 jk

a,&,

a 3 6 3 kk.

aj 6 3 ik

o 8 6j ki

a/V

a 2 6 3 jk

+ o,

kk.

corresponding coefficients are equal.

a 1 :a 2 :a s

:a 2 :a 3

which shows that the vectors a and a are

And

=ab

+ a^^

273

:,:6 8

= V-

which shows that the vectors b and

But

al bl

/ -

Hence

collinear.

V are collinear.

= a/ &/.

This shows that the product of the lengths (including sign) are equal and the theorem is proved. The proof may be carried out geometrically as follows. Since

equal to a

Let r be perpendicular to b. Then b r r. vanishes and consequently Vr also vanishes. This is true for any vector r in the plane perpendicular to b. Hence b and b are perpendicular to the same plane and are collinear. In for all values of

manner by using a b

to be parallel.

as a postfactor a

and a are seen

Also

ab-b =

a b -b,

which shows that the products of the lengths are the same. 18

VECTOR ANALYSIS

274

The indeterminate product ab imposes Jive conditions upon the vectors a and b. The directions of a and b are fixed and The scalar product likewise the product of their lengths. a

being a scalar quantity, imposes only one condition upon The vector product a x b, being a vector quantity, b.

a and

imposes three conditions. The normal to the plane of a and b is fixed and also th e area of the parallelogram of which they

The nine indeterminate products (12) of i,j, k themselves are independent. The nine scalar products

are the side. into

Only two

are not independent.

them

are different.

i.j=j.i=j.k = kj=ki = ik = 0.

and

The nine vector products

are mot independent either; for

ixi = jxj = kxk = and

ixj

jxk=

jxi,

kxj,

kxi

ixk.

b and a x b obtained respectively from the indeterminate product by inserting a dot and a cross be

The two products

tween the factors are functions of the indeterminate product. That is to say, when ab is given, a b and a x b are determined.

For these products depend solely upon the directions of a and b and upon the product of the length of a and b, all of which are if

known when ab ab

known.

That

b and a x b

x b

(17)

not hold conversely that if a b and a x b are known for taken together a b and a X b impose upon the vectors only four conditions, whereas a b imposes five. Hence It does

fixed

;

a b appears not only as the most general product but as the most fundamental product. The others are merely functions of

it.

Their functional nature

notation of the dot and the cross.

brought out clearly by the

LINEAR VECTOR FUNCTIONS

Definition:

scalar

known

275

the scalar of

be ob

may

tained by inserting a dot between the antecedent and conse quent of each dyad in a dyadic. This scalar will be denoted

by a subscript S attached

=a = &1 8

+ +

to 0.

a2 b 2

known

In like manner a vector

a3 b 3

...

(18)

as the vector of

may

obtained by inserting a cross between the antecedent and con sequent of each dyad in 0. This vector will be denoted by attaching a subscript cross to 0.

= aj

x b x + a2 x b2

x b3

be expanded in nonion form in terms of

#x =

)

Or <? x

=(j

i, j,

(19) k,

(20)

!3)

<Pj + k.

(P-k-k* ^.j) i+

= a n + a^ + a BZ + (^12 - a 2l) + 0*31 - a

- a 32 * 28 = i- 0-i + jS

(

(?-k,

(

21 )

(20)

(k- (P-i-i. (P.k)

0-j-j.0.i)k.

(i-

(21)

In equations (20) and (21) the scalar and vector of are in terms of the coefficients of when expressed expanded in the nonion form. Hence if and W are two equal dyadics, the scalar of

vector of

equal to the scalar of equal to the vector of

From

this it

and the

appears that

and

X X

= yx are

(22)

functions of

is uniquely determined when They may sometimes given. be obtained more conveniently from (20) and (21) than from

(18) 1

and

(19),

and sometimes

A subscript dot might be used

and free from

not.

for the scalar of

liability to misinterpretation.

if it

were

sufficiently distinct

VECTOR ANALYSIS

276

Products of Dyddics

In giving the definitions and proving the theorems concerning products of dyadics, the dyad is made the under 103.]

What

true for the dyad is true for the dyadic in general owing to the fact that dyads and dyadics obey the distributive law of multiplication.

lying principle.

The

Definition:

dyad

written

(ab)

and

That

is,

dyad (b

= a(b.c)d = b-c

the antecedent of the

second dyad

into the

(cd)

definition equal to the

(ab)-(cd)

product of the dyad

direct

first

ad.

(23)

and the consequent of the and consequent

are taken for the antecedent

respectively of the product and the whole is multiplied by the scalar product of the consequent of the first and the

antecedent of the second.

Thus the two

vectors which stand together in the product

(ab). (cd)

The

are multiplied as they stand.

new

The

other two are left to form

product of two dyadics may be dyad. defined as the formal expansion (according to the distributive law) of the product into a sum of products of dyads. Thus a

direct

*=(a

r^CCjdj +

and d>.

?T=(a 1 b 1

a 2 b2 c2

+ +

+a 2 b 2 +

a3 b 3

...)

-..)

a3 b3

)

+ C3 d3 = a b c 1 d + a b *e 2 d a + a x b x + a 2 b 2 -c d +a 2 b 2 .c 2 d 2 + a 2 b 2 + agbg-c^ + a 3 b 3 -c 2 d 2 + a 3 b 3 + (c^j +

The

parentheses

may

+ +

d3 H

be omitted in each of these three expressions.

(23)

LINEAR VECTOR FUNCTIONS

ajdj

a2 d x

ax d2

-f

277 ax d3

b 2 -c 2 a 2 d 2

b 3 .c 2 a 3 d 2

b 3 -c 3 a 3 d 3

a^j

+ -f-

(23)"

The product Theorem

regarded as

W is

and

two dyadics

a dyadic

W of two dyadics (P and W when The product an operator to be used as a prefactor is equiva

W followed

lent to the operator

by the operator 0.

=&..

Let

To show or

Let ab be any dyad of

cd) (c d

Hence

(a b

The theorem

and

= ab

W)*T

((?

(ab

(

= c

r),

0- (^ 0-

(24)

d any dyad of W.

(ad c (d

= (b c) (d r) a, r) = (b c) (d r) a,

(c d

r).

true for dyads. Consequently by virtue of it holds true for dyadics in general. If r denote the position vector drawn from an assumed origin is

the distributive law

P in

to a point

space, r

= W r will be = (^(3 r)

another point P and r" vector of a third point P n ,

That

the position vector of will be the position

to say,

W defines

a trans

formation of space such that the points P go over into the defines a transformation of space such that the points P P into the points P". over Hence W followed by points go f

carries

into

ff .

The

single operation

also carries

PintoP". Theorem: Direct multiplication of dyadics obeys the tributive law.

That

dis*

VECTOR ANALYSIS

278

and

Hence

= W=

(

)

W+

(25)

in general the product

(4>+

4>"

...).(

W+

?F"+...)

be expanded formally according to the distributive law. Theorem : The product of three dyadics <P, W, Q is associa-

may

Thatis

tive.

and consequently

o= t. (ma>

t.r).

(

either product

parentheses, as

may

(26)

be written without

(26 )

The

proof consists in the demonstration of the theorem for three dyads ab, cd, ef taken respectively from the three

dyadics

4>,

= (bc) ad ef = (bc) (d- e) af, = (d^e) ab -cf = (d e) (b c) af.

(abcd)

ab (cdef) The proof may

also be given

by considering

as operators

Let

Again

Hence

{^.(f.

{(*

for all values of

J)}

Q\ -r

*. [(f.

{(?

Consequently

J2).r].

T)\

0, W,

and

LINEAR VECTOR FUNCTIONS

279

The theorem may be extended by mathematical induction any number of dyadics. The direct product Parentheses may of any number of dyadics is associative. to the case of

be inserted or omitted at pleasure without altering the result. It was shown above (24) that r

(<P

(

(24)

Hence the product of two dyadics and a vector is associative. The theorem is true in case the vector precedes the dyadics and also when the number of dyadics is greater than two. But the theorem is untrue when the vector occurs between The product of a dyadic, a vector, and another the dyadics. is

dyadic

not associative.

(#.r).

0.(r-

Let ab be a dyad of $, and (a b

ab Hence

The

c d)

(ab

d a dyad of

= b r (a c d) = (b = ab d (r c) = b r)

results of this article

may

(27)

c) d,

r) (a

cd).

summed up

as follows

Theorem: The direct product of any number of dyadics number of dyadics with a vector factor at either

or of any

end or

laws of

both ends obeys the distributive and associative multiplication parentheses may be inserted or

omitted at pleasure. But the direct product of any number of dyadics with a vector factor at some other position than at

end is not associative parentheses are necessary to the a definite meaning. expression give Later it will be seen that by making use of the conjugate either

dyadics a vector factor which occurs between other dyadics may be placed at the end and hence the product may be

made

assume a form in which

it is

associative.

VECTOR ANALYSIS

280 104.]

Definition:

and

a vector r respectively

The skew products of a dyad ab into a vector r into a dyad ab are defined

by the equations

= a(b rx(ab) = (r x

(ab) x

r),

a)b.

The skew product of a dyad and a vector at dyad. The obvious extension to dyadics is

rrrajbj

a2 b2

xr +

a3 b 3

= r x (a b + a 2 b + a b + = r x ajbj + r x a 2 b 2 + r x

either

end

+ ...

(28)

a3 b3

...

Theorem: The direct product of any number of dyadics multiplied at either end or at both ends by a vector whether the multiplication be performed with a cross or a dot is associative.

But

position than the

in case the vector

end the product

occurs at any other

not associative.

= rx(0.y)=rx <P = <P.? xr, (<P ?F) xr=(P.(?P xr) =r x = r x <P s) (r x #) s r (0 x s) = (r x s = r <P x rx($xs) = (rx $)xs = rx $xs,

(rx

<P)

That

is,

(

</>)

but

(rX^)^(S

-r)

X*.

Furthermore the expressions s

and

can have no other meaning than s

<P),

(29)

LINEAR VECTOR FUNCTIONS

281

with a cross into a scalar

since the product of a dyadic

Moreover since the dot and the cross may is meaningless. be interchanged in the scalar triple product of three vectors it

appears that

x ^

5F)

(</>

0-(r x

and

r) (r

(31)

s),

The parentheses in the following expressions omitted without incurring ambiguity. <p.(r

(sx

(0-r) x

r).

0*sx(r-0),

(0-r) x

The formal skew product

cannot be

* of

(ab) x (cd)

x(r.

>).

two dyads a b and

= a(b

(31)

d would be

x c)d. b x

d are placed side by side with no sign of multiplication uniting them. Such

this expression three vectors a,

an expression

rst is

called a triad

The theory

;

and a sum of such expressions, a

of triadics

(32) triadic.

intimately connected with the theory

of linear dyadic functions of a vector, just as the theory of

connected with the theory of linear vector functions In a similar manner by going a step higher tetrads and tetradics may be formed, and finally polyads and

dyadics

of a vector.

But

the theory of these higher combinations of The dyadic vectors will not be taken up in this book.

polyadics.

furnishes about as great a generality as is ever called for in practical applications of vector methods.

VECTOR ANALYSIS

282

Degrees of Nullity of Dyadics 105.]

was shown

be reduced to a

sum

(Art. 101) that a dyadic could always

of three terms at most,

and

this reduction

can be accomplished in only one

way when

or the consequents are specified.

In particular cases

the antecedents it

may

be possible to reduce the dyadic further to a sum of two terms or to a single term or to zero. Thus let

= al + bm + cn.

m, n are coplanar one of the three in terms of the other two as If

be expressed

= x m + y n.

Then

may

= a#m + ayn + bm + cn, = (a# + b)m + (ay + c)n.

The dyadic has been reduced to two terms. If 1, m, n were all collinear the dyadic would reduce to a single term and if all the would vanished vanish. they dyadic Theorem

If a

be expressed as the

dyadic

sum

of three

terms <p

= al + bm +

which the antecedents

then the dyadic

may

a, b, c

are

known

be reduced to the

to be non-coplanar,

sum

two dyads

when and only when the consequents are coplanar. The proof of the first part of the theorem has

just been

To prove the second part suppose that the dyadic could be reduced to a sum of two terms

given.

= dp +

and that the consequents

were non-coplanar. 1, m, n of This supposition leads to a contradiction. For let 1 , m n be the system reciprocal to 1, m, n. That is, ,

_= mx n [Tmn]

n x

LINEAR VECTOR FUNCTIONS The

vectors

exist

283

and are non-coplanar because

m, n have been assumed to be non-coplanar. may be expressed in terms of them as

Any

= xl + ym + zn/ <p.r = (al + bm + en) (xl + ym + m = n n = 1, 1 = m 1 f

zn ).

But and

m=m

m =1

Hence

= xa +

n yb

=n +

giving to r a suitable value the vector equal to any vector in space. r

= (dp +

e q)

= d (p

= 0.

z,e.

But

vector r

may

e (q

made

r).

must be coplanar with d and e. Hence r can take on only those vector values which lie in the plane of d and e. Thus the assumption that 1, m, n are noncoplanar leads to a contradiction. Hence 1, m, n must be coplanar and the theorem is proved. This shows that

Theorem : If a dyadic terms

which the antecedents

be expressed as the

a, b, c

are

known

sum

to be non-coplanar,

the dyadic can be reduced to a single dyad when the consequents 1, m, n are collinear.

when and only

The proof of the first part was given above. the second part suppose <P could be expressed as

Let

of three

To prove

VECTOR ANALYSIS

284

From

the second equation postfactor for any vector r

where a

From

the

first

it is

= x a + y b + zc

used as a

the reciprocal system to

a, b, c

gives

expression

= 0.

#lxp+ymxp + znxp

Hence must be zero Hence y, z. 1

Hence

evident that

for every value of

x p

= 0,

mxp =

m, and n are

that

all parallel to

is,

for every value of x,

nxp = 0. p and the theorem has

been demonstrated. If the three consequents

m, n had been

known

to be non-

coplanar instead of the three antecedents, the statement of would have to be altered by interchanging the

the theorems

words antecedent and consequent throughout. There is a fur ther theorem dealing with the case in which both antecedents

and consequents of are coplanar. sum of two dyads.

Then

reducible to

the

106.]

the

sum

dyadic which cannot be reduced to of fewer than three dyads is said to be complete. Definition:

dyadic which may be reduced to the sum

two dyads, but

cannot be reduced to a single dyad is said to be planar. In case the plane of the antecedents and the plane of the con the dyadic is expressed as the sum of two dyads, the dyadic is said to be uniplanar. dyadic which may be reduced to a single dyad is said to be linear.

sequents coincide

when

In case the antecedent and consequent of that dyad are

col-

LINEAR VECTOR FUNCTIONS linear, the

dyadic

said to be unilinear.

285

If a dyadic

may

its terms vanish the dyadic is said to In this case the nine coefficients of the dyadic as expressed in nonion form must vanish.

so expressed that all of

be zero.

The

properties of complete, planar, uniplanar, linear, and when regarded as operators are as follows.

unilinear dyadics

Let

complete

and

r t

may

made

<P.

take on

any

desired

value by giving r a suitable value.

complete

reciprocal system s

In like manner

(xl

m, n are non-coplanar and hence have a m n ,

+ ym + zn

a, b, c

complete dyadic

planar

zc.

)

= xl + ym +

zn.

applied to a vector r cannot give zero

unless the vector r itself is

=#a + yb +

possess a system of reciprocals a

yb + zc

)

the vector s

zero.

may take

on any value in

the

plane of the antecedents and t any value in the plane of the consequents The dyadic but no values out of those planes. when ; of r in reduces vector to as a a used vector space prefactor every

In particular any vector r in the plane of the antecedents. of the the to is reduced plane consequents of perpendicular used as a postfactor reduces every to zero. The dyadic vector r in space to a vector in the plane of the consequents In particular a vector perpendicular to the plane of of <P. is reduced to zero. In case the dyadic the antecedents of

uniplanar the same statements hold. is linear the vector s may take on any value collinear If with the antecedent of and t any value collinear with the con-

VECTOR ANALYSIS

286

but no other values. The dyadic used as a r to the vector reduces line of the antecedent any prefactor In particular any vectors perpendicular to the con of 0. The dyadic are reduced to zero. used as a sequent of sequent of

;

postfactor reduces any vector r to the line of the consequent In particular any vectors perpendicular to the ante of 0. are thus reduced to zero.

cedent of

a zero dyadic the vectors matter what the value of r may be. Definition nullity.

nullity. lity

A A

A planar dyadic

and

are loth zero no

said to possess one degree of linear dyadic is said to possess two degrees of zero dyadic is said to possess three degrees of nul :

or complete nullity.

The

two complete dyadics is complete; of a complete dyadic and a planar dyadic, planar of a complete dyadic and a linear dyadic, linear. Theorem

107.]

direct product of

;

Theorem: The product of two planar dyadics except when

the plane of the consequent of the

planar

first

dyadic perpendicular to the plane of the antece dent of the second dyadic. In this case the product reduces and only in this case. to a linear dyadic

in the product

Let

Q= The vector and

The

= W

r takes

all

values in the plane of

vector g

a 2 b 2,

\x (bj

= c x)

takes on the values

(b x

c ) x

y (b x

c2)

(b2

C l)

y 0>2

(bj

c 2 )}

{x (b 2

c x)

E 2>

+ y

(b 2

c 2 )}

LINEAR VECTOR FUNCTIONS Let

where

and

x &i

+y +y

x (b x

Cj)

x (b 2

Cj) 4-

287

(b l

c 2 ).

y (b 2

These equations may always be solved for x and y when that is, when s has any desired values x and y are given 1

any desired value in the plane of determinant

But by

Chap.

(25),

\xb

vector

and

sequents of <P;

bj-c, b2

( ci

and a2

unless the

merely the product

II., this is

0>i

The

V^ b2

C 2)

perpendicular to the plane of the con x c 2 to the plane of the antecedents of

Their scalar product vanishes when and only

that

vectors are perpendicular

pendicular. plane of a x

Consequently

may

when

the

when

the planes are per take on any value in the

is,

and a2 and

is therefore a planar dyadic unless the planes of b x and b 2 , c x and c 2 are perpendicular. If however b x and b 2 , Cj and c 2 are perpendicular s can take f

on only values in a certain line of the plane of a x and a^ and hence <P W is linear. The theorem is therefore proved.

The product

of two linear dyadics is linear the consequent of the first factor is perpen In this case the dicular to the antecedent of the second.

Theorem

except when

and only in this case. product is zero Theorem : The product of a planar dyadic into a linear linear except

planar dyadic dyadic.

when is

the plane

of the

consequents of the

perpendicular to the antecedent of the linear

this case the

product

zero

and only

in this

case.

Theorem: The product of a linear dyadic into a planar dyadic

linear except

when

the consequent of the linear

VECTOR ANALYSIS

288

perpendicular to the plane of the antecedents of In this case the product is zero the planar dyadic. and

dyadic

only in this case.

immediately evident that in the cases mentioned the products do reduce to zero. It is not quite so apparent that It is

they can reduce to zero in only those cases.

The

similar to the one given above in the case of

They

dyadics.

are

the reader.

left to

theorem stated, page 286,

first

proofs are

two planar

The proof

the

also left to the reader.

The Idemfactor; 1 Reciprocals and Conjugates of Dyadics If a

dyadic applied as a pre factor or as a postfactor to any vector always yields that vector the dyadic is said to be an idemfactor. That is 108.]

Definition

then

r for all values of r,

for all values of

The capital I is used as the sym The idemfactor is a complete dyadic.

an idemfactor.

bol for an idemfactor.

For there can be no direction in which I r vanishes. Theorem : When expressed in nonion form the idemfactor I

Hence

+ jj +

kk.

(33)

idemfactors are equal. To prove that the idemfactor takes the form (33) it is merely necessary to apply the idemfactor I to the vectors i,

all

respectively. 1

Let

= a n ii+ ki

12 ij

a 32 kj

a 13 ik

a 33 kk.

In the theory of dyadics the idemfactor I plays a role analogous to unity in The notation is intended to suggest this analogy.

ordinary algebra.

LINEAR VECTOR FUNCTIONS I

. i

= an +

I-i

In like manner

and a 21

may ,

The

For

In like manner

Theorem:

If a

may V,

kk.

and

coefficients

Hence (33)

any dyadic and the idem-

1-0 = 0.

(I r

may

be.

Hence, page 266,

= 0.

be shown that I

the

is,

and

no matter what the value of

all

which are unity.

all of

That

= a 31 = 0.

direct product of

that dyadic.

a 31

= ii + jj +

Theorem

j 4-

be shown that

vanish except a n a 22 a 33

factor

a 21

289

a, b, c

be two reciprocal systems

of vectors the expressions I I

= aa + bb + cc =aa+b b+c c

(34)

are idemfactors.

For by (30) and (31) Chap.

II.,

= raa + r*bb + r cc r = ra a + r b b + r.c r

and

Hence the expressions must be idemfactors by Theorem

Conversely

the expression

= al + bm + is

an idemfactor

definition.

m, n must be the reciprocal system of

a, b, c.

VECTOR ANALYSIS

290

In the dyadic.

place since

first

(P is

possess a set of reciprocals a

hypothesis

Then

a, b, c are

is,

for all values of x, y,

Hence the

That

is,

be any two dyadics, and if the product l then the product W 0, is equal to the idemfactor the factors are taken in the reversed order, is also

Theorem

non-coplanar and

corresponding coefficients must be equal.

a complete

= xl + ym-\-zn = xsi + y b + zc

for all values of r, that

when

it is

Let

= #a + y V + 20 r $ = r.

the idemfactor,

Hence the antecedents

(Pand

;

equal to the idemfactor.

V=L

Let

To show

This relation holds for

must take on

all

= all

When

(?)

Hence by

=r is

complete r

definition

= I.

product of two dyadics be taken in either order.

If the

109.] Definition:

(

values of

desired values.

W may

an idemfactor, that product

two dyadics are so related that

their product is equal to the idemfactor, they are said to be 1

V to be complete. For the product incomplete and hence could not be equal to the

This necessitates both the dyadics * and

two incomplete dyadics

idemfactor.

LINEAR VECTOR FUNCTIONS The notation used

reciprocals.

algebra

0.y=I,

employed

Theorem:

for reciprocals in ordinary

denote reciprocal dyadics.

?F-i

291

and 5T=

That

0-i=L

is,

(35)

Reciprocals of the same or equal dyadics are

equal.

and

be two given equal dyadics, <J>~ 1 and JT"1 their reciprocals as defined above. By hypothesis

Let

0= ~i

and

= ~i. 0-1 = 1= 0- 1

To show 0.

0=,

I.0-i Hence

-1.

0.0~i=0.-\ 0-1.0

The

= l.

= I,

= 0-i = I. W~i = 0-i = ~\

-1.

the dyadic whose antecedents are the and whose conse reciprocal system to the consequents of are the antecedents to the of 0. reciprocal system quents reciprocal of

be written in the form

If a complete dyadic

its

reciprocal

For

(al

+ bm + cn)

Theorem into

0"1

= al + bm + en, = a +mV+n 1

V+

c )

(36)

=aa + bV +

If the direct products of a complete dyadic

two dyadics 1

W and Q

An incomplete

are equal as dyadics then

dyadic has no

(finite) reciprocal.

W and Q

292

VECTOR ANALYSIS

are equal.

and

If the

product of a dyadic

two vectors

into

(whether the multiplication be performed with a dot a or cross) are equal, then the vectors r and s are equal. r

That

is,

= d> J2, r = r = x

and

= Q, then r = then r =

then

This

may

(37)

be seen by multiplying each of the equations

through by the reciprocal of 0,

= 0-i r = r = 0~!

0-1.0. W = 0-i

0-1.

0xr = IXr=0"

= 0X8 = 1X8. s

1 -

reduce the last equation proceed as follows.

any vector,

Hence t is

any vector,

for

r is equal to

Equations (37) give what celation

Let

tIxr = tIxs, t

equivalent to the law of cancomplete dyadics. Complete dyadics may be is

canceled from either end of an expression just as if they were scalar quantities. The cancelation of an incomplete dyadic is not admissible. It corresponds to the cancelation of a zero factor in ordinary algebra. 110.]

number

Theorem:

The

reciprocal

of the

product of any

equal to the product of the reciprocals taken in the opposite order. It will be sufficient to give the proof for the case in which of dyadics

the product consists of two dyadics.

To show

LINEAR VECTOR FUNCTIONS

5F-1

0~l

Hence

y-1 }

( ?F

?F)

and W~

0- 1 )

JF- 1

(

(P"

0~ l

must be

293

0. 0-l

= I.

reciprocals.

That

The proof for any number of dyadics may be given same manner or obtained by mathematical induction. Definition

of times,

The products

of a dyadic

itself are called

and so

= .

in the

taken any number and are denoted in

<P,

powers of

the customary manner.

is,

0*,

forth.

Theorem

The

the reciprocal of

reciprocal of a

power

<P is

the power of

<P.

(0)-i

= (0- )" = 1

(37)

The

proof follows immediately as a corollary of the preced The symbol <P" n may be interpreted as the theorem. ing or as the reciprocal of nth power of the reciprocal of the nth power of 0. If be interpreted as an operator determining a trans formation of space, the positive powers of correspond to

repetitions of the transformation.

The negative powers of The idemfactor

correspond to the inverse transformations.

that is, no trans corresponds to the identical transformation formation at all. The fractional and irrational powers of CP will

not be defined.

single-valued.

square roots infinite

They

are seldom used

For instance the idemfactor 1.

But

has the two

in addition to these it has a

system of square roots of the form <P

and are not

= -ii +

kk.

doubly

VECTOR ANALYSIS

294

Geometrically the transformation

a reflection of space in the j k-plane. This transformation replaces each figure by a symmetrical figure, symmetrically situated upon the opposite side of the j k-plane. The trans is

formation

sometimes called perversion.

The idemfactor

has also a doubly infinite system of square roots of the form

Geometrically the transformation r

= V.T

This transformation replaces each about the i-axis through an angle rotated equal

a reflection in the i-axis. its

by 180. The idemfactor thus possesses not only two square roots but in addition two doubly infinite systems of square

figure

;

roots

and.

;

means

will be seen (Art. 129) that these are

by no

all.

111.]

The

conjugate of a dyadic has been defined (Art. 99)

as the dyadic obtained

by interchanging the antecedents and and the notation of a subscript

consequents of a given dyadic

C has

been employed.

The equation r

has been demonstrated.

= 4>

(9)

The following theorems concerning

conjugates are useful.

Theorem dyadics

The conjugate equal to the sum :

(d>

The conjugate

of the

sum

or difference of

two

or difference of the conjugates,

=0 C

of a product of dyadics is equal to the product of the conjugates taken in the opposite order.

Theorem

LINEAR VECTOR FUNCTIONS

295

It will be sufficient to demonstrate the theorem

the product contains

two

(d>.T)c =W c .0 Ct .

W) c

r = r

<P)

power

V) c =

(4>

The conjugate

= <P C

W = Vc

Hence Theorem

in case

To show

factors.

(40)

4>)

5F,

<P)

<^.

c .4> c

of the

power

of a dyadic is the

of the conjugate of the dyadic.

a corollary of the foregoing theorem. The expression c may be interpreted in either of two equal ways. Theorem The conjugate of the reciprocal of a dyadic is

This

equal to the reciprocal of the conjugate of the dyadic.

= For

(@~

The idemfactor

)c its

= (&

own

(42)

= Ic = I-

conjugate as

may

be seen from

the nonion form.

= ii +

Hence

(^c)"

Hence

two equivalent ways

+ kk

C^)

The expression ^ c-1 may

j j

= (*"%

therefore be interpreted in either as the reciprocal of the conjugate

or as the conjugate of the reciprocal. Definition:

If a

to be self-conjugate.

dyadic

If it is

equal to

its

conjugate,

it is

equal to the negative of

its

said

con-

VECTOR ANALYSIS

296

For se//-conjugate

said to be anti-self-conjugate.

it is

jugate,

dyadics.

For anti-self-conjugate dyadics

Theorem

way

Any

dyadic

two parts

into

may

00.

be divided in one and only one is self-conjugate and the

which one

other anti-self-conjugate.

But

0=5(0+0,) + 2 (0-0c). (& + c) c = c +<p cc =: 4> c +

and

For

|(^~

d>,

- 0<,) c = $ - <P CC = 0<, _ 0.

Hence the part |(0 4> c

(43)

&c)

anti-self-conjugate.

accomplished in one way.

and the part the division has been

self-conjugate;

Thus

Let

and

in another way were possible to decompose into a self-conjugate and an anti-self-conjugate part. Let

Suppose

then

Where Hence

Hence

(0 if

conjugate.

;

= (0 + J2) + (0"-). + c= + 0) = (0 + ), = -f J2) is

(0"

J2)

self-conjugate,

fi is

self-conjugate.

anti-self-conjugate

anti-self-

LINEAR VECTOR FUNCTIONS

Any gate

dyadic which is

negative and consequently vanishes.

its

and the division

zero

both self-conjugate and anti-self-conju

equal

Hence

297

into

two parts

unique.

Anti-self-conjugate Dyadics.

In case

112.]

The Vector Product

any dyadic the expression

If should be en gives the anti-self-conjugate part of 0. n is antinself-con 0". to Let therefore jugate equal tirely

be any anti-self-conjugate dyadic,

Suppose

$c = 20"

But

But by

lar-hbmr a r a r = bm r mb r = nc r = cn r r

definition

<P x

Hence r

+ bm

mb +

= 0" c

x m) x

nc,

(cxn)xr.

b x

4- c

= axl-fbxm4-cxn. r

=-~

= - 0"

= I0 X

may be stated in a theorem as Theorem The direct product of any results

mbr + cnr

Hence

The

= al-hbm-fcn,

=- \ r

follows.

anti-self-conjugate

dyadic and the vector r is equal to the vector product of minus one half the vector of that dyadic and the vector r.

VECTOR ANALYSIS

Theorem

anti-self-conjugate dyadic <P possesses one It is a uniplanar dyadic the plane of of nullity. degree :

Any

whose consequents and antecedents the vector of

perpendicular to

<P X ",

<P.

This theorem follows as a corollary from equations (44). Theorem : Any dyadic may be broken up into two parts

which one is self-conjugate and the other equivalent to used in cross multiplication. minus one half the vector of of

Any

= <P

or symbolically 113.]

<P X

(45)

vector c used in vector multiplication defines a

For

linear vector function.

cx(r + s)=cxr + cxs. Hence

dyadic.

must be

possible to represent the operator c

as a

This dyadic will be uniplanar with plane of

antecedents and consequents perpendicular

so that

be found as follows

I- (c x !)

(31) (I

=I Hence

and

This

may

= (I c = r

be stated in words.

(! r

= {I

1, -

I)}

= (e x I) c) = r (c x I). r

The dyadic may

will reduce all vectors parallel to c to zero.

its

(46)

LINEAR VECTOR FUNCTIONS Theorem : The vector a vector r

used as prefactors

and

If c precedes r the dyadics are to be

follows

if c

;

The dyadics

as postfactors.

I are anti-self-conjugate.

In case the vector

used in vector multiplication with x c or c X I used in direct

equal to the dyadic I

multiplication with

299

c is a unit vector the application of the

any vector r in a plane perpendicular to c is to equivalent turning r through a positive right angle about the axis c. The dyadic c X I or I x c where c is a unit vector operator c

therefore turns

any vector

perpendicular to

right angle about the line

c as

If r

axis.

c through a were a vector

lying out of a plane perpendicular to c the effect of the dyadic I X c or c x I would be to annihilate that component of r which is

and turn that component

parallel to c

of r

which

perpen

dicular to c through a right angle about c as axis. If the dyadic be applied twice the vectors perpendicular to r are rotated

direction.

through two right angles.

are reversed in

They

If it be applied three times they are turned

through

three right angles. Applying the operator I x c or c X I four times brings a vector perpendicular to c back to its original position.

The powers (I

x 5

of the dyadic are therefore

=-I

x 6

x I)

law as far as

its

- cc),

=-c x = I - c c,

It thus appears that the dyadic I

c or c

(47)

x I

obeys the same

powers are concerned as the scalar imaginary

1 in algebra.

The dyadic

Ixc orcxlisa

For vectors

parallel to c it acts

avoid this effect

and obtain a true

vectors perpendicular to as an annihilator.

quadrantal versor only for

VECTOR ANALYSIS

300 quadrantal versor for

vectors r in space it is merely neces sary to add the dyad c c to the dyadic I X c or c X I. all

X = Ixc +

= =

The dyadic

idemfactor.

The quadrantal

imaginary plete

and

conjugate

two

;

and i, j,

+ 2cc, iXc + cc,

The dyadic X is com x c is anti-self-

parts of which I

c c, self-conjugate.

k are three perpendicular unit vectors

xj-j

jk,

= ik-ki,

Ixk=k x I=ji may

(48)

appears as a fourth root of the versor is analogous to the

Ixi = ixl = kj

cc,

a scalar algebra.

1 of

consists of

114.]

therefore

= cxI +

(49)

ij,

be seen by multiplying the idemfactor

and k successively.

These expressions represent quadrantal versors about the axis i,j, k respectively combined into

i, j,

with annihilators along those axes. They are equivalent, i to in direct when used x, jx, k X respectively, multiplication,

jj,

The expression j,

an idemfactor for the plane of but an annihilator for the direction k. In a similar (I

ner the dyad k k

an idemfactor for the direction

and

man

but an

LINEAR VECTOR FUNCTIONS

301

These partial

annihilate! for the plane perpendicular to k.

idemfactors are frequently useful. If a, b, c are any three vectors and a , V,

the reciprocal

system,

aa + bb used as a prefactor is an idemfactor for all vectors in the plane of a and b, but an annihilator for vectors in the direc tion

Used

as a postfactor it

an idemfactor for

all vectors,

and V, but an annihilator for vectors In like manner the expression

in the plane of a

direction c

in the

used as a prefactor is an idemfactor for vectors in the direction c, but for vectors in the plane of a and b it is an annihilator.

Used

as a postfactor

direction c

an idemfactor for vectors

the

but an annihilator for vectors in the plane of a

and V, that is, for vectors perpendicular If a and b are any two vectors (a

x b) x

x (a x b)

= ba - ab.

(50)

For {(a

x b) x I}r =

x b) x

= bar

T = (ba

ab>r.

The

vector a x b in cross multiplication is therefore equal to the dyadic (b a a b) in direct multiplication. If the vector is

used as a prefactor the dyadic must be so used. (a xb) r

This

x b)

(b a

a b)

(ba

- ab).

a symmetrical and easy form in which to

the formula for expanding a triple vector product.

(51)

remember

VECTOR ANALYSIS

302

Reduction of Dyadics 115.]

be any complete dyadic and let r be a unit the vector r

Let

Then

vector.

a linear function of

Normal Form

When

r takes

values consis

all

that is, when the terminus being a unit vector the vector r of r describes the surface of a unit sphere,

tent with

its

and

varies continuously

surface

to a

sum

terminus describes a surface.

This

It is in fact

is closed.

Theorem

its

ellipsoid.

always possible to reduce a complete dyadic of three terms of which the antecedents among

themselves and the consequents among themselves are mutu This is called the normal form of 0. ally perpendicular. <P

= ai

+ ck

demonstrate the theorem consider the surface described

0-r.

this is a closed surface there

which makes

must be some

direction of r

maximum

or at any rate gives r as great a value as it is possible for r to take on. Let this direction of r be called i, and let the corresponding direction of r r

the direction in which r takes on a value at least as great as be called a. Consider next all the values of r which any in a plane perpendicular to

lie

of r lie in a plane 1

This

may

be proved as follows r

Hence

By expressing degree. is

Hence

the ellipsoid.

owing

.r=l=:

r, r

The corresponding values

to a fact that :

r^*-1 (*

r is a linear vector

= r .*c l

-i.* - 1)-

nonion form, the equation r

describes a quadric surface.

-1

=r V r r = is seen to be of the second .

The

only closed quadric surface

LINEAR VECTOR FUNCTIONS Of

303

must be at least as great this b and Call let the any corresponding direction of r be called j. Finally choose k perpendicular to i and j function.

these values of r one

other.

upon the

positive side of

plane of

and

Let

be the

r = k.

value of r which corresponds to Since the dyadic it k into c be a, b, i, j, changes may expressed in the form

= ai +

show

It remains to

ck.

that the vectors

as determined

a, b, c

above are mutually perpendicular.

= (ai + = (ai

When

-f-

r is parallel to

perpendicular to di perpendicular to

ck)-r,

ck) -dr,

Since r

j-dr +

di +

maximum and

ck-

r is parallel to

dr.

hence must be

a unit vector

Hence when

always

kdr = 0.

vanishes, and if Hence when r is But when parallel to i, r is perpendicular to both b and c. r is parallel to i, r is parallel to a. Hence a is perpendicular Consider next the plane of j and k and the to b and c. plane of b and c. Let r be any vector in the plane of j and k. If

further is

perpendicular to j, r b vanishes. perpendicular to k, r is

dr r

-dr

When

r takes the

hence

=r value

-b j,

+ ck)r,

(bj

+ ck)

(bj

perpendicular to dr

f .

-f r

dr, c

k-dr.

maximum

in this plane

and

Since r is a unit vector it is

VECTOR ANALYSIS

304

Hence when and

perpendicular to dr. is

perpendicular to

Hence

But when

c is zero.

parallel

r is parallel to

takes the

Consequently b is perpendicular to c. It has therefore been shown that a is perpendicular to b and

value

and that b

perpendicular to

denoted by

J, c

Then

where

Consequently the three

are mutually perpendicular.

antecedents of

= ai

the dyadic

+bj

takes the form

+ck

(52)

are scalar constants positive or negative.

Theorem: The complete dyadic

116.]

They may be

reduced to a

sum

of three dyads

may always

whose antecedents and

whose consequents form a right-handed rectangular system of unit vectors and whose scalar coefficients are either all positive or all negative.

&= The proof of made on page 20

(ai

fcj j

+ ck

k).

(53)

the theorem depends upon the statements if one or three vectors of a right-handed

that

system be reversed the resulting system is left-handed, but if two be reversed the system remains right-handed. If then one of the coefficients in (52)

other two axes are negative.

may

two

the directions of the

negative, the directions of the be reversed. Then all the coefficients is

of the coefficients in (52) are negative,

two vectors

which they belong may

be reversed and then the coefficients in

Hence the

are all positive.

any case the reduction to the form in which all are positive or all are negative has been

coefficients

performed.

As a limiting case between that in which the coefficients are all positive and that in which they are all negative comes

LINEAR VECTOR FUNCTIONS them

the case in which one of

takes the form <P

and

The

planar.

= ai +&j i

a and

coefficients

The dyadic then

zero.

305

(54)

may always

be taken

positive. By a proof similar to the one given above it is show that any planar dyadic may be reduced to to possible The vectors i andj are perpendicular, and the this form. vectors i and j are likewise perpendicular.

might be added that

in case the three coefficients a,

&, c

in the reduction (53) are all different the reduction can be

performed in only one way. If two of the coefficients (say a and 6) are equal the reduction may be accomplished in an

ways in which the third vector k is always the same, but the two vectors i j to which the equal coeffi cients belong may be any two vectors in the plane per

number

infinite

pendicular to k.

all

these reductions the

three scalar

have the same values as in any one of them. 6, c are all equal when $ is reduced

coefficients will

If the three coefficients a, to the

normal form

may

space.

are the

same

may be accomplished The three vectors ways.

(53), the reduction

in a doubly infinite

number

be any right-handed rectangular system in these reductions the three scalar coefficients

all of

as in

any one of them.

These statements

will

not be proved. They correspond to the fact that the ellipsoid which is the locus of the terminus of r may have three different principal axes or it

maybe an

ellipsoid of revolution,

or finally a sphere.

Theorem

Any

self -con jugate

the form

where Let

a, &,

= aii + and

dyadic

may

be expressed in

+ ckk

&jj

(55)

are scalars, positive or negative. <P

= ai

-f

+ 20

6jj

+ck

+ ckk

(52) ,

VECTOR ANALYSIS

306

0.0 c =a*i

&2j j

+c a k k + c 2 kk.

= 0^

Since

0* I

and

j j

+ kk

0= 0*. n +j j

were not parallel (& 2 a 2 !) would annihilate two vectors i and i and hence every vector in their plane. a 2 I) would therefore possess two degrees of (0* nullity If

and be

But

linear.

it is

this dyadic is not linear.

be parallel.

k and k

In like

are parallel.

if a, 6, c

are different

Hence i and i must planar. manner- it may be shown that j and j The dyadic therefore takes the form It

0= where

apparent that

aii

a, J, c are positive or

bjj

+ ckk

negative scalar constants.

Double Multiplication

Definition : The double dot product of two dyads is 117.] the scalar quantity obtained by multiplying the scalar product of the antecedents by the scalar product of the consequents.

The product

denoted by inserting two dots between the

ab:cd

= a-c bd.

(56)

This product evidently obeys the commutative law

ab:cd 1

The

= cd:ab,

researches of Professor Gibbs upon Double Multiplication are here first time.

printed for the

LINEAR VECTOR FUNCTIONS

307

to the dyads and with regard to the vectors in the dyads. The double dot product of two dyadics is obtained by multiplying the prod uct out formally according to the distributive law into the

and the distributive law both with regard

sum

number

of a

of double dot products of dyads.

a2 b2

a3 b3

W = G! d +

and

...

a 1 b 1 :c 2 d 2

a 1 b 1 :c 3 d 3

+ aab^Cjdj +

a 2 b 2 :c 2 d 2

a 2 b 2 :c 3 d 3

a 3 b 3 :c 2 d 2

a 3 b 3 :c 3 d s

b 1 :o 1 d 1

a 3 b 3 :c 1 d 1

+ a 3 -c 1 +

(56) .

b 2 d2

a 2 -c 3

bg.djH-a3.C2

b 3 -d 2

a 3 .c 3

b 3 .d 3

bg-d!

-f

...............

Definition:

The

double cross product of

which the antecedent

two dyads

---

(66)" is

the

the vector product of the dyad antecedents of the two dyads and of which the consequent is the vector product of the consequent of the two dyads. The of

product

dyads

denoted by inserting two crosses between the

abcd = axc

b x

(57)

This product also evidently obeys the commutative law

= cd

ab,

VECTOR ANALYSIS

308

and the distributive law both with regard to the dyads and with regard to the vectors of which the dyads are composed. The double cross product of two dyadics is therefore defined as the formal expansion

of the product according to the

law into a sum of double cross products of

distributive

dyads.

=a *F = c

a2 b 2

a3 b3

and *

(a^ +

a2 b2

aa b3

a2 b 2

a3b

d3 ) x

...

(c^ +

+ c3 a3 + 1

Ojdj

ax bx

a2 b 2

M! +

a3 b3

cjdj x

^ c3

ajbj

a2 b2

a3 b3

+ a 3 xc 3

xd +a3 xc 2 bgXdj +a 3 xc 3

-fa 2

xc 2

xd2 + a2 xc 3

............

Theorem

The double dot and double

...

...)

(57)

ba b3

...

(57)"

cross products of

two dyadics obey the commutative and distributive laws of But the double products of more than two multiplication. dyadics (whenever they have any meaning) do not obey the associative law. d>

W :0

$>*=*$ *

(<P

The theorem

(58)

(^x)-

sufficiently evident

without demonstration.

LINEAR VECTOR FUNCTIONS Theorem

dyads

The double dot product

309

two fundamental

equal to unity or to zero according as the two

dyads are equal or different.

ij:ki

= i-k

= 0.

Theorem: The double cross product of two fundamental dyads (12)

equal to zero

consequents are equal.

either the antecedents or the

But

neither antecedents nor con

sequents are equal the product is equal to one of the funda mental dyads taken with a positive or a negative sign.

That

is ij

There

*ik

=ix

*ki =i x k

x k

= +jk.

a scalar triple product of three dyads in which the multiplications are double. Let <P, 5T, Q be any exists

three dyadics.

The expression *

WiQ

a scalar quantity. The multiplication with the double must be performed first. This product is entirely in

cross

dependent of the order in which the factors are arranged or Let ab, cd, and ef be

the position of the dot and crosses. three dyads,

ab*cd:ef=[ace] That

[bdf].

(59)

the product of three dyads united by a double cross is equal to the product of the scalar triple product of the three antecedents by the scalar triple product From this the statement made of the three consequents. is,

and a double dot

the dots and crosses be interchanged the order of the factors be permuted cyclicly the two

above follows. or

For

scalar triple products are not altered.

If the cyclic order of

VECTOR ANALYSIS

310

reversed each scalar triple product changes Their product therefore is not altered.

the factors sign.

A dyadic

118.]

may

Let

cross.

= al + bm +

= ss=

+ bm +

(al

a x a

be multiplied by

en)

(al-f

itself

with double

bm +

en)

Ixm + axc Ixn

x b

+bxa mxl+bxb m x m + b + cxa nxl + cxb The products

n x

main diagonal Hence

in the

equal in pairs.

a, b, c

and

m, n are non-coplanar this

may

+ b/m + The product

fl>

a species of power of 0.

The

garded as a square of

notation

mxn

n x

The

vanish.

0<P = 2(bxc mxn+cxa nxl-faxb If

others are

Ixm).

(60)

be written c

will be

(60)

may be re employed been

to represent this product after the scalar factor 2 has

stricken out.

= (bxc mxn + cxa nxl + axb Ixm) (61). = 0*0 ^ J

The

product of a dyadic expressed as the three dyads with itself twice repeated is triple

</>*$: <P

sum

:0=(bxc mxn-fexa nxl + axb Ixm) :

(al

+ bm +

en).

In expanding this product every term in which a letter is repeated vanishes. For a scalar triple product of three vec-

LINEAR VECTOR FUNCTIONS tors two of which are equal reduces to three terms only

:0=[bca] [mnl] +

311

Hence the product

zero.

[cab] [nlm]

[abc] [linn]

= 3 [a b c] [Imn] 2 0*0:0 = 6 [abc] pmn].

The

product of a dyadic by

triple

itself

twice repeated

equal to six times the scalar triple product of its antecedents multiplied by the scalar triple product of its consequents. is a species of cube. It will be denoted by 8

The product

after the scalar factor 6 has

been stricken out.

0*0:0 (62) 119.]

be called the second of

and

;

the third of

0, the following theorems may be stated concerning the seconds and thirds of conjugates, reciprocals, and products.

Theorem

The second

of the conjugate of a dyadic

to the conjugate of the second of that dyadic.

the conjugate

The

equal

third of

equal to the third of the dyadic.

<*,).= <.),

Theorem: The second and third of the reciprocal of a dyadic are equal respectively to the reciprocals of the second

and

third.

<*-

(*,)-!=*,*

(f ). = (*,)->-*.1

Let <p- 1

= al + bm + cn = a + m b +n c l

+bm +cn n

]

(36)

VECTOR ANALYSTS

812

(*)

-1

c ] [!

[a b c]

But

=1 (0,)-* =

and

[a b c ] [a be]

Hence

]

(0-

+ m b + n c)

m n] = 1. [1 m n ] [Imn] = 0,-*. [1

).,

=[abc] [Imn], [abc] [Imn]

C^-Oa = IX W] [1 m n ]. (0,)- = (*-), = 0,-*. 1

Hence

Theorem: The second and third of a product are equal respectively to the product of the seconds and the product of the thirds.

(f.f),

= *,.*,

(0. *),= *,

^3-

Choose any three non-coplanar vectors 1, m, n as consequents and let 1 m n be the antecedents of W. ,

= al + b m + en, ?T = d + m e + n r = ad + be + cf, ( W\ =bxc exf + cxa fxd + axb dxe, <P = bxc mxn-fcxa nxl + axb Ixm, ?T = m x n e x f + n x f x d + x m dxe. = bxc exf + cxa fxd + axb dxe. Hence 2 2 ?F Hence (# 5T) = <? (^. JT) 8 = [abc] [def] <P

5P*

LINEAR VECTOR FUNCTIONS

313

= [abc] [Imn], rg = P m n ] [defj. = [a be] [def]. 8 z 8

Hence Hence

(0.F),=

Theorem

The second and

equal respectively to the

Theorem

The

power of a dyadic are and third of

of the second

(*"),

= W=0,"

The second

third of a

powers

the dyadic.

y,.

of the idemfactor

third of the idemfactor

the idemfactor.

unity.

(6T)

1=1 lg

Theorem: The product of the second and conjugate of a dyadic is idemfactor.

equal

0,=

a <P 2

$<,

m n]

The antecedents

a, b, c of

be non-coplanar.

Then

a b

Hence 120.]

Let a dyadic three dyads of

non-coplanar.

+ mb + c

[abc] <& c

be given.

Ixm,

nc,

x a b+

the dyadic

x b

sum

(68)

8 I,

mxn + cxa nxl + axb

C ;= <P

the third and the

the product of

may

[ab

x b

c).

be assumed to

c] (a a

c c)

^> 3 1

Let

be reduced to the

which the three antecedents are

VECTOR ANALYSIS

314

= al + b m +

cn,

mxn + cxa nxl + axb

x m,

[Imn]. Theorem: The necessary and sufficient condition that a be complete is that the third of be different from dyadic zero.

For the

was shown (Art. 106) that both the antecedents and

consequents of

Hence

two

the

a complete

are

dyadic

scalar triple products

non-coplanar.

which occur in

cannot vanish.

Theorem: The necessary and sufficient condition that a shall vanish but the dyadic $ be planar is that the third of second of <P shall not vanish.

was shown (Art. 106) that if a dyadic be planar its con n must be and 1, m, sequents planar conversely if the conse It

quents be coplanar the dyadic <P

dyadic

must vanish.

Hence

planar.

But $2 cannot

for a planar

vanish.

Since

have been assumed non-coplanar, the vectors b x c, c x a, a x b are non-coplanar. Hence if 2 vanishes each of the

b, c

vectors

mxn, nxl, Ixm vanishes

linear.

But

and not

linear.

that

is, 1,

m, n are

this is impossible since the dyadic

col-

planar

Theorem: The necessary and sufficient condition that a non-vanishing dyadic be linear is that the second of 0, and consequently the third of 0, vanishes. For if be linear the consequents

Hence <P

their vector products vanish

vanish.

each of

must be zero and hence these consequents of the third,

are collinear.

and the consequents

If conversely <P2 vanishes,

The vanishing

its

consequents

are collinear.

unaccompanied by the vanish

ing of the second of a dyadic, implies one degree of nullity.

The vanishing

of the second implies

two degrees

of nullity.

LINEAR VECTOR FUNCTIONS The vanishing results

may

of the dyadic itself

315

complete nullity.

The

be put in tabular form.

^0,

complete. is

0, <P

planar.

It follows immediately that the third of

(69)

is linear.

any anti-self-conjugate For any such

dyadic vanishes; but the second does not. dyadic is planar but cannot be linear.

Nonion Form.

Invariants of a Dyadic

be expressed in nonion form

121.]

Determinants. 1

= au ii + + The conjugate

a81 ki

a 12

i j

a 32 kj

a 18 ik

(13)

a 33 kk.

has the same scalar coefficients as 0, but

they are arranged symmetrically with respect to the main diagonal.

Thus

( 70 )

The second of $ may be computed. Let

term.

What

Take, for instance, one

be required to find the coefficient of ij in

C? . 2

can yield a double cross product equal to vector ij? product of the antecedents must be i and Hence the the vector product of the consequents must be j. terms in

The

antecedents must be

and k

These terms are 021J

a 31 k i

;

and the consequents, k and

= - a2l a 33 k = a 31 a 23

x33 kk J

a 23 j

i j.

1 The results hold only for determinants of the third order. determinants of higher orders is through Multiple Algebra.

The extension

VECTOR ANALYSIS

316

Hence the term

This

the

first

minor of a*12 19 in the determinant a

This minor

taken with the negative sign. That is, the coefficient of i j in 2 is what is termed the cofactor of the is

coefficient of

in the determinant.

The

cofactor

merely minor taken with the positive or negative sign according as the sum of the subscripts of the term whose first minor is under consideration is even or odd. The co the

first

any dyad in 2 is easily seen to be the cofactor of corresponding term in $. The cofactors are denoted

efficient of

the

generally by large letters. is

the cofactor of a*

the cofactor of a 12

the cofactor of a 32

n With

this notation the

second of

becomes

The value of the third of sum of three dyads

may

(71)

be obtained by writing

as the

(a n

a 21

a sl k)

+ (a 12 i + a 22 j + + (a 13 i + a 23 j +

a 32 k) j a 33 k)k

LINEAR VECTOR FUNCTIONS ^3

This

For and

[Oil

21 J

"31

(21 *

idea of the

most

a 33 k )

(72)

determinant

(72)

very natural when

the other hand

be expressed in that form the conception of

the third of $,

The

frequently called the determinant of

regarded as expressed in nonion form. unless

22 J

written <P 3

The

easily seen to be equal to the determinant

this reason is

317

reciprocal of a dyadic in nonion form

may

be found

easily by making use of the identity

.</> c

(68)

Hence 0"1

more natural.

= (73)

VECTOR ANALYSIS

318 If the

determinant be denoted by

(73)

a second dyadic given in nonion form as

6 31

& 32

& 33

kk,

W of

the two dyadics may readily be found by actually performing the multiplication

the product

On +

U+

a 12

6 32) :

621

a 18 a

6 81 ) ii

( a 31 6 12

W = an in +

(a u

a 32

6 23

a 12 6 12

6 12

a la 622

a 33

& 33>

o 18 J 18

6 21 31

Since the third or determinant of a product is equal to the product of the determinants, the law of multiplication of

determinants follows from (65) and

(74).

LINEAR VECTOR FUNCTIONS a.

"11

"21

"11

a 22

a23

Ojrtn

dinn

6 19 "12

"31

&12

&22

6 23

"21

& 32

6 33

^31^

a 32

*32

(

"11 "13

a 12

& 13

6 31 13

*21

6 22

319

a 33

(76)

The rule may be stated in words. To multiply two deter minants form the determinant of which the element in the mth row and nth column

the

sum

elements in the rath row of the

column

of the products of the

first

determinant and nth

of the second.

= al

If <?

= bxc mxn + cxa nxl + axb Ixm.

Then I

=(^2 ) 3 =

Hence

cxa axb]

[bxc

= (<P2 ) =

[a b c]

2 [1

[mxn nxl Ixm]

m n] 2 =

n "22 33

the third order

122.]

dyadic

<P,

has three scalar invariants

the scalar of the second of

or determinant of 0. quantities are

(77)

which are independent expressed. These are

the scalar of

of the cofactors of a given determinant of equal to the square of the given determinant.

three scalar quantities

which

*ia 22

The determinant

that

of the

<P,

form in

and the third

be expressed in nonion

form these

VECTOR ANALYSIS

320

(78) *11

*18

hi 32

matter in terms of what right-handed rectangular system may be expressed these quantities are

of these unit vectors

The

the same. in the

sum

scalar of

the

first

sum

the

The

main diagonal.

of the three coefficients

scalar of the second of

the

minors or cofactors of the terms in the

main diagonal. The third of is the determinant of the These three invariants are by far the most coefficients. important that a dyadic Theorem : Any dyadic the three invariants

may

0%&

x a 12

#22

#31

#32

a 13

#23

#33

be seen by actually performing the expansion. __

x i) 2

This equation scalar x.

That

x I) 3

(<p

dyadic

cubic equation of which are the coefficients.

(0-xI\*(0-xY) c = (0-xl\

(68)

Hence as

possesses. satisfies a

f<p

_ x !)<; =

0^ +

an identity holding for

It therefore holds,

all

x*.

values of the

in place of the scalar x, the

which depends upon nine scalars be substituted.

But the terms upon the

left are identically zero.

Hence

LINEAR VECTOR FUNCTIONS

321

This equation may be called the Hamilton-Cayley equation. Hamilton showed that a quaternion satisfied an equation analogous to this one and Cayley gave the generalization to A matrix of the Tith order satisfies an algebraic matrices. equation of the nth degree.

The analogy between

the theory

In fact, of dyadics and the theory of matrices is very close. a dyadic may be regarded as a matrix of the third order and conversely a matrix of the third order may be looked upon as addition and multiplication of matrices and dyadics are then performed according to the same laws. of higher generalization of the idea of a dyadic to spaces

The

a dyadic.

dimensions than the third leads to Multiple Algebra and the theory of matrices of orders higher than the third.

SUMMARY OF CHAPTER

vector r

said to be a linear function of a vector r

the components of r are linear homogeneous functions Or a function of r is said to be a of the components of r.

when

linear vector function of r

two vectors

the

sum (ri

when the function

of the

sum

of the functions of those vectors.

)=f(r

f(r a ).

(4)

These two ideas of a linear vector function are equivalent. A sum of a number of symbolic products of two vectors,

which are obtained by placing the vectors in juxtaposition without intervention of a dot or cross and which are called is called a dyadic and is represented by a Greek dyads,

function of dyadic determines a linear vector a vector by direct multiplication with that vector

capital.

= &1 -

a2 b2

a2 b2 21

+ r

a3 b3

r H

(7) (8)

VECTOR ANALYSIS

322

Two

dyadics are equal when they are equal as operators upon all vectors or upon three non-coplanar vectors. That is,

when <P

= W

r for all values or for three non-

coplanar values of or

(10)

for all values or for three non-

=s W

r for all values or for three

coplanar values of r and

non-

may be represented by a dyadic. the law of multiplication with distributive Dyads obey regard to the two vectors composing the dyad

Any

linear vector function

+ b + c+

) (1

+m+n+

...)

= al + am + an+ +

+ bm + bn +

+ cm + en + (11)

Multiplication by a scalar is associative. In virtue of these two laws a dyadic may be expanded into a sum of nine terms

by means of the fundamental dyads, ii,

ij,

ik,

ji,

Jj,

Jk,

ki,

kj,

kk,

(12)

= a n ii + a 12 + a 18 ik, = ai + <*22 + k = a 31 k + a 82 k + a 33 k k.

J *

<*23 J

( 13 )

two dyadics are equal the corresponding coefficients in their expansions into nonion form are equal and conversely If

LINEAR VECTOR FUNCTIONS Any

dyadic

which

the

323

may be expressed as the sura of three dyads of antecedents or the consequents are any three This expression of the dyadic

given non-coplanar vectors.

unique.

The symbolic product ab known general product of scalar

two vectors

as a dyad is the most which multiplication by a

It is called the indeterminate product.

associative.

The product imposes

upon the vectors a and

five conditions

Their directions and the product of their lengths are determined by the product. The scalar and vector products b.

indeterminate product. A scalar and from any dyadic by inserting a dot be obtained may and a cross between the vectors in each dyad. This scalar are functions of the

a vector

and vector are functions

of the dyadic.

= b8 + b + a *! + a x b + a x b + X = &1 x bj + a 0, = i-0.i + j*0-j + k 0*k = a n -f a 22 + #339 i - i k-k j) i + (k

= (j

(i-

0i)

0-j -j

(18) (19)

(20)

(21)

The direct product of two dyads is the dyad whose ante cedent and consequent are respectively the antecedent of the first dyad and the consequent of the second multiplied by the scalar product of the consequent of the the antecedent of the second.

(ab)

(c d)

first

dyad and

a/ *T

(23)

the formal expansion, into the according to the distributive law, of the product

The

direct product of

two dyadics

VECTOR ANALYSIS

324

sum

Direct multiplication of dyadics or of dyadics and a vector at either end or at both ends obeys the distributive and associative laws of multiplication. Con of products of dyads.

sequently such expressions as

Q.W.T,

s.0-?

$>.W.Q

s.^.^.r,

(24)-(26)

may be written without parentheses; for parentheses may be inserted at pleasure without altering the value of the product. In case the vector occurs at other positions than at the

end the product

The skew product

no longer associative. dyad and a vector may be defined

of a

by the equation (ab) x

x (ab)

The skew product

= a b x r, = r x a b.

of a dyadic

(28)

and a vector

equal to the

formal expansion of that product into a sum of products of dyads and that vector. The statement made concerning the associative law for direct products holds when the vector is

connected with the dyadics in skew multiplication.

The

expressions r

^ x r,

?F,

$ x

(29)

be written without parentheses and parentheses may be inserted at pleasure without altering the value of the product.

may

Moreover s

<P)

= (s

<p.(rx

?P)

<P,

(<P

(0 x

s),

(31)

But the parentheses cannot be omitted. The necessary and sufficient condition that a dyadic may be reduced to the sum of two dyads or to a single dyad or to zero is that, when expressed as the sum of three dyads of which the antecedents (or consequents) are known

LINEAR VECTOR FUNCTIONS

325

to be non-coplanar, the consequents (or antecedents) shall be respectively coplanar or collinear or zero. A complete dyadic is one which cannot be reduced to a sum of fewer

than three dyads.

planar dyadic

one which can be

reduced to a sum of just two dyads. A linear dyadic which can be reduced to a single dyad.

one

A is

complete dyadic possesses no degree of nullity. There no direction in space for which it is an annihilator. A -

planar dyadic possesses one degree of nullity. There is one direction in space for which it is an annihilator when used as a prefactor and one

when used

dyadic possesses two degrees

as a postfactor.

nuljity.

linear

There are two

independent directions in space for which it is an annihilator when used as a prefactor and two directions when used as a

A zero dyadic

possesses three degrees of nullity It annihilates every vector in space. or complete nullity. The products of a complete dyadic and a complete, planar, postfactor.

or linear dyadic are respectively complete, planar, or linear. The products of a planar dyadic with a planar or linear dyadic

where between the consequents of the first dyadic and the antecedents of the second introduce one more degree of nullity into the product. The product of a are respectively planar or linear, except in certain cases relations of perpendicularity

linear dyadic

by a

linear dyadic

in general linear

;

but in

case the consequent of the

first is perpendicular to the ante cedent of the second the product vanishes. The product of

any dyadic by a zero dyadic is zero. A dyadic which when applied to any vector in space re produces that vector is called an idemfactor. All idemfactors are equal

and reducible

= ii + I = aa + I

Or The product

to the

form jj

kk.

bb + cc

(33) (34)

any dyadic and an idemfactor

that dyadic.

VECTOR ANALYSIS

326

product of two complete dyadics is equal to the idemfactor the dyadics are commutative and either is called If the

the reciprocal of the other. complete dyadic may be canceled from either end of a product of dyadics and vectors as in ordinary algebra ; for the cancelation is equivalent to multiplication by the reciprocal of that dyadic.

dyadics possess no reciprocals.

They correspond

The reciprocal

of a product

Incomplete to zero in

equal to the ordinary algebra. product of the reciprocals taken in inverse order.

(0.

5F)-

5F-1

0-i.

(38)

The conjugate

of a dyadic is the dyadic obtained by inter The order the of the antecedents and consequents. changing the of the con a is to of product equal product conjugate

jugates taken in the opposite order.

(0.

9%=

(40)

The conjugate of the reciprocal is equal to the reciprocal of the conjugate. dyadic may be divided in one and only one way into the sum of two parts of which one is self-

conjugate and the other anti-self-conjugate.

anti-self-conjugate dyadic or the anti-self-conjugate any dyadic, used in direct multiplication, is equivalent to minus one-half the vector of that dyadic used in skew

Any

part of

multiplication.

T=-j0

xr, (44)

A dyadic of the form c

I or I

used in direct multiplication used in skew multiplication.

c is anti-self-conjugate

and

equivalent to the vector o

LINEAR VECTOR FUNCTIONS Also

= (I x c) <P = (I x c)

= (c x I) = (c x I)

327 (46)

The dyadic c X I or I x c, where c is a unit vector is a quadran tal versor for vectors perpendicular to c and an annihilator for vectors

parallel to

The dyadic Ixc + ccisa true The powers of these dyadics

quadrantal versor for all vectors.

behave like the powers of the imaginary unit V^l, as ma y be seen from the geometric interpretation. Applied to the unit vectors

i, j,

= kj - j k,

etc.

The vector a x b in skew multiplication (a x b) X I in direct multiplication.

(ax

(a r

x (ax

x b)

(49) equivalent to

b)=ba-ab

(50)

= (b a a b) r = r (b a - ab).

(51)

complete dyadic may be reduced to a sum of three dyads of which the antecedents among themselves and the

among themselves each form

a right-handed rectangular system of three unit vectors and of which the scalar coefficients are all positive or all negative.

consequents

0= This

(ai

called the normal

plete dyadic

may

Any

+ ck

(53)

k).

form of the dyadic.

incom

The reduction

unique in case In case they are not is

be accomplished in more than self-conjugate dyadic may be reduced to

the normal form 4>

ftj j

are different.

different the reduction

one way.

be reduced to this form but one or more of

the coefficients are zero. the constants a,

may

= aii +

which the constants

a, S, c

6jj

+ ckk,

(55)

are not necessarily positive.

VECTOR ANALYSIS

328

The double dot and double

cross multiplication of dyads

defined by the equations

ab:cd =

abcd = axc

b.d,

(56)

bxd.

(57)

The double dot and double

cross multiplication of dyadics obtained by expanding the product formally, according to the distributive law, into a sum of products of dyads. The double dot and double cross multiplication of dyadics is com is

mutative but not associative. One-half the double cross product of a dyadic If called the second of 0.

<Px

itself

= b xc mxn + cxa nxl+axb Ixm.

(61)

One-third of the double dot product of the second of is

called the third of

and

equal to the product of the

scalar triple product of the antecedents of

and the

scalar

triple product of the consequent of 0.

The second The third of

= \0$ 0:

<P=[abc] [Imn].

of the conjugate

the conjugate

original dyadic.

is is

(62)

the conjugate of the second. equal to the third of the

The second and

third of the reciprocal are

the reciprocals of the second and third of the second and The second and third of a product are the

third of a dyadic.

products of the seconds and thirds.

(*c\

= (*.)*

(65)

LINEAR VECTOR FUNCTIONS The product to the

and conjugate

of the second

329

of a dyadic

^^c=^

conditions for the various degrees of nullity expressed in terms of the second and third of 0.

= 0,

0, <P

equal

(68)

The

and the idemfactor.

product of the third

complete is

= 0, $2 = 0, 3

may

planar

(69)

is linear.

The

closing sections of the chapter contain the expressions (70)-(78) of a number of the results in nonion form and the

deduction therefrom of a number of theorems concerning determinants. They also contain the cubic equation which is satisfied

by a dyadic

03 This

_ Qa

+ 0^

^ _ [

called the Hamilton-Cayley equation.

cients

4>.

<P<i

variants of

and

The

coeffi

are the three fundamental scalar in

<P.

EXERCISES ON CHAPTER

Show

(79)

that the

two

given in Art. 98 for

definitions

a linear vector function are equivalent

Show

that the reduction of a dyadic as in (15) can be accomplished in only one way if a, b, c, 1, m, n, are given. 2.

Show Show

from

zero,

linear 5.

(<P

that

Show

a) c

<Pxr=

line of their

that

= 0.

= -a XT

must equal

then

and the

then

for

(1>

any value

unless both

consequents for

of r different

and

parallel to

are

any three non-coplanar values

330

VECTOR ANALYSIS Prove the statements made in Art. 106 and the con

verse of the statements.

Show

and

that

Q is

are equal.

complete and

Q= W Q

then

Give the proof by means of theory

developed prior to Art. 109. 8.

Definition

Two

dyadics such that

that

two dyadics that are commutative are said to be Show that if homologous. any number of dyadics are homo logous to one another, any other dyadics which may be obtained from them by addition, subtraction, and direct multiplication is

to say,

are

homologous to each other and to the given dyadics. Show homologous dyadics are homolo ~l ~l the statement that if

also that the reciprocals of

gous.

Justify

which are equal, be called the quotient of rules

addition,

governing

subtraction,

?F,

(P,

then the

multiplication

division of homologous dyadics are identical

and

with the rules

it being governing these operations in ordinary algebra understood that incomplete dyadics are analogous to zero, and the idemfactor, to unity. Hence the algebra and higher

analysis of homologous dyadics

practically identical with

that of scalar quantities.

Show

10.

that (I

Show

X $ and

that whether or not

a, b, c

& = c X #.

be coplanar

abxc+bcxa+caxb = [abc]I bxca+cxab+axbc=[abc]L

and

If a, b, c are coplanar use the above relation to prove 11. the law of sines for the triangle and to obtain the relation with scalar coefficients which exists between three coplanar

vectors.

This

may

be done by multiplying the equation by a

unit normal to the plane of a, b, 12.

What

and

the-condition which

coefficients in the

must

subsist

between the

expansion of a dyadic into nonion

form

LINEAR VECTOR FUNCTIONS the dyadic be self-con jugate self-conjugate 13.

What,

331

the dyadic be anti-

Prove the statements made in Art. 116 concerning the of ways in which a dyadic may be reduced to its

number

normal form. 14.

The necessary and

sufficient condition that

be zero

self-conjugate dyadic

an anti-

that the vector of the

dyadic shall be zero. 15.

Show

that

be any dyadic the product

<P C is

self-conjugate. 16.

Show how

make

use of the relation

demonstrate that the antecedents and consequents of a self conjugate dyadic are the same (Art. 116). 17.

Show

that

and

18.

Show

that

+ W\ =

2 </>

= 2

+ 4>*V +

the double dot product

the dyadic vanishes. by for a linear condition dyadic in the f orin

20.

Show

that

Show

that

= <P3 + e- 2 (0 + ?T) 3 = 8 + <P V + (<P

of a dyadic

Hence obtain the

itself vanishes,

19.

= 0.

ef) 3

+ V*

that the scalar of a product of dyadics is changed by cyclic permutation of the dyadics. That is 21.

CHAPTER VI ROTATIONS AND STRAINS 123.]

IN the foregoing chapter the analytical theory of been dealt with and brought to a state of

dyadics has

completeness which

nearly final for practical purposes.

There are, however, a number of new questions which present themselves and some old questions which present themselves under a new form when the dyadic is applied to physics or geometry. Moreover it was for the sake of the applica

them was developed. It is then the object of the present chapter to supply an extended

tions of dyadics that the theory of

application of dyadics to the theory of rotations

and to develop, as

far as

may

and

strains

appear necessary, the further

analytical theory of dyadics.

That the dyadic

$ may

be used to deuote a transformation

of space has already been mentioned.

knowledge of the

precise nature of this transformation, however, was not needed at the time. Consider r as drawn from a fixed origin, and r as

drawn from the same

origin. r

Let now

= 0-r.

This equation therefore may be regarded as defining a trans formation of the points P of space situated at the terminus of r into the point

remains fixed.

Points in the

situated at the terminus of r

the finite regions of space. r

becomes a point

finite

Any

The

regions of space

point upon a line

=b+ xa =$ b+#$

origin

remain in

ROTATIONS AND STRAINS Hence

straight lines go over into

same

333

straight lines

and

lines

line a

go over by the transformation into lines parallel to the same line a. In like manner planes and into the over of planes go quality parallelism is invariant. parallel to the

Such a transformation strain

known

as a homogeneous strain.

of frequent occurrence in Homogeneous physics. For the of deformation the infinitesimal instance, sphere in a fluid

(Art. 76)

a homogeneous strain.

geneous strain

generally

known by

In geometry the homo different names. It is

called an affine collineation with the origin fixed. Or it is known as a linear homogeneous transformation. The equa tions of such a transformation are

= an x + y< = x

Theorem

124.]

If the dyadic

of the points of space

which

gives the transformation due to a homogeneous strain,

the second of 0, gives the transformation of plane areas 2 which is due to that strain and all volumes are magnified by ,

that strain in the ratio of

the third or determinant of

to unity.

= al + bm + cn r = <P.r = al-r-f bm r -f cnr. into a, b, Hence The vectors 1 m n are changed by and m are the planes determined by m and n n and Let

transformed into the planes determined by b and c, c and a and b. The dyadic which accomplishes this result is

$2 Hence due to

b x

mxn + cxa nxl + axb

Ixm.

denote any plane area in space, the transformation replaces s by the area s such that

if s

VECTOR ANALYSIS

334

important to notice that the vector s denoting a plane not transformed into the same vector s as it would

It is

area

if it

denoted a

the latter case

upon

This

line.

acts

evident from the fact that in

whereas in the former case

acts

To show

that volumes are magnified in the ratio of <P Z to choose unity any three vectors d, e, f which determine the volume of a parallelepiped [d e f]. Express with the vec tors

which form the reciprocal system

The dyadic

to d,

f as consequents.

changes d, e, f into a, b, c (which are different m n ). a, b, c above unless d, e, f are equal to 1 Hence the volume [d e f ] is changed into the volume [a b c]. <P

from the

= [abc][dVf] [d e fr^Cdef]. = [d e f] $3 [a b c] 8

Hence

The But

volume [a b cj to [d e f] is as <P3 is to unity. d, e, f were any three vectors which deter mine a parallelepiped. Hence all volumes are changed by ratio of the

the vectors

the action of

in the

same

ratio

and

this ratio is as

Eotations about a Fixed Point.

to 1.

Versors

Theorem : The necessary and sufficient condition that 125.] a dyadic represent a rotation about some axis is that it be reducible to the form

= i i+j j + where

k and

i, j,

are

k k

two right-handed rectangular

systems of unit vectors.

Let

(1)

= #i-f-f-3k

ROTATIONS AND STRAINS Hence

335

reducible to the given form the vectors

if C? is

i, j,

and any vector r is j to its relative i, j, k into the same posi position changed from Hence by the transformation no tion relative to i ,j ,k are changed into the vectors

of shape

change which carries

body

suffers

i, j,

The

effected.

k into

The

vectors

Conversely suppose the

no change of shape

to a rotation.

strain reduces to a rotation

that

is,

suppose

k must be carried

subjected

into another

right-handed rectangular system of unit vectors. Let these be i j , k The dyadic <P may therefore be reduced to the ,

form

Definition

+ j j+k

i i

dyadic which i i

reducible to the form

+jj + k k

and which consequently represents a rotation

called a

versor.

Theorem: The conjugate and reciprocal of a versor are

and conversely

the conjugate and reciprocal of a dyadic are equal the dyadic reduces to a versor or a versor equal,

multiplied by the negative sign.

= i+j

Let

Hence the second part

Hence

first

part of the theorem

proved.

let

= ai + b j + ck, <p c = i*+j b + kc, 4>-i

=<P C

aa4-bb

To prove

the

VECTOR ANALYSIS

336

Hence (Art. 108) the antecedents a, b, c and the consequents Hence (page 87) they a, b, c must be reciprocal systems. must be either a right-handed or a left-handed rectangular system of unit vectors. The left-handed system may be changed to a right-handed one by prefixing the negative

Then

sign to each vector.

(iy

#.*rff,tnt). The unity

;

Hence

third or determinant of a versor

is evidently equal to that of the versor with a negative sign, to minus one. the criterion for a versor may be stated in the form

$ Or inasmuch

= I.

3> n 3

as the determinant of

if (P* (P C =I, it is

(%\ \ /

plus or minus one

only necessary to state that if C

a versor.

There are two geometric interpretations of the transforma such that tion due to a dyadic

__.

j (J/ 1

The transformation due

_j_

k k)

_j_

(3)

one of rotation combined with

The dyadic

reflection in the origin.

i+j

k causes a

it is a versor. The negative in space and vector of direction the reverses then every sign replaces each figure by a figure symmetrical to it with respect

rotation about a definite axis

to the origin.

reversing the directions of

and

the

system i j k still remains right-handed and rectangular, but the dyadic takes the form ,

= i+j j-k i

(i i

-k

) .(i

k).

ROTATIONS AND STRAINS Hence

the

transformation due to

337

a rotation due to

a reflection in the plane of i and j + k k followed by k k causes such a transfor i i + j j the For j dyadic mation of space that each point goes over into a point sym

i+j

metrically situated to it with respect to the plane of i and j Each figure is therefore replaced by a symmetrical figure. transformation that replaces each figure by Definition : .

a symmetrical figure is called a perversion and the dyadic which gives the transformation is called a perversor.

The

criterion for a perversor

that the conjugate of a and that the determi

dyadic shall be equal to its reciprocal

nant of the dyadic shall be equal to minus one. 4>.<P C

Or inasmuch

= I, = I,

C? c

or minus one the criterion -

I0I=-1. the determinant

must be plus

take the form

may

= I,

(3)

a perversor.

from geometrical considerations that the prod uct of two versors is a versor of two perversors, a versor It is evident

;

but of a versor and a perversor taken in either order, a perversor. .

126.]

If the axis of rotation

be the i-axis and

measured positive

of rotation be the angle q

the angle

in the positive

trigonometric direction, then by the rotation the vectors such that i, j, k are changed into the vectors i ,j ,k i

= k =

The dyadic $ tion

i i

cos q j

sin q

sin y,

k cos

k k which accomplishes

this rota

VECTOR ANALYSIS

338

= ii +

cos q (jj

+ kk) +

sin q (k

- jk).

(4)

jj +kk = I-ii, kj-jk = I x i.

Hence

i i

cos q (I

sin q I

,( 5 )

more generally in place of the i-axis any axis denoted unit vector a be taken as the axis of rotation and if as the by before the angle of rotation about that axis be denoted by q, If

which accomplishes the rotation

the dyadic

=aa + To show

that this

rotation apply

a a)

cos q (I

sin q I

(6)

dyadic actually does accomplish the r. The dyad a a is an idemfactor

to a vector

for all vectors parallel to a; but an annihilator for vectors

a a is an idemfactor perpendicular to a. The dyadic I for all vectors in the plane perpendicular to a; but an annihilator for all vectors parallel to a. The dyadic I x a a quadrantal versor (Art. 113) for vectors perpendicular

to a; but

an annihilator for vectors parallel to a

r be parallel to

0.r =

Hence

leaves unchanged which are parallel to vectors) .

Hence

= cos

aar =

If then

all

vectors (or components of

If r is perpendicular to a

q r

sin q a

the vector r has been rotated in

its plane through the in its vector were space component parallel angle q. any but its component perpendicular to a to a suffers no change The whole is rotated about a through an angle of q degrees.

If r

;

therefore rotated about a through that angle. Let a be given in terms of i, j, k as

vector

a l a z ik

ROTATIONS AND STRAINS -r

a 2 a x ji

a z a l ki

Hence

$ = {&J 2

cos #)

a s (1

{

a 1 (1

kk,

- cos 2) + C1

{

{^3^2 (1

cos q)

cosg)

ik,

+ Okk. i i

S i n 2} lj

a a sin ^^ ik

a 3 sin q} ji

S l n 2l J

a 2 sin q}

cos ^)

cos q}

c s ?)

kk,

cos 2)

cos^)

cos #}

"" COS ( a 2 a 3 (1 2)

+ {3

a x kj

a i a 2 (1

{ 2

3 ij

a2 a3

= 0ii-a

+ +

339

j j

= ii + jj +

ajsin^} kj

+ cosg} kk.

(7)

be written as in equation (4) the vector of and the scalar of may be found. If

127.]

+ cos q

xj <P X

<2> s

The

+ cosg

axis of rotation

+k x

2 sin q

sin q (k

x k)

j+k -k) +sing a = 1 + k cos q.

i is

seen to have the direction of

-k),

<P X ,

the negative of the vector of 0. This is true in general. The direction of the axis of rotation of any versor is the

negative of the vector of (P. The proof of this statement depends on the invariant property of $ x Any versor may be reduced to the form (4) by taking the direction of i .

VECTOR ANALYSIS

340

coincident with the direction of the axis of rotation. this reduction has

been made the direction of the axis

After is

seen

But <P X is not altered by the to be the negative of <P X nor is the axis of reduction of <P to any particular form Hence the direction of rotation altered by such a reduction. .

the axis of rotation

The tangent

always coincident with

tion of the negative of the vector of

of one-half the angle of version q sin q

the direc

<?.

cos q

(8) 4> s

The tangent of one-half the angle of version is therefore determined when the values of <# x and <P S are known. The

$ x and

vector

the scalar

(P s ,

which are invariants of

mine completely the versor

<?.

Let

deter

Let the magnitude

in the direction of the axis of rotation.

<P,

be a vector drawn

be equal to the tangent of one-half the angle q of

version.

The

vector

determines the versor

completely,

will be

called the vector semi-tangent of version. By (6) a versor $ was expressed in terms of a unit vector parallel to the axis of rotation. <p

Hence

if ft

There

= aa +

cos q (I

a a)

sin q I

be the vector semi-tangent of version

a more compact expression for a versor

in terms

of the vector semi-tangent of version. Let c be any vector in The version carries represented by ft space. c

c into c

ROTATIONS AND STRAINS show

It will be sufficient to

For

ft.

if c

341

this in case c is perpendicular to

component of it) were parallel to ft the ft x would be zero and the statement

(or any

result of multiplying by

would be that

In the

c is carried into c.

nitudes of the two vectors are equal. (c (c

x c)-=

and

c are

The term

+ftxc-ftxc

ftft

mag

2c-ftxc

ft.c.

by hypothesis perpendicular

+ ftxc.ftxc=:c 2

c-c

place the

cc-hftxc ftxc + 2cftxc

cc + ftxcftxc = cc + Since

first

For

c vanishes.

tan 2

Hence the

\ q).

equality.

second place the angle between the two vectors

In the

equal to

ftxc)(c + ftxc)_cc ft x c ft x c 2 2 c c* (1 + tan 2 i j) (1 + tan -2 q) 2

= cos c

(c c

tan 2

+ ft x c) 1

Hence the cosine and and c + ft x c are equal the angle q

tan 2 i q)

tan i 2

2 c

_ c

= sin

(ft

tan

I j) *

sine of the angle between c ft X c and sine of respectively to the cosine

and consequently the angle between the vectors

must equal the angle

Now

VECTOR ANALYSIS

342

ftXC=(I

and

x Q)

+a x

-I

Ixft)-C

(I 1

tt)-

ft)

-(I-Ixft) =

Ixft.

Multiply by c (I

x a)

x Q)- 1

-Q

-I

X Q)- 1

=c+

a x

Hence the dyadic

carries the vector c

what the value

ft c.

tt)

(10)

x c into the vector c + ft X c no matter Hence the dyadic determines the

version due to the vector semi-tangent of version The dyadic I + 1 x ft carries the vector c

ft.

into

+ ft.ft)c.

+ I X Q)

=c+

ftxc)

ft)

ftxc

=c+Q

ftxc

Qc=

ftx(ftxc)

Q) C

Hence the dyadic 1 carries the vector c

dicular to

ftft ft

c into

the vector

as has been supposed.

c, if

be perpen

Consequently the dyadic

+ Ixft) 2 1 + ft-ft

produces a rotation of all vectors in the plane perpendicular to ft. If, however, it be applied to a vector x ft parallel to ft the result is not equal to x ft.

+ IXQ)-(I + IXQ) *V-* i + O-O

(I I

+ IXQ) + Q.Q

Q v "l

+ Q-

ROTATIONS AND STRAINS To

obviate this difficulty the dyad

for all vectors perpendicular to

merator.

The

versor

may

ft)

(Ixft)-(I xft) Hence substituting

ft)

an annihilator

be added to the

IXft)*

ft-ft

2 1

ft)

= l.ftft-ft.ftl.

may

xft) x

^(l-ft.ft)I This

=i+

which

then be written

ftft+CI 1

Q, ft,

ft,

343

+ +

2ftft ft

+ 2Ixft

be expanded in nonion form.

Let

(11)

128. ]

If a is a unit vector a dyadic of the

form

= 2aa-I

(12)

a liquadrantal versor. That is, the dyadic turns the a of about the axis two space points right angles. through is

This may be seen by setting q equal to

in the general

expression for a versor

cos q (I

a a)

sin q I

it may be seen directly from geometrical considerations. The dyadic <P leaves a vector parallel to a unchanged but re

verses every vector perpendicular to a in direction. Theorem: The product of two biquadrantal versors

versor the axis of which

perpendicular to the axes of the

VECTOR ANALYSIS

344

biquadrantal versors and the angle of which is twice the angle from the axis of the second to the axis of the first

Let a and b be the axes of two biquadrantal

versors.

The

product

=(2bb-I).(2aa-I) is

any two versors Consider the common perpendicular to a and b.

certainly a versor;

a versor.

for the product of

The biquadrantal versor 2 a a I reverses this perpendicular in direction. (2bb I) again reverses it in direction and con sequently brings it back to its original position. Hence the product Q leaves the common perpendicular to a and b un

changed.

The a

therefore a rotation about this line as axis.

cosine of the angle from a to

2 b

a b a

Hence the angle from a to

of the versor

2 (b

- 1 = cos

(b, a).

equal to twice the angle

Theorem

Conversely any given versor

as the product of lie in

-a

two biquadrantal

may be

versors, of

expressed

which the axes

the plane perpendicular to the axis of the given versor to one half the

and include between them an angle equal angle of the given versor. For let Q be the given versor.

Let a and b be unit vectors

Furthermore perpendicular to the axis let the angle from a to b be equal to one half the angle of Then by the foregoing theorem this versor. J? x

of this versor.

J2=(2bb-I).(2aa-I).

(14)

two biquad rantal versors affords an immediate and simple method for compounding two finite rotations about a fixed point. Let

The

and

resolution of versors into the product of

be two given versors.

Let b be a unit vector per-

ROTATIONS AND STRAINS

345

and W. Let a be a unit vector pendicular to the axes of perpendicular to the axis of <P and such that the angle from a to b is equal to one half the angle of 0. Let c be a unit vector perpendicular to the axis of W and such that the angle from b to c is equal to one half the angle of Then .

</>

= (2bb-I).(2aa-I)

$T=(2ec-I).(2bb-I)

V. But

- I)

1)2. (2

- I).

equal to the idemfactor, as may be seen from represents a rotation through four right angles

the fact that

(2 cc

or from the expansion

(2bb-I).(2bb-I) = 4b.b bb-4bb + W <P = (2 c c - I) (2 a a - I). Hence The product

into

a and

and

tangents of

= I.

a versor the axis of which

and the angle of which

perpendicular to one half the angle from a to

equal to

two versors of which the vector semiversion are respectively QJ and ft^ the vector are

semi-tangent of version

a ~

of the product

q 1 + a 2 +a 2

<P is

i-a.-a,

0=(2bb-I) (2aa-I) = (2 c c - I) (2 bb - I). <P = (2cc-I) (2aa-I).

Let

and

iff

ba x

-2aa -2b

= 4a

b b

X a,

<?)

VECTOR ANALYSIS

346

= 4(a.b)

=4c

b cb

rx = 4

(ST.

But

[abc]

= bxc

axb

b b

axe b b

b b

abbc

r-^r

[abc]J b

a b

a* b b

bxc

= axe

+cx

a-bb-c

x (axb)

x b) = " (a

2 aa

2 cc

b b

= (bxc)

Q 2 x Qj

Hence

^=T b

x Q

2 c c

<p) x

= axb-,

= 4 ca c x a, 2 0)^ = 4 (c-a) -l.

(?F.

Hence

2 bb

-l,

= 4(c.b) 2 -l

?r5

a*bb*c

ROTATIONS AND STRAINS

847

This formula gives the composition of two

finite rotations.

If the rotations be infinitesimal

infinitesi

Neglecting infinitesimals of the second order the for

mal.

mula reduces

The

ftj

and Q^ are both

combine according to the law of This demonstrates the parallelogram law for

infinitesimal rotations

vector addition.

angular velocities.

The

If the

60.

Right Tensors, Tonics, and Cyclotonics

icSy

129.]

subject was treated from different

and

standpoints in Arts. 51

dyadic

be a versor

may

be written in the

form (4)

= ii +

cos q (jj

The

axis of rotation

axis

is q.

is i

+ kk) +

and the angle

= ii + .

This

- jk).

of rotation about that

be another versor with the same axis and

Let

an angle of rotation equal to q

Multiplying

sin q (kj

cos q

(j j

+ kk) +

sin q r (kj

+ ? ) (j j + k k) + Bin(j+ 9 )(kj-jk). i i

cos (g

the result which was to be expected

two versors

jk).

the product of

which the axes are coincident

and with an angle equal the two given versors.

the same axis

(16)

a versor with

to the

sum

of the

angles of If a versor be multiplied by itself, geometric and analytic considerations alike make it evident that 2

and

i i

= ii +

cos

(j j

+ kk) +

sin 2 q (k j sin

(kj

- j k), j

k).

VECTOR ANALYSIS

348

On the kj-jk.

hand

other

equal jj

+ kk;

and

<P 2

equal

Then

The product of ii. Hence

= ii +

(i i

cos q

ii into either

let

= ii +

cos n q (PS

(cos q

cos 11

sin q 2

$ 2)n

zero and into itself

sin q

n 2)

1 q sin # fl^""

^ raised to any power reproduces

The dyadic The dyadic

<P 2

itself.

(Pf

= <Pr

raised to the second

2 power gives the negative raised to the third the power, <#! negative of <P2 ; raised to the fourth power, raised to fifth power, <PZ and so the l

;

The dyadic

on (Art. 114).

multiplied by

equal to

Hence

<P . 2

i i

cos n q

nfnl) V ; &n

But Equating

i i

coefficients of

cos

= cos

sm 7i q = TI cos "^

n (n q

and $2

for

~~^TI

sin

q sin q

<P%

two expressions

COS>1

(71-1) (71-2) :

in these

-- 71

j sin q

-2

cos <P

cos n

? sin q

cos n

"# sm^ +

Thus the ordinary expansions for cos nq and sin 715 are obtained in a manner very similar to the manner in which they are generally obtained. The expression for a versor

Let

a,b, c be

may be generalized as follows. three non-coplanar vectors and a V, c , the any ;

reciprocal system. <p

= aa

cos q

Consider the dyadic

(bb

)

sin q

(cb

(17)

ROTATIONS AND STRAINS

349

This dyadic leaves vectors parallel to a unchanged. Vectors in the plane of b and c suffer a change similar to rotation.

Let r

= cos p b +

= cos

This transformation

may

sin

q) b

sin (p

be given a definite geometrical The vector r, when p is regarded

interpretation as follows. as a variable scalar parameter, describes an ellipse of

b and

c are

this ellipse

two conjugate semi-diameters (page

r is,

the ellipse

cylinder.

Let

117).

be regarded as the parallel projection of the

unit circle

That

which

= cos p + i

and the

sin q

same

are cut from the

circle

The two semi-diameters

and

ject into the conjugate semi-diameters a

of the circle pro

and b

of the ellipse.

The radius

vector r in the ellipse projects into the radius vector f in the unit circle. The radius vector r in the ellipse which is

equal to

such that f

Thus the vector

projects into a radius vector r in the circle

= cos (p + q) + i

sin

r in the ellipse is so

(jp

+ q) j.

changed by the applica

as a prefactor that its projection f in the unit circle rotated through an angle q. This statement may be given a neater form by making use

tion of is

of the fact that in parallel projection areas are definite constant ratio.

The vector

changed in a

r in the unit circle

may

be regarded as describing a sector of which the area is to the area of the whole circle as q is to 2 TT. The radius vector f

then describes a sector of the is

to the area of the

dyadic

$ applied as

of which b and

whole

ellipse.

The

area of this sector

ellipse as q is to 2

a prefactor

TT.

Hence

a radius vector r in an

the

ellipse

two conjugate semi-diameters advances that vector through a sector the area of which is to the area of c are

VECTOR ANALYSIS

350

the whole ellipse as q is to

radius vector r sector q from it is

similarity to

its

Such a displacement

be called an

of the

elliptic rotation through a

an ordinary rotation of which

the projection.

Definition

= aa + is

may

2-Tr.

dyadic cos q (bb

of the form

cc )

The

called a cyclic dyadic.

sin q (c

versor

V - be )

(17)

a special case of a

cyclic dyadic. It is evident

from geometric or analytic considerations that

the powers of a cyclic dyadic are formed, as the powers of a

versor were formed, by multiplying the scalar q by the power to which the dyadic is to be raised. n

= aa +

cos

If the scalar q is

nq (b b

c c )

sin

an integral sub-multiple of 2 27T

TT,

that

is, if

= m,

1 it is

possible to raise the dyadic

namely, the

may then

power w, that

such an integral power,

becomes the idemfactor

be regarded as the

mth

root of the idemfactor.

q and 2 TT are commensurable it is possible to raise to such a power that it becomes equal to the idemfactor and even if q and 2 TT are incommensurable a power of d> may be found which differs by as little as one pleases from

In like manner

the idemfactor.

Hence any cyclic dyadic may be regarded

a root of the idemfactor. 1

It is evident that fixing

the result of the application of to all radii vectors it for all vectors in the plane of b and c. For any

ellipse practically fixes vector in that plane may be

the ellipse.

regarded as a scalar multiple of a radius vector of

ROTATIONS AND STRAINS

<Z>

where dyadic

The transformation represented by

Definition:

130.]

itself is called

right tensor

The order

+ &jJ+ckk

are positive scalars

6, c

351

may

the

(18)

The

called a ^rare strain.

rt^Atf tensor.

be factored into three factors

which these factors occur

The

immaterial.

transformation

such that the

and

components of a vector remain un

altered but the k-component

altered in the ratio of c to 1.

The transformation may therefore be

described as a stretch or

elongation along the direction k. If the constant c is greater than unity the elongation is a true elongation but if c is less :

than unity the elongation is really a compression, for the ratio Between these two cases of elongation is less than unity.

comes the case in which the constant

unity.

The lengths

of the k-components are then not altered.

The transformation due as

the successive

to the dyadic

may

be regarded

or simultaneous elongation of the i, j, and k respectively in the If one or more of the constants

com

ponents of r parallel to

ratios

a to

a, 6, c

1, b

to 1, c to 1.

than unity the elongation in that or those directions becomes a compression. If one or more of the constants is is less

unity,

The

components parallel to that direction are not altered. i, j, k are called the principal axes of the strain.

directions

Their directions are not altered by the strain whereas, constants #,

The

the

be different, every other direction is altered. scalars a, 6, c are known as the principal ratios of &, c

elongation.

was seen that any complete dyadic was the normal form

In Art. 115 reducible to

VECTOR ANALYSIS

352

where

constants.

a, J, c are positive

This expression

factored into the product of

two dyadics.

(ai

The

+ ck

ftj j

i+j

factor

which It

(i i

k k)

(aii

6jj

k),

turns the vectors

(19)

+ ckk).

k k

the same in either method of factoring

may

k into the vectors

a versor.

The vector

semi-tangent of the versor

i>i

The other

^ + k kls ixi + i

+j xj + k xk +j .j + k.k"

i.i-

factor

+ ck k

aii

a right tensor and represents a pure strain. In the first case the strain has the lines i , j , k for principal axes: in is

the second,

i, j,

the same,

a to

In both cases the ratios of elongation are If the negative sign occurs b to 1, c to 1.

the product the version and pure strain must have associated with them a reversal of directions of all vectors in before

space

that

is,

a perversion.

Theorem:

Hence

reducible to the product of a Any dyadic versor and a right tensor taken in either order and a positive or negative sign. Hence the most general transformation representable by a dyadic consists of the product of a rota is

tion or version about a definite axis

through a definite angle a strain either with or without perver pure accompanied by sion. The rotation and strain may be performed in either

In the two cases the rotation and the ratios of elonga tion of the strain are the same ; but the principal axes of the

order.

strain differ according as

it is

performed before or after the

ROTATIONS AND STRAINS

353

system of axes being derivable from the other

rotation, either

by the application of the versor as a prefactor or postfactor respectively.

and its conjugate be given the product of ratios of is a right tensor the elongation of which are the ratios of of the elongation of (P and the axes of which squares If a dyadic

are respectively the antecedents or consequents of

ing as

in the product.

follows or precedes 4>

(ai

C .

6 j

(aii i i

accord

+ ck

+ ckk ),

6 jj

62 j

k),

k k

(20)

kk.

The general problem

of finding the principal ratios of elonga the tion, antecedents, and consequents of a dyadic in its normal form, therefore reduces to the simpler problem of find

ing the principal ratios of elongation and the principal axes of a pure strain.

The

131.]

natural and immediate generalization of the

right tensor

the dyadic

where and a

a, 6, c ,

= aaa + &bb +

ccc

(21)

are positive or negative scalars and where a, b, c Neces are two reciprocal systems of vectors.

and a

sarily a, b, c

Definition

are each three non-coplanar.

dyadic that

may

be reduced to the form (21)

called a tonic.

The

effect of a tonic is to leave

directions a, b, c in space.

coplanar into its components parallel to 23

unchanged three nonIf a vector be resolved b, c

respectively these

VECTOR ANALYSIS

354

components are stretched in the ratios a to 1, & to 1, c to 1. more of the constants a, &, c are negative the com

If one or

ponents parallel to the corresponding vector a, b, c are re versed in direction as well as changed in magnitude. The tonic

may be

factored into three

factors

which each

stretches the components parallel to one of the vectors

a, b, c

but leaves unchanged the components parallel to the other two.

The value

cc )

(aa

of a tonic

+ &bb + ccXa not altered

any three vectors respectively

in place of a, b, c

collinear with

them be sub

provided of course that the corresponding changes are necessary be made in the reciprocal system a , b , c .

stituted,

which

But with the exception

of this change, a dyadic

which

expressible in the form of a tonic is so expressible in only one way if the constants a, 6, c are different. If two of the

constants say J and c are equal, any two vectors coplanar with the corresponding vectors b and c may be substituted in place of b

and

If all the constants are equal the tonic

reduces to a constant multiple of the idemfactor. non-coplanar vectors may be taken for a, b, c.

The product same

two

which the axes

tonics of

commutative and

Any

a, b, c

three

are the

a tonic with these axes and

with scalar coefficients equal respectively to the products of the corresponding coefficients of the two dyadics.

+ \bV + ^ cc c2

0. y = The

a 2 aa

+ ^^bV-f c^cc

(22)

generalization of the cyclic dyadic

aa is

cos q (b

= a aa

-1-

V+

1 (b

c c )

V+

cc )

sin q (c b

bc)

V - be ),

(23)

ROTATIONS AND STRAINS where

355

which a

are three non-coplanar vectors of

a, b, c

the reciprocal system and where the quantities a, 6, c, are This dyadic may be changed positive or negative scalars. into a more convenient form by determining the positive is

p and

the positive or negative scalar q (which TT) so that always be chosen between the limits

may

and

(24)

scalar

c=psinq.

That

is,

and

tan

(24 y

Then

+ This

may

The

be ).

(cV

(25)

be factored into the product of three dyadics

0= (aaa + bV + cc {aa

+ p sin

)

cos q (b b 4- o c

order of these factors

(a a

)

+ p bV + jpcc )

sin q

(cV

immaterial.

- be )}.

The

first is

a tonic

parallel to b and c but stretches those parallel to a in the ratio of a to 1. If a is

which leaves unchanged vectors

negative the stretching must be accompanied by reversal in direction. The second factor is also a tonic. It leaves

unchanged vectors

parallel to a

but stretches

the plane of b and c in the ratio cyclic factor.

to 1.

all

The

vectors in third

Vectors parallel to a remain unchanged ; but which b and c are conjugate

radii vectors in the ellipse of

semi-diameters are rotated through a sector such that the area of the sector is to the area of the whole ellipse as q to 2

TT.

Other vectors in the plane of b and

may

be regarded

as scalar multiples of the radii vectors of the ellipse.

VECTOR ANALYSIS

356 Definition :

dyadic which

= a aa + p cos

owing

(bb

reducible to the form

)

sin q (c

(25)

to the fact that it combines the properties of

the

cyclic dyadic and the tonic is called a cyclolonic. The product of two cyclotonics which have the same three vectors, a, b, c as antecedents and the reciprocal system

for consequents

a third cyclotonic and

com

mutative.

cc 5F

= a2 aa

5P*=

+jp2 cos j2 (bb <?

)

sinq l (cb

be )

+ jpa

sin q 2 (cb

)

Reduction of Dyadics 132.]

+ p p 2 cos (q l + j a ) (bb + cc ) sin ( 2l + & (c b - b c ). (26)

a 2 aa

+ Pi P*

)

Canonical Forms

Theorem : In general any dyadic

may

be reduced

The dyadics for which be regarded as limiting cases impossible may be represented to any desired degree of approxi

either to a tonic or to a cyclotonic.

the reduction

which may

mation by tonics or cyclotonics. From this theorem the importance of the tonic and cyclo tonic which have been treated as natural generalizations of the right tensor and the cyclic dyadic may be seen. The proof of the theorem, including a discussion of all the

long and somewhat tedious. The method of proving the theorem in general however is If three directions a, b, c may be found which are patent.

special cases that

left

may

arise, is

unchanged by the application of $ then

must be a

If only one such direction can be found, there exists a plane in which the vectors suffer a change such as that due to the cyclotonic and the dyadic indeed proves to be such.

tonic.

ROTATIONS AND STRAINS The question

to find the directions

357

which are unchanged

by the application of the dyadic 0. If the direction a is unchanged, then

=aa al).a = 0. a

The dyadic

(27)

therefore planar since

reduces vectors

In special cases, which are set aside for the present, the dyadic may be linear or zero. In in the direction a to zero.

any case

the dyadic

<P-aI reduces vectors collinear with a to zero

possesses at least

one degree of nullity and the third or determinant of

vanishes.

(0-aI) 8 = Now

- a I) 8 =

(page 331)

Hence

(4>

W) z Z

-a

<P B

<Z>

+ 1

(28)

2 :

W+ :

^ - a3 1 8

= I and I3 = 1.

But

Hence the equation becomes a3

The value

- a2

a which

0^ -03 = 0.

satisfies

(29)

the condition that

Let x replace

a solution of a cubic equation.

The

cubic equation becomes

- x*

d> 3

= 0.

(29)

VECTOR ANALYSIS

358

value of x which

Any

equation will be such

this

satisfies

that

(*-aI), = 0. That

to say, the dyadic

(28)

planar.

vector per

pendicular to its consequents is reduced to zero. Hence The further discussion leaves such a direction unchanged. of the reduction of a dyadic to the form of a tonic or a cyclotonic depends merely upon whether the cubic equation in x has one or three real roots.

Theorem

133.]

If the cubic equation

- x*

4> s

has three real roots the dyadic to a tonic.

For

let

= a,

are

a I,

61,

Let

in general planar.

vectors

drawn perpendicular

The dyadics

be the three roots of the equation. <P

(29)

in general be reduced

may

cl be respectively three

to the planes of the consequents

of these dyadics.

= 0, (0-cI).c = 0. <P a = a a, </>-b = &b, c = cc. <p b

Then

(30)

If the roots a,

coplanar.

&, c

are distinct the vectors

For suppose c

= ma +

?ib

a, b, c

are non-

ROTATIONS AND STRAINS

raca-ffl>0b

But

= a a,

m (a

Hence and

m(a

Hence

or a

vectors

a, b, c

n(b TI

= 0,

or b

still

are coplanar, the roots are the roots are distinct, the

In case the roots

always possible to choose three in such a manner that the equa

non-coplanar vectors a, b, c This being tions (30) hold. reciprocal to a, b, c

a, b, c

are necessarily non-coplanar.

are not distinct

a a,

= 0.

= 6 b.

= c,

Consequently if the vectors not distinct; and therefore

ncb

359

so, there exists a

and the dyadic which

system a

carries a, b, c into

b b, c c is the tonic

Theorem

If the cubic

equation

- x* 4> a +

has one real root the dyadic a cyclotonic.

The cubic equation has one tive or negative according as

the root be a.

may

in general be reduced to

This must be posi Let positive or negative.

real root. <P

Determine a perpendicular

the consequents of

(29)

to the plane of

(<P-aI) .a

= 0.

Determine a also so that a

and

(0- a I) =

a and a be so adjusted that a a l. This cannot be accomplished in the special case in which a let the lengths of

VECTOR ANALYSIS

360

and a are mutually perpendicular. ;

the plane perpendicular to a

Let b be any vector in

- a I)

= 0.

Hence <Pb is 3 <P b, $ b, and perpendicular to a l 2 <P~ b, 0~ b, etc., will all be perpendicular to a and lie in one plane. The vectors <P b and b cannot be parallel or would have the direction b as well as a unchanged and thus the cubic would have more than one real root. 2 The dyadic a, b, b into b, <P b re changes a, of The volume the parallelepiped spectively.

Hence

al)b

(<P

perpendicular to a In a similar manner

</>

[<p.a

But Hence

The

a a

vectors

(<P <0

= 3 [a $a = aa. (0 b) = <P3 a

and

Inasmuch

as a

and

x <P

(</>.b)

and

b_!

=;r #-b

(&-

b b2

The

(b 2 vectors b 2

= 27ib 2

b x -f

b_!

=2nb

Their

Hence (31)"

let

(32)

= /r #~ 2

etc.

(33)

etc.

X b, = 0.

b) x b x

b and b x are b2

Then

(31)

b_ 2

><Pb xb.

have the same sign, s

to each other.

^ = a-i* Let also

(31)

b].

the same plane.

all lie in

vector products are parallel to a

<P-b

</>

</>.b]

parallel.

Let

= 27ib r b + b = 2nb 3

b_!

+ b_ = 2 n b_ 2

(34) etc., etc.

ROTATIONS AND STRAINS Lay

off

861

from a common origin the vectors b, bj,

Since

b_j,

etc.,

not a tonic, that

b_2, etc.

since there

is,

no direction

the plane perpendicular to a which is left unchanged by takes these vectors b OT pass round and round the origin as

all positive and negative values. The value of Let therefore lie between plus one and minus one.

= cos b-j + bj = 2 cos q b. n

Then Determine

n must

(36)

from the equation

= cos q b + sin q c. sin q c. b_j = cos q b bx

Then Let a , b

f ,

be the reciprocal system of a, was so determined that a

sible since a

Then

pos

= cos q (bV + ccO + sin q (c V - be ). ra = 0, ?F.b = b .})_ = b. & = a a = $ a, (a *& + p b, (a aa + p W) b = p b = b_ =p b = d> b_r (a aa + p ) 1,

Hence

and since

Let

are non-coplanar.

a, b, c

This

b, c.

)

The dyadic a a a the vectors

+pW

changes the vectors

= (a aa + p

and

b and

= a aa + ^ cos j (bb + 4- ^?

The dyadic

b_ x respectively.

sin q (c

into

Hence cc7 ) b o ).

in case the cubic equation has only one real reducible except in special cases to a cyclotonic. The theorem that a dyadic in general is reducible to a tonic or cyclotonic has therefore been demonstrated.

root

VECTOR ANALYSIS

362

There remain two cases 1 in which the reduction impossible, as can be seen by looking over the proof. In

134.] is

the

first

place

tonic form be

place

the constant n used in the reduction to cyclofalls through. In the second

1 the reduction

the plane of the antecedents of

and the plane of the consequents are perpendicular the and a used in the reduction to cyclotonic form are

vectors a

perpendicular and it a a shall be unity.

impossible to determine a such that

The reduction

n=l,

b_1

Let

Choose

Consider the dyadic

through.

2b.

= 2b. b =b b_r

=b W = a aa + p (bV + co ) y.a = aa=<P.a,

+ pc

?p*.o=jt)c=

Hence

falls

= a aa + p

The transformation due

pbi

pb 1

V+

cc )

+ p cb r

(37)

may be

seen best by into three factors which are independent of the to this dyadic

factoring it order or arrangement

.(aa + bb

In these cases it will be seen that the cubic equation has three real roots. In one case two of them are equal and in the other case three of them. Thus these dyadics may be regarded as limiting cases lying between the cyclotonic in

which two of the roots are imaginary and the tonic in which all the roots are real The limit may be regarded as taking place either by the pure distinct. imaginary part of the two imaginary roots of the cyclotonic becoming zero or by two of the roots of the tonic approaching each other.

and

ROTATIONS AND STRAINS The ratio

863

factor represents an elongation in the direction a in a 1. The plane of b and c is undisturbed. The

first

a to

second factor represents a stretching of the plane of b and the ratio

The

to 1.

last factor takes the

+ cb

+ oV)

c in

form

= #a,

+ c V) x b = x b + x o, x c = x c. (I + c V)

A dyadic of the form I + cb leaves vectors parallel to a and c unaltered. A vector #b parallel to b is increased by the vec tor c multiplied

by the

ratio of the vector

# b to

In other

words the transformation of points in space is such that the plane of a and c remains fixed point for point but the points in planes parallel to that plane are shifted in the direction c

by an amount proportional to the distance of the plane in which they lie from the plane of a and c. Definition

dyadic reducible to the form I

+ cb

is called a shearing dyadic or shearer and the geometrical transformation which it causes is called a shear. The more

general dyadic <P

= a aa + p

V+

;

c c )

+ oV

(37)

The trans will also be called a shearing dyadic or shearer. formation to which it gives rise is a shear combined with elongations in the direction of a and is in the plane of b and c. If n 1 instead of n +1, the result is much the same.

The dyadic then becomes

$ $=

= a aa

aa + bb + cc )

-,p (bV f

{aa

+ c<0 - c V

-p

(b D

cc )>

(37) (I

cV).

VECTOR ANALYSIS

364

The

factors are the

same except the second which now repre

sents a stretching of the plane of b and c combined with a reversal of all the vectors in that plane. The shearing dyadic then represents an elongation in the direction a, an elonga

combined with a reversal of direction in the plane of b and c, and a shear. tion

Suppose that the plane of the antecedents and the plane of Let the consequents of the dyadic are perpendicular.

0al

these planes be taken respectively as the plane of j and the plane of i and j. The dyadic then takes the form

The

a I

coefficient

must vanish.

j j

+ Ck +Dk i

k and

For otherwise the dyadic

Bk)

planar and the scalar a + B is a root of the cubic equation. With this root the reduction to the form of a tonic may be is

carried on as before.

new

This

case occurs.

Nothing new Let

may be reduced

arises.

as follows to the

But

vanishes

form

ab + bc where Square

V=a

=b c = and b V = 1. W = A D ki = ac c

Hence a must be chosen parallel to k The dyadic W may then be transformed

Then

and

into

V= CiA+DDi

=AD*, b

;

parallel to

ROTATIONS AND STRAINS With

the dyadic and hence the dyadic

this choice of a, b, V, c

ab + be

desired form

= al +

= aaa + abb +

This

may

reduces to the

reduced to

<P is

ab + bc

ace

(38)

+ aV +

be factored into the product of two dyadics the

order of which

The

365

immaterial.

factor al represents a stretching of space in all directions in the ratio a to 1. The second factor first

what may be called a complex

represents r

= IT +

r+ bc -r=

r-t-aV-r + bc

If r is parallel to a it is left unaltered

tional to the if

by the dyadic Q.

it is

magnitude of the vector

In like manner

changed by the addition of a term equal to b and which in magnitude is

r is parallel to c it is

which in direction

proportional to the magnitude of the vector

= (I + ab -f bc ).zb= Q *xc = (I + ab + be ) xc = -zb

Definition

dyadic which <P

-r.

changed by the addition of a term in direction equal to a and in magnitude propor

r is parallel to

which

For

shear.

= aI +

may

zb + a a xc

#b.

be reduced to the form

ab + bc

(38)

called a complex shearer.

The complex shearer as well as the simple shearer men tioned before are limiting cases of the cyclotonic and tonic dyadics.

VECTOR ANALYSIS

366 135.]

of dyadics

more systematic treatment which may

arise

may

of the various kinds

be given by means of the

Hamilton-Cayley equation

and the cubic equation in x x*

S x*

= 0.

(29)

are the roots of this cubic the Hamilton-Cayley equation may be written as If a,

&, c

- al)

(<P

- JI)

- <?I) = 0.

(40)

however, the cubic has only one root the Hamilton-Cayley equation takes the form

If,

(0_al).(0 2 - 2^0082

+ p*I) =

In general the Hamilton-Cayley equation which

(41)

an equa

is the equation of lowest tion of the third degree in degree which is satisfied by 0. In general therefore one of the above

equations and the corresponding reductions to the tonic or cyclotonic form hold. In special cases, however, the dyadic

may

satisfy

an equation of lower degree. That equation which may be satisfied by a dyadic is called

of lowest degree its characteristic

(<P

equation.

- a I)

II.

(0-aI)

III.

(

IV.

(<P

V. VI. VII.

The following

- b I)

possibilities occur.

- c I) = 0.

(

- a !).(</>- 61) = 0. (0-aI) 3 = 0. (<P

al)

(<P-aI)

= 0.

ROTATIONS AND STRAINS In the to the

first

case the dyadic

a tonic and

In the second case the dyadic reduced to the form

= a aa +p

cos q

(bb

In the third case the dyadic reduced to the form </>

= aaa +

case the dyadic so expressed that

= al +

sin q (eb

cc )

+ cb

may

Two

of the

The following reduction number of ways.

(bb

a complex shearer and

al + ab

In the sixth case the dyadic may be reduced to the form 4>

again a tonic.

be accomplished in an infinite

fifth

may

(bb

= aaa +

a simple shearer and

In the fourth case the dyadic

In the

be reduced

a cyclotonic and

cc )

ratios of elongation are the same.

may

form

6bb + ccc

367

-f

may

again a simple shearer which

(aa

bb + cc

)

+ cb

In the seventh case the dyadic is again a tonic which may be reduced in a doubly infinite number of ways to the form

= al = a(aa + /

bb + cc

These seven are the only essentially different forms which a dyadic may take. There are then only seven really different kinds of dyadics

three tonics in which the ratios of elonga two alike, or all equal, and the cyclo

tion are all different,

tonic together with three limiting cases, the the one complex shearer.

two simple and

VECTOR ANALYSIS

368

Summary

of Chapter

The transformation due to a dyadic is a linear homogeneous The dyadic itself gives the transformation of the in The second of the dyadic gives the trans space. points formation of plane areas. The third of the dyadic gives the strain.

ratio in

which volumes are changed.

The necessary and

sufficient condition that a dyadic repre

sent a rotation about a definite axis the form

or that

i i

4> c

or that

=I c= I

+ 3

that

be reducible to

k k

=+ 8 >

(1)

(2)

The

necessary and sufficient condition that a dyadic repre sent a rotation combined with a transformation of reflection

by which each

figure is replaced

by one symmetrical

to it is

that

= -(i + $ <P C = I* 0.00 = 1, i

or that or that

A dyadic of the form (1)

(1)

k k)

^3 = 3

(iy

<0.

called a versor

(3) ;

one of the form

a perversor.

If the axis of rotation of a versor be chosen as the i-axis

the versor reduces to

= ii + cos q (j + kk) + sin q (kj - j k) = ii + cos q (I ii) + sin q I x

(4)

or If

any unit vector a <p

The

= aa +

(5)

directed along the axis of rotation

cos q (I

a a)

sin q 1

axis of the versor coincides in direction with

(6) X.

ROTATIONS AND STRAINS If a vector be

drawn along the

axis

and

369

the magnitude of

the vector be taken equal to the tangent of one-half the angle of rotation, the vector determines the rotation completely.

This vector

called the vector semi-tangent of version.

In terms of

the versor

(9)

be expressed in a number of

may

was. <P

dft

a-ft

/ cos q (I

)

a-ay

aa\

sin q

ft)

-I

(10)

x Q)- 1

J^ + axQ) tf

= 1

(10)

(loy

Q-ft

a unit vector a dyadic of the form <P

= 2aa-I

(11)

a biquadrantal versor. Any versor may be resolved into the product of two biquadrantal versors and by means of such resolutions any two versors may be combined into is

another. of version

for the vector semi-tangents

dyadic reducible to the form

The law of composition

= aa + cos q (bb + cc + sin q )

called a cyclic dyadic.

simple rotation

(cb

-W)

(17)

produces a generalization of

elliptic rotation, so to speak. 24

The

pro-

VECTOR ANALYSIS

370

duct of two cyclic dyadics which have the same antecedents a, b, c and consequents a b c is obtained by adding their

cyclic dyadic

angles q. idemfactor.

where

may

be regarded as a root of the

A dyadic reducible to the form = aii +bjj + ckk

#, &, c are positive scalars is called

(18)

a right tensor.

represents a stretching along the principal axis i, j, k in the a to 1, b to 1, c to 1 which are called the principal ratios

ratio

This transformation

is a pure strain. as the product of a versor, be expressed Any dyadic may and a or a right tensor, positive negative sign.

of elongation.

= <P=

(a i

(i i

k k)

(i i

j j

Jjj

ckk).

k k).(aii

k k) (19)

Consequently any linear homogeneous strain may be regarded as a combination of a rotation and a pure strain accompanied or unaccompanied by a perversion. The immediate generalizations of the right tensor and the cyclic dyadic is to the tonic

= aaa + &bb + ccc

(21)

and cyclotonic cc or

)

+ c(cV-bc)

= aaa + p cos q (bb + cc )+^sing (cV

where

= + V 62 + c 2

and tan I q *

=2?

(23)

be )

(25)

(24)

dyadic in general may be reduced either to the form (21), and is therefore a tonic, or to the form (25), and is The condition that a dyadic be a therefore a cyclotonic.

Any

tonic

that the cubic equation

+ 0^ x -

<J>

(29)

ROTATIONS AND STRAINS have three real

shall

roots.

Special cases

371 in

which the

may be accomplished in more ways than one arise The condition that a the equation has equal roots.

reduction

when

dyadic be a cyclotonic is that this cubic equation shall have only one real root. There occur two limiting cases in which the dyadic cannot be reduced to cyclotonic form. cases it may be written as 4>

and

=aaa

+jp (bb

a simple shearer, or

a complex shearer.

cc )

+ cb

(37)

takes the form

= al + and

In these

ab + bc

Dyadics

(38)

may

be classified accord

ing to their characteristic equations

=0 1) =

(<P-aI).(0-&I).(<P-cI) (#

a I)

(<P

cos q

+ jp

tonic

cyclotonic

= simple shearer special tonic (0_ <*!)($ &I) = 8 complex shearer (0 a I) = 2 special simple shearer (0 a I) = special tonic. (0 a I) =

a I)

& I)

CHAPTER

VII

MISCELLANEOUS APPLICATIONS Quadric Surfaces If

136.]

be any constant dyadic the equation r

= const.

(1)

The constant, in case it be not zero, may quadratic in r. and hence the equation takes be divided into the dyadic is

the form

= 1, r = 0.

The dyadic

may

be assumed to be

(2)

For

self -conjugate.

an anti-self-conjugate dyadic, the product

r is

The proof of this state identically zero for all values of r. ment is left as an exercise. By Art. 116 any self-conjugate dyadic is reducible to the form -- t

Hence the equation

represents a quadric surface real or imaginary. The different cases which arise are four in number.

sign

negative

it is

If the

a real ellipsoid. If one an hyperboloid of one sheet; if two are

signs are all positive, the quadric

QUADRIC SURFACES

373

If the three signs are negative, a hyperboloid of two sheets. is the quadric imaginary. In like manner the negative

all

equation r

seen to represent a cone which may be either real or imaginary according as the signs are different or all alike. is

Thus

the equation 0-

= const. The

represents a central quadric surface. a cone in case the constant is zero.

quadric surface

may

Conversely any central be represented by a suitably chosen self-

form

in the

conjugate dyadic

surface reduces to

= const.

evident from the equations of the central quadric surfaces when reduced to the normal form. They are

This

2 7/

The corresponding dyadic

The most general the vector r

z = const. c

<Pis

scalar expression

which

and which consequently when

quadratic in set equal to a con is

stant represents a quadric surface, contains terms like r

where

a, b, c, d,

a) (b

e are constant vectors.

The

first

two terms and

are of the second order in r ; the third, of the first order

the

;

independent of r. Moreover, it is evident that these four sorts of terms are the only ones which can occur in a scalar expression which is quadratic in r. last,

But and

r (r

a) (b

=r I =r r)

VECTOR ANALYSIS

374

Hence the most general quadratic expression may be reduced to

where

a constant dyadic,

a constant scalar.

a constant vector, and

The dyadic may be regarded

desired. conjugate To be rid of the linear term r

self-

replacing r

-t). 0-

Since equal.

of origin

A+ C=0 +

(7=0. and third terms are

Hence r

now

2 r

or t

incomplete

is self -conjugate.

as the case

may

+C = f

t) t

may

= 5L 0-

be chosen so that

reducible to the central form r

In case

A-

complete the vector

Hence the quadric

-t)

-A-t- A +

= 0-t IA L

self-conjugate the second

r If

-t) +

(r t

make a change

r it is

= const. untplanar or unilinear because

lies in

the plane of

or in the line

be the equation

and the reduction to central form is still pos But unless A is so situated the reduction is impossible.

soluble for t

sible.

The quadric surface is not a central surface. The discussion and classification of the various quadrics here.

an interesting

The present

object

exercise. is

It will not

to develop so

much

non-central

be taken up of the theory

QUADRIC SURFACES

375

of quadric surfaces as will be useful in applications to mathe matical physics with especial reference to non-isotropic

media. Hereafter therefore the central quadrics and in par ticular the ellipsoid will be discussed. 137.]

The tangent plane may be found by r

di Since

<P is

= 1.

self-conjugate these

differentiation.

<t>

= 0.

two terms are equal and

dr.0-r = 0.

(5)

perpendicular to <P r. Hence normal to the surface at the extremity of the vector

The increment d r this

normal be denoted by

r is

and

let the unit

(6)

Let

normal be

Let p be the vector drawn from the origin perpendicular to

The perpendicular the tangent plane, p is parallel to n. distance from the origin to the tangent plane is the square root of p p. It is also equal to the square root of r p. r

Hence

= r cos r

Ll! p.p

(r,

= p2

p and

= , JL = L .

p.p

r0r = rH =

But Hence inasmuch

IT are parallel,

0.r = !T=-^-. p.p

they are equal.

(T)

VECTOR ANALYSIS

376

page 108 it was seen that the vector which has the direc tion of the normal to a plane and which is in magnitude equal

from the origin to the plane

to the reciprocal of the distance

may

be taken as the vector coordinate of that plane.

Hence

the above equation shows that <P r is not merely normal to the tangent plane, but is also the coordinate of the plane.

That

the length of <P r is the reciprocal of the distance to the plane tangent to the ellipsoid at the extremity of the vector r. is,

from the origin

The equation

of the ellipsoid in plane coordinates

may be

found by eliminating r from the two equations. (

Hence

= H 0-

0- 1

Hence the desired equation

= 1,

= JT 0"

H-0-i-H = l.

(8)

*4 +y+" c

kk.

= #i-t-yj+3k, N = ui + v + wk,

Let

and where

u, v,

the reciprocals of the intercepts of the the axes i, j, k. Then the ellipsoid may be are

plane N upon written in either of the two forms familiar in Cartesian

geometry.

K 0- 1 .N = a 2

2 Z>

w 2 = 1.

(10)

QUADRIC SURFACES

377

The locus of the middle points of a system of in an ellipsoid is a plane. chords This plane is parallel called the diametral plane conjugate with the system of 138.]

drawn tangent

to the

ellipsoid at the extremity of that one of the chords

which

chords.

It is parallel

the plane

passes through the center. Let r be any radius vector in the ellipsoid. Let n be the vector drawn to the middle point of a chord parallel to a.

Let

= s + x a.

If r is a radius vector of the ellipsoid

Hence

= (B + 2 #

a) .

2? a

= 1. # a = 1. x a)

Inasmuch as the vector s bisects the chord parallel to a the two solutions of x given by this equation are equal in mag nitude and opposite in sign. Hence the coefficient of the linear

term x vanishes.

Consequently the vector locus of the terminus of

s is

= 0.

perpendicular to

the center of the ellipsoid, perpendicular to to the tangent plane at the extremity of a. If b is

with

The

therefore a plane passed through

s is

and

parallel

any radius vector in the diametral plane conjugate _

The symmetry of this equation shows that a is a radius vector in the plane conjugate with b. Let c be a third radius vector in the ellipsoid and let it be chosen as the line of intersection

with a and

of the diametral b.

Then a

planes conjugate respectively

= 0, c = 0, a = 0.

(11)

VECTOR ANALYSIS

378

The

vectors

a, b, c

The

vectors a , b

For

V=a

may

b, <P

-b

= 1,

V=b

=c b = 0, a =c

= 1, c = b a = 0.

a, b, c.

= 1,

= 0,

be therefore expressed in the forms

= a a + b b + cV, 0" = aa + bb + cc.

(12)

and If for

a, <D

form the system reciprocal to

The dyadic

and

Let

the dyadic 0.

are changed into

convenience the three directions

be called a

system of three conjugate radii vectors, and if in a similar manner the three tangent planes at their extremities be called a system of three conjugate tangent planes, a number of geometric theorems may be obtained from interpreting the

system of three conjugate radii vectors in a be obtained may doubly infinite number of ways. The volume of a parallelepiped of which three concurrent invariants of 0.

edges constitute a system of three conjugate radii vectors is constant and equal in magnitude to the rectangular parallele piped constructed upon the three semi-axes of the ellipsoid. For let a, b, c be any system of three conjugate axes.

0- 1 = aa + bb + The determinant

or third of

pendent of the form in

0" 1

which

is is

-i=[abc]

cc.

an invariant and inde expressed. 2 .

QUADRIC SURFACES But

0-!:=a 2

=a6

[a b c]

Hence

and

<Ps~~\

The sum

it is

kk,

In like manner by inter

This demonstrates the theorem. preting

& 2 jj

379

possible to

show that: drawn to an

of the squares of the radii vectors

ellipsoid in a

system of three conjugate directions

constant

and equal to the sum of the squares of the semi-axes. The volume of the parallelepiped, whose three concurrent edges are in the directions of the perpendiculars upon a system of three conjugate tangent planes and in magnitude equal to the reciprocals of the distances of those planes from the

center of the ellipsoid,

constant and equal to the reciprocal upon the semi-axes of the

of the parallelepiped constructed ellipsoid.

The sum

of the squares of the reciprocals of the three per

pendiculars dropped from the origin upon a system of three conjugate tangent planes is constant and equal to the sum of the squares of the reciprocals^ the semi-axes. If i, j, k be three mutually perpendicular unit vectors 4>

tf^-i

Let

a, b, c

= i*

0-i

i .

0-i

j .

+k j

0"1

be three radii vectors in the ellipsoid drawn

respectively parallel to

Hence

</>

<P a

i, j,

-i

But

j *

the three terms in this expression are the squares of the reciprocals of the radii vectors drawn respectively in the i, j,

directions.

Hence

VECTOR ANALYSIS

380

The sum

of the squares of the reciprocals of three mutually perpendicular radii vectors in an ellipsoid is constant. And

in a similar

manner: the sum

of the squares of the perpen

diculars dropped from the origin

dicular tangent planes 139.]

The equation

mined by the vector a

upon three mutually perpen

constant.

of the polar plane of the point deter l

= 1.

(13)

For let s be the vector of a point in the polar plane. The vector of any point upon the line which joins the terminus of s and the terminus of a is y

If this point lies

upon the surface

8*0*8+ If the terminus of

of the ratio

+ #a x + y s

2 xy

s lies

two values must be equal

in the polar plane of a the

x:y determined by

in magnitude

and opposite in

this equation

Hence the term

sign.

in x y

vanishes.

Hence

the desired equation of the polar plane of the terminus

Let a be replaced by

The polar plane becomes

z a.

= 1,

1 -

z 1

It is evidently

immaterial whether the central quadric determined by * be

real or imaginary, ellipsoid or hyperboloid.

QUADRIC SURFACES

When

381

the polar plane of the terminus of z a approaches the origin. In the limit when z becomes infinite the polar plane becomes increases

= 0.

Hence the polar plane of the point at infinity in the a is the same as the diametral plane conjugate with

direction

This

frequently taken as the definition of the diame In case the vector a is a radius tral plane conjugate with a. vector of the surface the polar plane becomes identical with

statement

the tangent plane at the terminus of s

The equation

therefore represents the tangent plane. The polar plane may be obtained from another standpoint are given, which is important. If a quadric Q and a plane

P = r c C = 0, r - 1) + k (r c -

and the equation

represents a quadric surface which passes through the curve and is tangent to Q along that of intersection of Q and

curve.

In like manner

Q Q the equation

=T = T* r

two quadrics Q and Q are given,

= -r-l = r

represents a quadric surface which passes through the curves of intersection of Q and Q and which cuts Q and Q at no f

other points.

In case this equation

factorable into

two

equations which are linear in r, and which consequently rep resent two planes, the curves of intersection of Q and Q

become plane and

lie

in those

two planes.

VECTOR ANALYSIS

382

A is any point outside of the quadric and if all the tangent planes which pass through A are drawn, these planes envelop If

a cone. This cone touches the quadric along a plane curve the plane of the curve being the polar plane of the point A. For let a be the vector drawn to the point A. The equation of any tangent plane to the quadric is s

If this plane contains

the conditions which

plane passes through

= 1.

A, its equation is satisfied by a. Hence must be satisfied by r if its tangent

are

a r

= 1, r = 1. r

The points r therefore lie in a plane r (<P a) = 1 which on comparison with (13) is seen to be the polar plane of A. The quadric which passes through the curve of intersection of this polar plane with the given quadric and

the quadric along that curve r

Hence

k (a

through the point

If this passes

- 1) +

a - 1) + k r - 1) (a

which touches

- I) 2 = 0.

-4,

- 1) -

= 0. r - I) = 0.

this is easily seen

transforming the origin to the point whose vertex is at that point

to be a cone 140.]

Let

sible in the

be any self-conjugate dyadic.

It is expres

form

where A, 5, more let

are positive or negative scalars.

A<B<G - Bl =

(<7- B) kk

- A)

ii.

Further-

QUADRIC SURFACES

VC B

Let

Then

and

0- Bl = cc-aa = \

Let

Then

j(c

-(pq

The dyadic $ has been expressed

The reduction has assumed are different

= q.

sum of the sum

as the

(14)

a constant

qp.

tacitly that the constants -4, B,

from each other and from

This expression for

= a.

qp).

multiple of the idemfactor and one half

pq +

+ a)(c-a)+(c-a)(c+a)

and

= 51 +

VB A

383

zero.

closely related to the circular

sections of the quadric surface r

Substituting the value of $,

Let

+ r

r r

= 1 becomes r = 1. r

p q

be any plane perpendicular to

= 1.

r.r-ftt q

substitution

= 0.

This

is a sphere because the terms of the second order all the have same coefficient B. If the equation of this sphere be subtracted from that of the given quadric, the resulting

that of a quadric which passes through the inter The difference section of the sphere and the given quadric.

equation

r (r

Hence the sphere and the quadric

= 0. intersect in

curves lying in the planes

and

= n.

two plane

VECTOR ANALYSIS

384

Inasmuch as these curves lie upon a sphere they are circles. Hence planes perpendicular to p cut the quadric in circles. In like manner it may be shown that planes perpendicular to q cut the quadric in follows

The proof may be conducted

circles.

.5 r

= 1.

p q

If r is a radius vector in the plane passed

through the center

of the quadric perpendicular to p or q, the term r

Hence

ishes.

p q

van

the vector r in this plane satisfies the equation

B r-r = l and

of constant length.

section.

mean

The

may

section

radius of the section

therefore a circular

equal in length to the

semi-axis of the quadric.

For convenience stants

The

A, B,

let the

quadric be an ellipsoid.

The

are then positive.

The con

1 reciprocal dyadic (P"

be reduced in a similar manner.

B Let

Then

1-*

and

= f f - dd = \

-i.

= (f

d) (f

- d)

+ (f-d)(f + Let

Then

+ =u d

0-i

and

= 4 1 + I* >

d)j

= v.

O v + vu).

(15)

QUADRIC SURFACES

385

The vectors u and v are connected intimately with the cir cular cylinders which envelop the ellipsoid r

-N N+N

For

(t>~

now N

be perpendicular to u or v the second term, namely, N, vanishes and hence the equation becomes

N N = B. -

That

is,

the vector

of constant length.

But

the equation

the equation of a cylinder of which the elements and tan gent planes are parallel to u. If then N N is constant the is

The cylinder is a circular cylinder enveloping the ellipsoid. radius of the cylinder is equal in length to the mean semi-axis of the ellipsoid.

There are consequently two planes passing through the origin and cutting out circles from the ellipsoid. The normals to these planes are p

extremities of the

two

and

mean

The

through the There are also

circles pass

axis of the ellipsoid.

circular cylinders enveloping the ellipsoid.

of the axes of these cylinders are

n and

The

Two

direction

elements of

these cylinders pass through the extremities of the

mean

axis

of the ellipsoid.

These results can be seen geometrically as follows. Pass a plane through the mean axis and rotate it about that The section is an axis from the major to the minor axis. ellipse.

One

axis of this

ellipse is the

mean

axis of

the

This remains constant during the rotation. The ellipsoid. other axis of the ellipse varies in length from the major to the

minor axis of the ellipsoid and hence at some stage must pass through a length equal to the mean 25

axis.

this stage of

VECTOR ANALYSIS

386

the rotation the section

In like manner consider

circle.

the projection or shadow of the ellipsoid cast upon a plane parallel to the mean axis by a point at an infinite distance

from that plane and in a direction perpendicular to it. As the ellipsoid is rotated about its mean axis, from the position in which the major axis is perpendicular to the plane of projec tion to the position in to that plane, the

mean

which the minor axis

axis of the ellipsoid as one axis.

The

ellipsoid to the

some stage

mean

other axis changes

major and hence at

passes through a value equal to this stage the shadow and projecting

of the rotation axis.

perpendicular

shadow and the projecting cylinder have the

from the minor axis of the the

cylinder are circular.

The

necessary and sufficient condition that r be the major 1 or minor semi-axis of the section of the ellipsoid r $ r

by a plane passing through the center and perpendicular to a r be coplanar. is that a, r, and Let

and :

Furthermore if

r is to

maximum

= 0. d r $ r = 0, d r a = 0. d r r = 0, r

Differentiate

be a major or minor axis of the section; for r is a or a mininum and hence is perpendicular to dr.

r are all ortho These three equations show that a, r, and are dr. Hence vector same to the coplanar. they gonal

Conversely

[a r

= 0. = 0, r]

dr may be chosen perpendicular "

(16)

to their

common

plane.

QUADRIC SURFACES Hence

maximum

r is a

or a

minimim and

387 is

one of the prin

cipal semi-axes of the section perpendicular to 141.]

of an ellipsoid in the

form ?T 2

instead of

This

may

be done ; because

= 1, r = 1. r

(17)

ii kk =+ jj + 2 ^ ir>

a dyadic such that W*

a square root of

equal to

may

<P.

But

and written as $*.

an advantage to write the equation

It is frequently

membered that there

are

be regarded as must be re

other square roots of

for

example,

and

For

this reason it is necessary to bear in

root which

meant by

been denoted by

The equation

mind that the square

that particular one which has

of the ellipsoid

.r.

Let

<P* is

may

be written in the form

be the radius vector of a unit sphere.

the sphere

= 1.

The equation of

VECTOR ANALYSIS

388

?Trit becomes

evident that an ellipsoid may be transformed into a unit sphere by applying the operator If r

r, and vice versa, the unit sphere may be transformed into an ellipsoid by applying the inverse oper ~l Furthermore if a, b, c are to each radius vector r ator

to each radius vector

a system of three conjugate radii vectors in an ellipsoid

moment

a a

Hence the three

=c c = c

a If for the

=b b =b

?F 2

denote respectively

=V V=c V=V c =c a

^.0 = 1, a = 0.

radii vectors

a c of

= 1, = 0. the unit sphere into

which three conjugate radii vectors in the ellipsoid are trans ~ formed by the operator W 1 are mutually orthogonal. They form a right-handed or left-handed system of three mutually perpendicular unit vectors.

Theorem : ellipsoid

Any

ellipsoid

be transformed into any other

may

by means of a homogeneous

strain.

Let the equations of the ellipsoids be r

and

By means ellipsoid are

of the strain

01.

of the strain

= 1, r = 1. r

0* the

changed into the r

By means

radii vectors r of the first

radii vectors r

of a unit sphere

= l.

~l the radii vectors r

of this unit

sphere are transformed in like manner into the radii vectors f Hence by the product r is changed

of the second ellipsoid.

into f . f

= r-

(19)

QUADRIC SURFACES

389

The transformation may be accomplished in more ways The radii vectors r of the unit sphere may be transformed among themselves by means of a rotation with or than one.

without a perversion. vectors in the sphere

Hence the semi-axes

Any three mutually orthogonal unit may be changed into any three others. of the first ellipsoid may be carried over

strain into the semi-axes of the second.

by a suitable

The

strain is then completely determined and the transformation can be performed in only one way. 142.] The equation of a family of confocal quadric sur

faces

--ir-

a* If r

and

1 are two surfaces of the

family, 2

.-..

0-1

y-i

Hence

- 71^11+ -

i i

0- - r- 1 ^

(7i 2

r is

-n

- 74)

+ ((^-w^kk,

2) j j

(ii

7l*

- n2 )

two quadrics

=1 r = 1

that the reciprocals of

multiple of the idemfactor

+ j j + kk)

sufficient condition that the r

and

Tin

-tti)jj 2

The necessary and

be confocal,

and

differ

by a

390

VECTOR ANALYSIS

two confocal quadrics

Let the quadrics be

and

Let

Then

the quadrics

= 1, r = 1. r and s = s and r =

1 .

be written in terms of

may s

and

do so at right angles.

= 0-

intersect, they

and

= l, s = 1,

0- 1

W~ l

where by the confocal property,

0-i_ W~* = xl. If the quadrics intersect at r the condition for perpendicularity is that the normals d> r and r be perpendicular. That is, s

But

W~l

= 0. r- + s =

= 4> rl = ?r-i s + rs - s .

W~l

(

1 *

=1-s

5T-1

In like manner r

= 0X

Add:

= FS = S f

2 a

= (0- 1

s 1

(P""

If the parameter

<P~l

- P" s = 0,

(0~

Hence and the theorem

)

= 0- s <P~ S = S S s s = x s 1

proved.

n be allowed

to vary

from

oo to

+ oo

the

resulting confocal quadrics will consist of three families of which one is ellipsoids another, hyperboloids of one sheet ; ;

and the

third, hyperboloids of

two

sheets.

the foregoing

QUADRIC SURFACES

391

theorem each surface of any one family cuts every surface The surfaces form a triply of the other two orthogonally. orthogonal system. The lines of intersection of two families (say the family of one-sheeted and the family of two-sheeted hyperboloids) cut orthogonally the other family

the family The points in which two ellipsoids are cut by of ellipsoids. these lines are called corresponding points upon the two ellip soids.

may

be shown that the ratios of the components of

the radius vector of a point to the axes of the ellipsoid through that point are the same for any two corresponding points.

For

let

any

ellipsoid be given

The neighboring

ellipsoid in the family is represented

dyadic a

y-\

= $-i

and

by the

Inasmuch as

by the dyadic

ldn.

homologous (see Ex.

are

8, p.

330)

dyadics they may be treated as ordinary scalars in algebra. Therefore if terms of order higher than the first in dn be omitted,

0+&dn.

The two neighboring

ellipsoids are

and

r f

(19)

+# (0 + (#

<Z>

r= f

then

= 1,

d n)

d n)-i

I<Pdn)

VECTOR ANALYSIS

392

The

vectors r and r differ

perpendicular to

r which is by a multiple of the ellipsoid 0. Hence the termini of r and

upon one

r are corresponding points, for they lie

of the lines

which cut the family of ellipsoids orthogonally. The com ponents of r and f in the direction i are r i = x and f

=x=r

The

ratio of these

The axes of the Their ratio

A/a

,.,

components

ellipsoids in the direction is

V& 2

manner

"l ..

fj\

A a i

are

Va2

~* dn

dn = y- j V^ 2 and

d n and

dn = z

Hence the ratios of the components of the vectors r and r drawn to corresponding points upon two neighboring ellip soids only differ at most by terms of the second order in d n from the

ratios of

the axes of those ellipsoids.

It follows

immediately that the ratios of the components of the vectors

drawn

to corresponding points

upon any two

ellipsoids, sepa

rated by a finite variation in the parameter n, only differ at most by terms of the first order in dn from the ratios of the

axes of the ellipsoids and hence must be identical with them. This completes the demonstration.

The Propagation of Light in Crystals 1 143.]

The

electromagnetic equations of the ether or of any

infinite isotropic

netic 1

medium which

transparent to electromag

waves may be written in the form

The

To treat

following discussion must be regarded as mathematical not physical. the subject from the standpoint of physics would be out of place here.

THE PROPAGATION OF LIGHT IN CRYSTALS d2

Pot where

.FD

and

units,

the electrostatic force

In case the medium

due to the function F.

(i)

the electric displacement satisfying the hydrodya constant of the dielectric meas 0,

namic equation V D = ured in electromagnetic constant

V.D = O

+ VF=O,

393

not isotropic the becomes a linear vector function 0. This function is

self-conjugate as is evident from physical considerations. it will be taken as 4 TT <D. The equations

For convenience then become

-4-7T0.D + U/

Operate by

V-D=0.

VF=0,

(2)

x Pot

47rVxVx0.D =

(3)

CL t

The

last

term disappears owing to the fact that the curl of

the derivative

VF vanishes

(page 167).

The equation may

also be written as

Pot

-r-y

47rVxVx<P.D =

(3)

VxVx=VV.-V.V.

But

Remembering that V D and consequently -n _ V -V-TT2 vanish and that Pot V V is equal a

and (t t

the

equation reduces at once to ,72 Tk

dt*

0.D

VD =

(4)

Suppose that the vibration D is harmonic. Let r be the vector drawn from a fixed origin to any point of space.

VECTOR ANALYSIS

394

D=A

Then where

and

cos

and n a constant

are constant vectors

scalar

The vibrations take place in represents a train of waves. That is, the wave is plane polarized. The the direction A. wave advances in the direction m. The velocity v of that ad n by m, the magnitude of the vector m. If this wave is an electromagnetic wave in the medium considered it must satisfy the two equations of that medium. vance

the quotient of

Substitute the value of

The value

in those equations.

V V

and

D may

obtained most easily by assuming the direction i to be coinci r then reduces to i r which is equal to

dent with m.

m x.

The

variables y

D=i

V V

VV Hence

3D D

m2 i

m A

sin

(m x

n f)

cos

(m x

n f)

sin

cos r

(mx

VV* 0.D = -mm.

</>.D.

nf).

nf)

the harmonic vibration

tions (4) of the

m m

Hence

A m

(m x

-j--^

no longer occur in D.

# D

Moreover

to satisfy the

equa

medium

n2 D and

-z

V V

Hence

= A cos

and

= m-m

m A = 0.

(5)

(6)

THE PROPAGATION OF LIGHT IN CRYSTALS The

once that the vibrations must

latter equation states at

m of

be transverse to the direction

propagation of the waves. be put in the form

The former equation may

The vector s

s is

(5)

in the direction of advance m.

the quotient of velocity of the wave. of

Introduce

395

This

The

vector

The magnitude

the reciprocal of the may therefore be called is

the wave-slowness.

D This

may

s s

also be written as

x (0

It is evident that the

wave slowness

(7)

depends not at

upon the phase of the vibration but only upon

The motion

n t\

Dividing by the scalar factor cos (m x

A = sx(0A)xs = ss

its

all

direction.

wave not plane polarized may be discussed by wave into waves which are plane polarized. the decomposing a Let be a vector drawn in the direction A of the 144.] of a

displacement and

let

the magnitude of a be so determined

that

(8)

The equation (7) then becomes reduced a = sx

(<2>*a)

to the

Xs = s-s #-a a a = 1.

form

ss-#-a

(9)

(8)

These are the equations by which the discussion of the velocity or rather the slowness of propagation of a wave in different directions in a non-isotropic

medium may a

be carried on.

(10)

VECTOR ANALYSIS

396

Hence the wave slowness direction a

equal in

magnitude (but not in direction) to the

drawn

radius vector

due to a displacement in the

in the

ellipsoid

in that

direction.

axa = = ss

=s But the

s(aX #

first

a)-

a x

The wave-slowness

a twice and vanishes.

aXs #

term contains a x

= 0.

Hence (11)

therefore lies in a plane with the direction a of displacement and the normal a drawn to the ellipsoid a

at the terminus of a.

Since s is perpen a dicular to a and equal in magnitude to it is evidently com determined as pletely except regards sign when the direction

known.

Given the direction of displacement the

line of

advance of the wave compatible with the displacement is com pletely determined, the velocity of the advance is likewise

known.

The wave however may advance

in either direction

along that line. By reference to page 386, equation (11) is seen to be the condition that a shall be one of the principal axes of the ellipsoid formed by passing a plane through the ellipsoid

perpendicular to there are

Hence

two possible

for

any given direction of advance These are the

lines of displacement.

principal axes of the ellipse cut

from the

ellipsoid a

by a plane passed through the center perpendicular line of advance.

minateness of

to the

To these statements concerning the deterwhen a is given and of a when s is given just

such exceptions occur as are obvious geometrically. If a and a are parallel s may have any direction perpendicular to a. This happens when a is directed along one of the principal axes of the ellipsoid.

If s is perpendicular to

circular sections of the ellipsoid a

plane of the section.

one of the

may have any direction in the

THE PROPAGATION OF LIGHT IN CRYSTALS

When slowness

397

the direction of displacement is allowed to vary the To obtain the locus of the terminus of s, a varies.

must be eliminated from the equation

=s -s (I a

The dyadic

s s

(P)

a a

= 0.

(12)

in the parenthesis is planar because

vectors parallel to

The

third or determinant

annihilates

This

is zero.

gives immediately

(0-

This

#) 8

ss)

= 0,

= 0.

a scalar equation in the vector

(13) It is the locus of

the extremity of s when a is given all possible directions. number of transformations may be made. By Ex. 19, p. 331, (<P

+ ef) 8 =

00-

</>

Hence

Dividing out the

common

factor

and remembering that $

self-conjugate.

Hence

-r

-8

Let

= 0.

8-1.8 S

1 si)- -8

(CM-s-

8*8

= 0.

(14)

VECTOR ANALYSIS

398

/^_W/_J_\ JJ++

[i_g

i-i.i*-^_-jj Let

Then

= xi +

+ zk

and

[ir^J

= #2 +

z 2.

the equation of the surface in Cartesian coordinates

j?/

i-_

l-fl2

-72

(14)

The equation

in Cartesian coordinates

may

be obtained

rp n f1 v f rr\ from

directly

The determinant a2

x y x

By means

of this dyadic

x y 6

of the relation s 2

= x2 +

(13)

z 2 this

assumes the

forms n

~T~

"22 s

~~" -^

This equation appears to be of the sixth degree. It is how ever of only the fourth. The terms of the sixth order cancel out.

The

Suppose that a represents the wave-slowness. polarized in the direction a passes the origin at a

vector

plane wave

THE PROPAGATION OF LIGHT IN CRYSTALS At

certain instant of time with this slowness.

unit of time

will

have travelled in the direction

equal to the reciprocal of the

magnitude

be in this position represented by the vector If

= ui +

399

the end of a s,

a distance

The plane

will

(page 108).

wit

the plane at the expiration of the unit time cuts off intercepts upon the axes equal to the reciprocals of u, v, w. These quantities are therefore the plane coordinates of the plane. They are connected with the coordinates of the points in the

plane by the relation

ux + vy + wz = If different plane

waves polarized

in all possible different

directions a be supposed to pass through the origin at the

same instant they will envelop a surface at the end of a unit of time. This surface is known as the wave-surface. The perpendicular upon a tangent plane of the wave-surface is the reciprocal of the slowness and gives the velocity with which the

wave

The equation

travels in that direction.

surface in plane coordinates u,

of the wave-

identical with the

equa

tion for the locus of the terminus of the slowness vector

The equation

= (15)

where

-f

+ w2

This may be written in any of the

The The equations

forms given previously. Wave-Surface.

page 397

the variable vector

surface

in vector s

known

as FresneVs

form are given on

be regarded as determining a

plane instead of a point. 145.]

In an isotropic medium the direction of a ray of

light is perpendicular to the wave-front.

the direction of the

wave

advance.

The

It is the

same as

velocity of the ray

VECTOR ANALYSIS

400

In a non-isotropic The ray does not travel per

equal to the velocity of the wave.

medium

this is

no longer

true.

pendicular to the wave-front

that

is,

in the direction of the

And

the velocity with which the ray travels wave s advance. In fact, whereas the is greater than the velocity of the wave.

wave-front travels off always tangent to the wave-surface, the ray travels along the radius vector drawn to the point of tan-

The wave-pknes envelop the wave-plane. wave-surface; the termini of the rays are situated upon it. Thus in the wave-surface the radius vector represents in mag the

gency of

nitude and direction the velocity of a ray and the perpen upon the tangent plane represents in magnitude and

dicular

direction the velocity of the wave.

surface the surface which

If instead of the

wave-

the locus of the extremity of the

wave slowness be considered

seen that the radius vector

it is

represents the slowness of the wave; and the perpendicular upon the tangent plane, the slowness of the ray.

Let v be the velocity of the

Then

ray.

because

the extremity of v lies in the plane denoted by s. Moreover the condition that v be the point of tangency gives d v per pendicular to s. In like manner if a be the slowness of the 1 and the condition ray and v the velocity of the wave, s v of tangency gives d s perpendicular to v. Hence r

and v

may

= 0,

and v

be expressed in terms of a

s <P

= 2s.tfs<P.a sds-

= 1,

(16)

= 0,

a, s,

and

s s

= 0,

as follows.

^arfs + ss<?-rfa ss # d a.

Multiply by a and take account of the relations a a 4> . d a and a a = s s. Then

and

THE PROPAGATION OF LIGHT IN CRYSTALS ds

But

since v

= # (s

Hence

= 0,

v and

direction.

= 0.

a),

(17)

=xs

8*8 a

a have the same

401

= 0.

Hence the ray velocity v

is perpendicular to a, that is, the the to the tangent plane ellipsoid at the ray extremity of the radius vector a drawn in the direction of the displacement. Equation (17) shows that v is coplanar with

velocity lies in

a and plane.

The

vectors

In that plane

angle from

Making use it is

(16) (17),

s to

a, s,

s is

and v therefore

lie

in one

;

perpendicular to a ; and v , to equal to the angle from a to $

the relations already found (8) (9) (11) easy to show that the two systems of vectors

and

x v

are reciprocal systems. tions take

(<P

a be replaced by a the equa

on the symmetrical form

= -a = a = s x a s = a x v

=a .v =a s

-a

x a

Thus

=v =a 0-1

a .

= 1, = 1,

x a xv x

= 1.

(18)

a dual relation exists between the direction of displace on the one hand ; ment, the ray-velocity, and the ellipsoid 26

VECTOR ANALYSIS

402

and the normal to the ellipsoid, the wave-slowness, and the ~l on the other. ellipsoid 146.]

was seen that

cular sections of

was normal

to one of the cir

the displacement a could take place in any For all directions in

direction in the plane of that section. this plane the

wave-slowness had the same direction and the

same magnitude.

Hence the wave-surface has a singular

plane perpendicular to s. This plane is tangent to the surface along a curve instead of at a single point. Hence if a wave travels in the direction s the ray travels along the elements of

the cone drawn from the center of the wave-surface to this

curve in which the singular plane touches the surface. The two directions s which are normal to the circular sections of are called the

wave

primary

These are the axes of equal

optic axes.

but unequal ray velocities. In like manner v being coplanar with a and velocities

[4>

The

a v

v <P~l

]

a 0.

equation states that if a plane be passed through the center of the ellipsoid <P~l perpendicular to V, then a a will be directed along one of the prin which is equal to last

Hence if a ray is to take a definite It is more con direction a may have one of two directions. venient however to regard v as a vector determining a plane. The first equation cipal axes of the section.

a v a]

states that a is the radius vector

drawn

in the ellipsoid

the point of tangency of one of the principal elements of the if by a principal parallel to v cylinder circumscribed about element element is meant an passing through the extremities :

the major or minor axes of orthogonal plane sections Hence given the direction v of the ray, the

of that cylinder.

two

are those radii vectors possible directions of displacement

VARIABLE DYADICS of the ellipsoid

which

lie in

403

the principal planes of the cylin

der circumscribed about the ellipsoid parallel to v If the cylinder is one of the two circular cylinders which the direction of displacement may be circumscribed about .

may

be any direction in the plane passed through the center and containing the common curve of tangency

of the ellipsoid

The ray-velocity for all these directions of displacement has the same direction and It is therefore a line drawn to one the same magnitude. of the cylinder with the ellipsoid.

At this singular of the singular points of the wave-surface. number of infinite an there are tangent planes envelop point ing a cone. The wave-velocity may be equal in magnitude and direction to the perpendicular drawn from the origin to any of these planes. The directions of the axes of the two are circular cylinders circumscriptible about the ellipsoid the directions of equal ray-velocity but unequal wave-velocity. They are the radii drawn to the singular points of the wave-

surface

and are called the secondary

optic

axes.

travels along one of the secondary optic axes the

If a ray

wave planes

travel along the elements of a cone.

Variable Dyadics.

The Differential and Integral Calculus

Hitherto the dyadics considered have been constant. vectors which entered into their make up and the scalar

147.]

The

coefficients

which occurred in the expansion in nonion form For the elements of the theory and for

have been constants.

elementary applications these constant dyadics suffice. The introduction of variable dyadics, however, leads to a simplifica

and unification of the differential and integral calculus of vectors, and furthermore variable dyadics become a necessity in the more advanced applications for instance, in the theory tion

of the curvature of surfaces

body one point of which

and in the dynamics of a

is fixed.

rigid

VECTOR ANALYSIS

404

W be

a vector function of position in space. Let r be the vector drawn from a fixed origin to any point in space.

Let

= dx i + dy j +

5W 3W + dy ^dx-^ $# c?y

Hence

dz-Ti c/

---h 3W + k 5W) W = d r H SW dx dz (

The expression enclosed

a dyadic.

It thus

a linear function of

The

first partial

)

in the braces

appears that the differential of the differential change of position.

and the consequents the

(

antecedents are

derivatives of

c?r,

i, j,

W with re

The expression is found in a manner precisely spect of x, y, z. W. analogous to del and will in fact be denoted by

=i-

k-.

(1)

dW = dr.VW.

Then This equation

(2)

like the one for the differential of a scalar

function F.

dV=dr VF. It

may be regarded

nonion form

as defining

VW.

expanded into

VW becomes

VW = 11 3x .

+ if

.5X

.9Y

-+ -+kj T kk-^--, 3z z dz

VARIABLE DYADICS

405

and x which were applied to a vector The operators function now become superfluous from a purely analytic For they are nothing more nor less than the standpoint.

and the vector of the dyadic

scalar

V W.

W = V W = (V W)* curl W = V x W = (V W) div

(4) (5)

The

analytic advantages of the introduction of the variable

dyadic ator

In the first place the oper to a vector function just as to a scalar applied In the second place the two operators and x are therefore these.

V may be

function.

V W. On

are reduced to positions as functions of the dyadic

the other hand from the standpoint of physics nothing is to be gained and indeed much may be lost if the important in and as the divergence and curl x terpretations of

W be forgotten

of the scalar and vector of If

and their places taken by the analytic idea

VW.

the vector function

be the derivative of a scalar

function V^

dW = dVF=e?r V V F=

where

i i

+ 75x <y

Qty

**dy dx + The

zero

;

= <y

=- + c/

k^ <y x

T dy d z

TT &y

- + kj -g- + k k dzdx 9zSy

result of applying

be a dyadic. is

VVF",

This dyadic

twice to a scalar function

is self -conjugate.

Its vector

V V V is evidently 3 V 92 V 3 F V-VF= (VVF)*= 0-22 + T-2 + TT

its scalar

seen to

Vx VV

VECTOR ANALYSIS

406 If

V three

cally to a scalar function

sum

times, the result

symboli

would be a

of twenty-seven terms like *

r c/

This

an attempt were made to apply the operator

a triadic.

;r-

vx dy &z

,etc.

Three vectors are placed in juxtaposition

without any sign of multiplication. Such expressions will not be discussed here. In a similar manner if the operator

be applied twice to a vector function, or once to a dyadic func tion of position in space, the result will be a triadic and hence outside the limits set to the discussion here.

x and

may however be applied respectively a dyadic and a vector.

V-

30 = i._ + 3x

50 + ^Sy

30 dy

u, v,

w are vector functions

x u

V 0= V

and

Vx 0= x*

i{-

X)v V ^

^-) +

I* j^

to yield

(7)

30 -. T 3z

(8)

of position in space,

+V w

Vxvj

operators

k x ^-, 9z

+ k-

where

T r

=u i+v +w

+ ^dx

The

to a dyadic

v v

-f

(7) (8)

k w, W

-++

< 8 >"

In a similar manner the scalar operators (a V) and (V V) may be applied to 0. The result is in each case a dyadic,

VARIABLE DYADICS

(a.V)<P

(V The

operators a

tions are ators.

= ^32

and

+ a2

30 ^- + $

z 2f

407

+ c?

5d> a8

32 -

V V as

The

individual steps

(10)

applied to vector func

no longer necessarily to be regarded

the dyadic

(9)

as single oper

be carried out by means of

may

VW. (a

W=a V) W = V V)

= a V W, (V W) = V V W.

(V W)

But when applied to a dyadic the operators cannot be inter preted as made up of two successive steps without making use of the triadic V 0. The parentheses however may be removed without danger of confusion just as they were removed in case of a vector function before the introduction of the dyadic.

Formulae similar to those upon page 176

may

differentiating products in the case that the lead to dyadics.

(u v)

= >V u v +

be given for

differentiation

V v,

V(vxw)=Vvxw Vwxv, Vx

= w V v V v w v V w + V w v, V (v w) = V v w + V w v, V (v w) = V v w + v V w.

x w)

Vx V

(v .

(u #)

VxVx

The

principle in these ciated before, namely :

x v

=Vu

= VV. and

all

w <P

v x

V w,

V V V

0, <P,

etc.

similar cases is that

The operator

V may be treated

enun sym-

VECTOR ANALYSIS

408

The

bolically as a vector.

must be to which

x (vw)

Hence (v

Hence 148.]

(v w)] v

x w)

[V [V

[V x

x vw

=Vv

V w.

v x

x w)] T + [V

x w)] w

x w)]^

x w,

V (w x v)] = V w x V(vxw) = Vvxw v w x =

x w)] v

implies

+ [V x (vw)]^

[Vx (vw)] w = V xvw, - [v x V w] v = - v (v w)] v =

[V x

which

upon each factor of a product

Thus

it is applied.

Again

differentiations

carried out in turn

[

was seen (Art. 79) that

curve of which the

initial

of that function at r

and r

denote an arc of a

and the

final point is r point the line integral of the derivative of a scalar function taken along the curve is equal to the difference between the values

Cd r Jc

may

by no means the same

vector

V W = 0.

be well to note that the integrals

fdr.VW are

F(r)- F(r ).

V W = W (r) - W (r ),

Cd r

and

VF=

In like manner

is r

dx cannot be

and thing.

fvw

placed arbitrarily

dr is

upon

a dyadic.

The

either side of

it.

VARIABLE DYADICS

409

to the fundamental equation (2) the differential di W. The differentials must be written necessarily precedes

Owing

before the integrands in most cases. For the sake of uni they always will be so placed.

formity of

Passing to surface integrals, the following formulae, some which have been given before and some of which are new,

may

be mentioned.

ff ax

VW=

fdr

ff da. Vx W=

fdr*

r/daVx0=

The

line integrals are taken over the complete bounding curve of the surface over which the surface integrals are taken. In

manner the following

relations exist

between volume and

surface integrals.

fff dv VW=rfa W

///

VECTOR ANALYSIS

410

The

surface integrals are taken over the complete bounding which the volume integrals

surface of the region throughout are taken.

Numerous

formulae of integration by parts like those upon The reader will rind no difficulty in

page 250 might be added. obtaining also be

them

The

for himself.

extended to other cases.

integrating operators may the potentials of scalar

and vector functions the potential, Pot added.

The Newtonian

cian and Maxwellian of dyadics

Pot

</>,

of a dyadic

may

of a vector function and the Lapla-

may

be defined.

New W =

// ^^I^> dV d,

Max * = The

analytic theory of these integrals may be developed as The most natural way in which the demonstrations

before.

may sum

be given

by considering the vector function

W as

the

of its components,

W = Xi+

Fj + ^k

as expressed with the constant consequents and the dyadic i, j, k and variable antecedents u, v, w, or vice versa,

These matters will be tering

upon them

at all

left at this point.

was

The

object of en

to indicate the natural extensions

which occur when variable dyadics are considered. These ex tensions differ so slightly from the simple cases which have

THE CURVATURE OF SURFACES

411

gone before that it is far better to leave the details to be worked out or assumed from analogy whenever they may be needed rather than to attempt to develop

them

in advance. It

mention what the extensions are and

cient merely to

is suffi

how they

treated.

maybe

The Curvature of Surfaces 1

There are two different methods of treating the cur vature of surfaces. In one the surface is expressed in para149. ]

metic form by three equations

x =/i

=/a O>

=/8 <X

")>

(u, v).

analogous to the method followed (Art. 57) in dealing with curvature and torsion of curves and it is the method

This

employed by Fehr in the book to which reference was made. In the second method the surface is expressed by a single equation connecting the variables x,y,z ,

The

latter

method

thus

= 0.

of treatments affords a simple application of

the differential calculus of variable dyadics. Moreover, the most to the results connected important dyadics lead naturally

with the elementary theory of surfaces. Let r be a radius vector drawn from an arbitrary fixed The increment d r origin to a variable point of the surface. lies in

the surface or in the tangent plane

at the terminus of

Hence the surface.

drawn

to the surface

derivative

V^is

collinear with the

Moreover, inasmuch as

normal to the

F and the negative of F when

1 Much of what follows is practically free from the use of dyadics. especially true of the treatment of geodetics, Arts. 155-157.

This

VECTOR ANALYSIS

412

may be equated to zero give the same geometric surface, In considered as the normal upon either side of the surface. case the surface belongs to the family defined

F (#, y, z) = const. the normal increases.

V F lies upon that side upon which the constant Let V F be represented by N the magnitude of

which may be denoted by N, and in the direction of

IT.

let

n be a unit normal drawn

Then

(1)

If s is the vector

drawn

any point in the tangent plane

r and n are perpendicular. r, s the equation of the tangent plane is

the terminus of

Consequently

(s-r) and in

manner the equation

of the normal line is

(s-r)x VjF=0, or

where k

= r + & V JP

a variable parameter.

These equations may be

translated into Cartesian form 150.]

and give the familiar results. The variation dn of the unit normal to a surface

plays an important part in the theory of curvature, perpendicular to n because n is a unit vector.

THE CURVATURE OF SURFACES

N The dyadic dicular to

iv 2

413

an idemfactor for

vectors perpen

all

n and an annihilator for vectors

parallel

Hence

= d n,

n n)

V^.(I-nn)=0,

and

N Hence

But

dr <*

.-*,.

Hanco

N N

VV .F

= dr

nn).

n n).

~"> V

^ P-").

(2)

> = (I-..

Let

dn = dr

Then

<P.

(4)

In the vicinity of any point upon a surface the variation d n of the unit normal radius vector

The dyadic

is self-con

N4> c = Evidently (I is

and in

Hence

self-conjugate.

jugate.

For

- nn), (VV F) c

- n ri) c =

a vector parallel to fore planar

a linear function of the variation of the

- n n) <P C is

and by (6) Art. 147

equal to 0.

the dyadic

- nn)^

produces

When applied zero.

lie

It is there

fact uniplanar because self-conjugate.

antecedents and the consequents

VV F The

in the tangent plane to

VECTOR ANALYSIS

414 the surface.

It is possible

form 4>

(Art. 116) to reduce

i i

to the

b j j

(5)

and j are two perpendicular unit vectors lying in the tangent plane and a and b are positive or negative scalars. where

dn = dr The

vectors

and the scalars

The dyadic

of the surface. 151.]

The

conic r

a, 6

C? is

a and

is, if

the point

;

but

have the same the conic

j j ).

vary from point to point

variable.

1 is called the indicatrix of the

surface at the point in question.

that

i i

If this conic is

an hyperbola, that

ellipse,

convex at

sign, the surface is

is, if

a and

have opposite signs the surface is concavo-convex. The curve r . r = 1 may be regarded as approximately equal to the intersection of the surface with a plane drawn parallel to the tangent plane and near to

it.

r be set equal to zero

If r

a pair of straight lines. These are the asymp If they are real the conic is an hyperbola ; totes of the conic.

the result

imaginary, an

ellipse.

Two

directions on the surface

which

are parallel to conjugate diameters of the conic are called

con

The directions on the surface which coin jugate directions. cide with the directions of the principal axes i j of the ,

indicatrix are

known

as the principal directions.

They are a The directions upon the

special case of conjugate directions.

surface which coincide with the directions of the asymptotes of the indicatrix are

the

surface

known

as asymptotic directions.

convex, the indicatrix

ellipse

In case

and the

asymptotic directions are imaginary. In special cases the dyadic may be such that the cients a and

form

b are equal.

= a(i

may i

coeffi

then be reduced to the

)

(5)

THE CURVATURE OF SURFACES in an infinite

any two

convex.

cible to the

form <p

The i

The

and j may be becomes a

indicatrix

The surface in the neighborhood of an The asymptotic directions are imaginary.

In another special case the dyadic

The

perpendicular diameters of this circle Such a point is directions upon the surface.

an umbilic.

umbilic

directions

pair of

give principal called

The

of ways.

perpendicular directions.

Any

circle.

number

415

$ becomes

i i

linear

and redu (5)"

indicatrix consists of a pair of parallel lines perpendicular is called a parabolic point of the surface. Such a

point further discussion of these

and other

special cases will be

omitted.

The quadric surfaces afford examples of the various kinds The ellipsoid and the hyperboloid of two sheets of points. The indicatrix of points upon them is an ellipse. are convex. The hyperboloid of one sheet is concavo-convex. The in The indicatrix dicatrix of points upon it is an hyperbola. a is a The circle. of any point upon points are all sphere umbilies. The indicatrix of any point upon a cone or cylinder The points are parabolic. A sur is a pair of parallel lines. face in general

may have upon

hyperbolic, parabolic, 152.]

points of

all

types

elliptic,

and umbilical.

line of principal curvature

upon a surface

curve which has at each point the direction of one of the prin The direction of the curve at a cipal axes of the indicatrix. point is always one of the principal directions on the surface at that point. Through any given point upon a surface two per pendicular lines of principal curvature pass. Thus the lines of curvature divide the surface into a system of infinitesi

mal rectangles. An asymptotic line upon a surface is a curve which has at each point the direction of the asymptotes of the indicatrix.

The

direction of the curve at a point

one of the asymptotic directions upon the surface.

always

Through

VECTOR ANALYSIS

416

any given point of a surface two asymptotic lines are

the surface

lines pass.

These

Even when real The angle angles.

convex.

imaginary they do not in general intersect at right

between the two asymptotic lines at any point is bisected by the lines of curvature which pass through that point.

The necessary and

sufficient condition that a

curve upon a

surface be a line of principal curvature is that as one advances along that curve, the increment of d n, the unit normal to the

surface

rfn= 0. dr

Then

For

to the line of advance.

is parallel

i i

+ yj

b j j )

evidently d n and d r are parallel when and only when The statement is therefore proved. parallel to i or j .

It is frequently taken as the definition of lines of curvature.

The

differential equation of a line of curvature is

dnxdr = 0. Another method of statement

(6)

that the normal to the surface,

the increment d n of the normal, and the element d r of the surface is

lie

one plane when and only when the element d r

an element of a

line of principal curvature.

The

differential

equation then becomes [n

rfr]

= 0.

(7)

The

necessary and sufficient condition that a curve upon a surface be an asymptotic line, is that as one advances along that curve the increment of the unit normal to the surface is

perpendicular to the line of advance.

= dr dn dr = dr dn

If then

zero

asymptotic direction.

tf>

For

dr.

r is zero.

The statement

Hence d r

therefore proved.

an It

THE CURVATURE OF SURFACES is

417

The

frequently taken as the definition of asymptotic lines.

an asymptotic

differential equation of

line is

= 0.

(8)

Let P be a given point upon a surface and n the 153.] normal to the surface at P. Pass a plane p through n. This

p is normal to the surface and cuts out a plane section. Consider the curvature of this plane section at the point P. Let n be normal to the plane section in the plane of the section, n coincides with n at the point P. But unless the

plane

cuts the surface everywhere orthogonally, the normal n to the plane section and the normal n to the surface will not

plane

coincide,

d n and d n will also be

of the plane section lying in

The curvature

different.

(Art. 57).

____ ds

far as numerical value is concerned the increment of the

unit tangent t and the increment of the unit normal n are Moreover, the quotient of d r by d s is a unit vector equal.

in the direction of

Consequently the scalar value of C

d n

hypothesis

d (V

Hence

dn dr

dr ds

ds 2

ndr = n

d r)

= dn

and

n d

-dr

= 0.

= 0,

Since n and n are equal at P,

Hence

- - --

dn*dr = dr ds^2

j-^2 ds 27

= dr3

dr dr 3

( 9)

VECTOR ANALYSIS

418 C7

= a --

Hence

tf= a

cos 2 (i

dr)

& cos 2

(7= a

cos 2 (i , rfr)

b sin 2 (i

The

dr),

dr).

(10)

interpretation of this formula for the curvature of a

normal section

as follows

When

the normal to the surface from

the plane p turns about C of the , the curvature

plane section varies from the value a when the plane passes through the principal direction i , to the value b when it

The values passes through the other principal direction j of the curvature have algebraically a maximum and minimum .

in the directions of the principal lines of curvature. b

have unlike signs, that

at Pj there exist

the surface

The sum

and

concavo-convex

which the curvature of a

for

These are the asymptotic directions. two normal sections

normal section vanishes. 154.]

is, if

two directions

If a

of the curvatures in

at right angles to one another is constant and independent of the actual position of those sections. For the curvature in

one section

Cl = and

a cos 2

dr)

b sin 2 (i

dr),

in the section at right angles to this (7 2

Hence

sin 2 (i

di)

O l + C2 =

b cos 2 (i

dr).

(11)

which proves the statement.

show that the

invariant $% s is equal to the pro duct of the curvatures a and b of the lines of principal curv It is easy to

ature. 4>t

Hence the equation

<P a

= ab x

0^ 3

(12)

THE CURVATURE OF SURFACES is

the quadratic equation which determines the principal curv

atures a and I at any point of the surface.

and

equation the scalar quantities a of

419

F(x,

By means

of this

be found in terms

may

(I-nn)

y, z).

N (nn

VV^ )^ =

VV.F- nn)^ = (nn nn

Hence

(nn. Hence

= -(VV^)

(nn

(nn^

VV^)^ = nn: VVJ^=n. VV F <^^

= V.V^

VFVF:WF --

T--- --

(13)

CIS)

These expressions may be written out in Cartesian coordinates, but they are extremely long. The Cartesian expressions for are even longer. The vector expression may be obtained 2/5 as follows:

nn) 2

= nn.

Hence S-IA\

Given any curve upon a surface. Let t be a unit tangent to the curve, n a unit normal to the surface and m a 155.]

VECTOR ANALYSIS

420

x t. The three vectors n, t, m constitute The vector t is parallel to the element d r.

vector defined as n

i, j, k system. Hence the condition

for a line of curvature t

x d n

= 0.

(15)

Hence

m d n= = = m dn + n d m = 0.

Moreover

Hence

Hence d (m

becomes

= 0.

x dm

= 0,

d m.

(16)

dmxdn = Q.

The increments

m and of n

and of

a line of principal curvature. geodetic line upon a surface

(16)

r are all parallel in case of

a curve whose osculating That the

plane at each point is perpendicular to the surface.

is the shortest line which can be drawn between two points upon a surface may be seen from the following Let the surface be smooth and considerations of mechanics.

geodetic line

let a

smooth

elastic string

which

surface be stretched between any

acting under

constrained to

two points of

it.

lie

in the

The

string

own

tensions will take a position of equili brium along the shortest curve which can be drawn upon the surface between the two given points. Inasmuch as the its

string is at rest

surface

must

lie

upon the surface the normal

reactions of the

in the osculating plane of the curve.

Hence

that plane is normal to the surface at every point of the curve and the curve itself is a geodetic line.

The

vectors t and

mine that plane.

lie

in the

osculating plane and deter is a geodetic, the normal

In case the curve

to the osculating plane lies in the surface

perpendicular to the normal

Hence

and consequently

THE CURVATURE OF SURFACES

421

n*tx<2t = 0, n x

The

(17)

t = 0.

differential equation of a geodetic line is therefore

dr d 2 r] =0.

(18)

Unlike the differential equations of the lines of curvature and the asymptotic line, this equation is of the second order.

The

surface

therefore covered over with a doubly infinite

system of geodetics. Through any two points of the surface one geodetic may be drawn.

one advances along any curve upon a surface there is necessarily some turning up and down, that is, around the axis m, due to the fact that the surface

curved.

There may

may not be any turning to the right or left. If one advances along a curve such that there is no turning to the right or left, but only the unavoidable turning up and down, it is to be or

expected that the advance that total t,

is,

along the shortest possible route

Such

along a geodetic.

amount

form an

of deviation i, j,

Hence is

in fact the case.

from a straight

line is

d t.

The

Since

k system I

Since t

= tt +

nn + mm.

= tt*dt + nndt + mm*dt.

a unit vector the

first

term vanishes.

The second

term represents the amount of turning up and down; the third term, the

amount

to the right or left.

Hence

the proper measure of this part of the deviation from a In case the curve is a geodetic this term straightest line.

vanishes as was expected. curve or surface 156.]

may

mapped upon

a unit

fixed origin is sphere by the method of parallel normals. of a assumed, from which the unit normal n at the point

VECTOR ANALYSIS

422 given surface

of a sphere.

upon the surface

points P of a curve the points P will r

sphere.

The terminus

is laid off.

If the

of this normal lies

normals to a surface at all

are thus constructed from the trace a curve

This curve

upon

same

origin,

the surface of a unit

called the spherical image of the given

In like manner a whole region be mapped upon a region T the sphere.

curve.

T of

the surface

may

T upon region T upon

The

region

the sphere has been called the hodogram of the the surface. If d r be an element of arc upon the surface the

corresponding element upon the unit sphere

dn= If

da be an element

dr.

of area

upon the surface, the corre is d*! where (Art. 124).

sponding element upon the sphere

=a <P

= a6

The

ratio of

hodogram

curvature at

i i

d a.

& j j

dd = ab nn

Hence

its

= ab

nn.

d a.

an element of surface at a point is

(19)

P to the area of

equal to the product of the principal radii of

P or to

the reciprocal of the product of the prin

cipal curvatures at P.

was seen that the measure of turning to the right or left is m d t. If then is any curve drawn upon a surface the total amount of turning in advancing along the curve is the It

integral.

dt.

(20)

For any closed curve this integral may be evaluated in a manner analogous to that employed (page 190) hi the proof of Stokes s theorem.

Consider two curves

and

near

THE CURVATURE OF SURFACES The

together.

which the integral undergoes when changed from C to C is

variation

the curve of integration

f m-

fm.dt= Cs

dt.

di)= fSm dt+ Cm*Sdt

d(m-8t) = S

dmSt-hmd

Cm* dt= C Sm* dt- C dm*

+ C d (m

The integral of the perfect differential d (m when taken around a closed curve. Hence S

mdt=

The idemfactor

8m f or t

and

(2t

= tt +

= m

= m nn

dt,

Si).

S t) vanishes

nn + mm, e2t

m m vanish. A dm S

be effected upon the term S

423

similar transformation

mdt= /(Smn n^t rfmn

differentiating the relations

may

Then

n-St).

and n

it is

seen that

c?mn=:--m^n Hence

/m^t=/

Sn t*dn

Sim rft=/(mxt-8nx^n)=

m dn I

Sn)

Sn x dn.

VECTOR ANALYSIS

424

The the

differential

Sn x dn

hodogram upon

represents the element of area in the unit sphere. The integral

represents the total area of the hodogram of the strip of surface which lies between the curves C and C Let the f

C start at

curve

any desired quired in

2 TT.

The

a point upon the surface and spread out to

The

size.

total

amount

making an infinitesimal

of turning which

circuit

is re

about the point

total variation in the integral is

f 8 fin. rft=f m.dt-27T.

But

H denote the total area of the hodogram.

Hence

= 2 TT

iT=27r

H+

The

area of the

Cm*

fm dt

-JET,

rft,

(22)

= 27r.

hodogram of the region enclosed by any amount of turning along that curve

closed curve plus the total is

equal to 2 TT.

If the surface in question is

convex the area

will appear positive when the curve upon the so described that the enclosed area appears positive.

upon the sphere surface If,

however, the surface

concavo-convex the area upon the This matter of the sign of the

sphere will appear negative.

hodogram must be taken above.

into account in the statement

made

THE CURVATURE OF SURFACES 157.]

If the

closed curve

geodetic lines the

The

amount

425

a polygon whose sides are

of turning along each side is zero.

total turning is therefore equal to the

sum

of the exterior

The statement becomes : the sum of angles of the polygon. the exterior angles of a geodetic polygon and of the area of the hodogram of that polygon (taking account of sign) is equal to 2 TT. Suppose that the polygon reduces to a triangle. If the surface is

convex the area of the hodogram

and the sum of the exterior angles of the triangle 2 TT. TT.

positive

is less

than

The sum of the interior angles is therefore greater than The sphere or ellipsoid is an example of such a surface.

If the surface is

negative.

concavo-convex the area of the hodogram

The sum

this case less

than

of the interior angles of a triangle is in

TT.

Such a surface

sheet or the pseudosphere.

There

the hyperboloid of one

an intermediate case in

which the hodogram of any geodetic triangle is traced twice in opposite directions and hence the total area is zero. The sum of the interior angles of a triangle upon such a surface is equal TT. Examples of this surface are afforded by the cylinder,

cone, and plane. surface is said to be developed

when

it is

so deformed that

upon the surface retain their length. Geodetics remain One surface is said to be developable or applicable geodetics. upon another when it can be so deformed as to coincide with lines

the other without altering the lengths of lines. Geodetics upon one surface are changed into geodetics upon the other.

The sum

any geodetic triangle remain un From this it follows the of process developing. changed by that the total amount of turning along any curve or the area of the

of the angles of

hodogram

any portion of a surface are also invariant

of the process of developing.

VECTOR ANALYSIS

426

Harmonic Vibrations and

The

158.]

motion

The

differential

Bivectors

equation of rectilinear harmonic

integral of this equation

be reduced by a suitable

may

choice of the constants to the form

sin

This represents a vibration back and forth along the X-axis 0. Let the displacement be denoted by about the point x D in place of x. The equation may be written

This

=i A

Consider

=i A sin

n t.

cos

m x.

a displacement not merely near the point x

but along the entire axis of

sin

points x

Or,

where

a positive or negative integer, the displacement is at all times equal to zero. The equation represents a stationary is

wave with nodes at these points. At points midway between these the wave has points of maximum vibration. If the equation be regarded as in three variables x, y, z it repre wave the plane of which is perpendicular to the axis of the variable x. sents a plane

The displacement given by

Dx is

the equation

cos

(m x

(1)

wave perpendicular to the axis of x but The vibration is harmonic and advances

likewise a plane

not stationary.

along the direction

with a velocity equal to the quotient of

HARMONIC VIBRATIONS AND BIVECTORS n by m.

If v be the velocity;

the period; and

the

427

wave

length,

n = -, m

2?r I

= -.

(2)

The displacement

=j A

D2 differs

cos

(m x

nt)

from Dj in the particular that the displacement takes

The wave as place in the direction j, not in the direction i. before proceeds in the direction of x with the same velocity. This vibration is transverse instead of longitudinal. By a simple extension

it is

seen that

= A cos

(m x

a displacement in the direction A. The wave advances along the direction of x. Hence the vibration is oblique to is

the wave-front.

A still

by substituting m

r for

more general form may be obtained

m x.

= A cos

Then r

(3)

t).

This is a displacement in the direction A. The maximum amount of that displacement is the magnitude of A. The wave advances in the direction m oblique to the displace ment; the velocity, period, and wave-length are as before. So much for rectilinear harmonic motion. Elliptic har

monic motion may be defined by the equation

The general

integral r

The

obtained as

= A cos

discussion of waves

viously.

(p. 117).

may

The general wave

advancing in the direction

sin

be carried through as pre

harmonic

elliptic

seen to be

motion

VECTOR ANALYSIS

428

= A cos

(

A sin

sin

(4)

t).

}

+ B cos (m

(5)

the velocity of the displaced point at any moment in the This is of course differ ellipse in which it vibrates. is

entirely

ent from the velocity of the wave. An interesting result is obtained by adding up the dis placement and the velocity multiplied by the imaginary unit

The

and divided by

= A cos

A sin (m

sin

+ B cos (m

expression here obtained, as far as

its

form

n f)

n t) }.

concerned,

an imaginary vector. It is the sum of two real vectors of which one has been multiplied by the imaginary scalar 1. is

Such a vector

ordinary imaginary scalars

may

be called biscalars.

found very convenient in the discussion of harmonic motion. Indeed any undamped elliptic har

of bivectors elliptic

The The use

called a bivector or imaginary vector.

monic plane wave may be represented as above by the pro duct of a bivector and an exponential factor. The real part of the product gives the displacement of

pure imaginary part gives divided by n.

any point and the

the velocity of

displacement

The

analytic theory of bivectors differs from that of much as the analytic theory of biscalars It is unnecessary to have differs from that of real scalars. 159.]

real vectors very

any distinguishing character

for bivectors just as

it is

need-

HARMONIC VIBRATIONS AND BIVECTORS less to

vector

429

have a distinguishing notation for biscalars. The bimay be regarded as a natural and inevitable extension It is the

of the real vector.

formal

sum

two

real vectors

which one has been multiplied by the imaginary unit 1. There The usual symbol i will be maintained for of

not

much

likelihood of confusion with the vector

1is

for the

reason that the two could hardly be used in the same place and for the further reason that the Italic i and the Clarendon i

Whenever

differ considerably in appearance.

becomes

especially convenient to have a separate alphabet for bivectors the small Greek or German letters may be called upon.

bi vector

may be

expressed in terms of

i, j,

k with com

plex coefficients. If

= TJ +

and

i r2

= #i +

yj + z.

Two

bivectors are equal when their real and their imaginary Two bivectors are parallel when one is the parts are equal.

product of the other by a scalar (real or imaginary). If a bivector is parallel to a real vector it is said to have a real direction.

In other cases

direction.

The value

has a complex or imaginary

the sum, difference, direct, skew,

and indeterminate products of two bivectors is obvious with out special definition. These statements may be put into analytic form as follows. Let

Then if

r if

= TJ + r = s,

= s2 + i s2 r = B and r 2 = s 2 r = x s = (x + i # 3 ) and

VECTOR ANALYSIS

480 r

+s=

(r 1

<>!

= (r

r r

i(r 2

- r2

(r l s l

s 1)

s ) 2

r2 s2)

s ), 2

(r l

+ i (r

(T I s 2

Sl ),

r 2 Sj).

Two

bivectors or biscalars are said to be conjugate when their real parts are equal and their pure imaginary parts differ is

The conjugate

only in sign.

of a real scalar or vector

The conjugate

equal to the scalar or vector itself.

and

sort of product of bivectors

any

biscalars is equal to the pro

similar duct of the conjugates taken in the same order. made and differences. be sums may concerning

statement

Oi +

i r ) 2

Oi + Ol +

r2> ( r l

- * r2 ) = r

(r i

*2 )

(r x

~ * F2> =

If the bivector r

= TJ

- i r2 ) =

(rl rl

i r2

F2 r2>

r2 ,

x TV ( r2 F l

-r

r2>-

multiplied by a root of unity or cyclic factor as it is frequently called, that is, by an imagi nary scalar of the form cos q

where the conjugate

4- i r2 be*

= a + ib, a 2 + & 2 = 1,

sin q

multiplied by a

i 6,

(7)

and hence the four

products

are unaltered

Thus

by multiplying the bivector

by such a

r TI

= r/ + TI

i r 2

= r2

+ x

iV) (r x T!

i r2 ),

r2 , etc.

factor.

HARMONIC VIBRATIONS AND BIVECTORS

160.]

bivector by

closer examination of the effect of multiplying a a cyclic factor yields interesting and important

Let

geometric results. ri

* r a

Then

( cos ?

in the real

cos ^

= r2

cos q

reference to Art. 129

duced

431

sin 2) ( r i r2 sin

ra)-

( 8)

T I sin j.

will be seen that the

and imaginary vector parts

change pro

of a bivector

multiplication with a cyclic factor, is precisely the same as would be produced upon those vectors by a cyclic dyadic d>

=aa +

cos q

used as a prefactor.

(bb

c c )

q (c b

- be )

b and c are supposed to be two vectors and r2 a is any vector not in

collinear respectively with r x their plane.

- sin

Consider the ellipse of which TJ and r2 are a It then appears that r^

pair of conjugate semi-diameters.

and

ellipse.

are also a pair of conjugate

They

semi-diameters of that

are rotated in the ellipse

a sector of which the area

from

to the area of the

toward

whole

r 1$

ellipse

?r. Such a change of position has been called an rotation elliptic through the sector q. The ellipse of which T I and r2 are a pair of conjugate semi-

as q

to 2

diameters

When

called

the directional

ellipse

of the bivector

the bivector has a real direction the directional ellipse

reduces to a right line in that direction. When the bivector has a complex direction the ellipse is a true ellipse. The angular direction from the real part T I to the complex part r2 considered as the positive direction in the directional If the real and imagi ellipse, and must always be known.

nary parts of a bivector turn in the positive direction in the if in the negative direc ellipse they are said to be advanced tion they are said to be retarded. Hence multiplication of a ;

VECTOR ANALYSIS

432 bivector by

cyclic

factor retards

sector equal to the angle

its directional

ellipse

of the cyclic factor.

It is always possible to multiply a bivector

by such a cyclic and imaginary parts become coincident the ellipse and are perpendicular.

factor that the real

with the axes of r

(cos q

sin q) (a

b) where a

accomplish the reduction proceed as follows r

If a

(cos 2 q

i sin

2 q) (a

Form

b).

= 0, r

(cos 2 q

Let

and

sin 2 q) (a

=a+

tan 2 q

With

= 0.

b).

i 6,

this value of q the axes of the directional ellipse are

given by the equation a

-f i

(cos q

sin q)

In case the real and imaginary parts a and b of a bivector are equal in magnitude and perpendicular in direction both a and b in the expression for r r vanish. Hence the angle q

The

indeterminate.

directional ellipse

bivector whose directional ellipse lar bivector.

The necessary and

condition r

called a circu

= 0, r circular. r = zi + 2/j + *k, 2 r = x* + y* + z = 0. r

The

a circle.

sufficient condition that

non-vanishing bivector r be circular

a circle

= 0, which for real vectors

implies r

not sufficient to ensure the vanishing of a bivector.

= 0, The

HARMONIC VIBRATIONS AND BIVECTORS bivector

The

circular, not necessarily zero.

a bivector vanish

condition that

that the direct product of

433

its

con

jugate vanishes.

Oi +

i r2 )

(r x

then

- t r2 ) = r

= r2 =

and

conjugate and

that

= 0,

becomes equal to

their product

The condition

161.]

= 0.

In case the bivector has a real direction its

two bivectors be

parallel is that

the product of the other by a scalar factor. Any biscalar factor may be expressed as the product of a cyclic If factor and a positive scalar, the modulus of the biscalar.

one

two bivectors ellipses

by only a cyclic factor their directional Hence two parallel vectors have their are the same. differ

directional ellipse similar

and

similarly placed

the ratio of

similitude being the modulus of the biscalar. It is evident that any two circular bivectors whose planes coincide are parallel.

A circular

vector and a non-circular vector cannot

be parallel. The condition that two bivectors be perpendicular r

Consider

first

coincide.

= 0, s2

= 0.

the case in which the planes of the bivectors

Let r

The

(TJ

i r2 ),

dicular to r2 ,

and b are biscalars. and s l may be taken

condition r

scalars a

(s 1

may

i g2 ).

be chosen perpen

in the direction of ra

then gives 82

and

rx 28

= 0.

The

VECTOR ANALYSIS

434

The

equation shows that r2 and

first

hence

and

are perpendicular and

are

perpendicular. Moreover, the second shows that the angular directions from rx to r2 and from s to 1 s are the same, and that the axes of the directional 2 ellipses of r

and

are proportional.

Hence the conditions

for perpendicularity of

whose planes coincide are that similar, the

two bivectors

their directional ellipses are

is the same, and the 1 axes of the are If both vectors major ellipses perpendicular. have real directions the conditions degenerate into the per

angular direction in both

pendicularity of those directions. The conditions therefore hold for real as well as for imaginary vectors.

Let r and

be two perpendicular bivectors the planes of Resolve T I and r2 each into two com

which do not coincide.

ponents respectively parallel and perpendicular to the plane

The components perpendicular to that plane contribute nothing to the value of r s. Hence the components of rx of

and

r2 parallel to the plane of s

perpendiqular to

form a bivector

this bivector

and

which

the conditions

stated above apply. The directional ellipse of the bivector r is evidently the projection of the directional ellipse of r upon

the plane of

Hence, ellipse of

two bivectors are perpendicular the directional either bivector and the directional ellipse of the if

other projected upon the plane of that one are similar, have the same angular direction, and have their major axes per pendicular.

Consider a bivector of the type

162.]

where

and

are bivectors

and

position vector of a point in space.

a biscalar.

r is the

It is therefore to be con-

1 It should be noted that the condition of perpendicularity of major axes is not the same as the condition of perpendicularity of real parts and imaginary parts

HARMONIC VIBRATIONS AND BIVECTORS sidered as real,

be considered as

real.

the scalar variable time and

435 also to

Let

A = Aj +

A*p

m = ni + i mj

D As

has been seen before, the

represents a train of plane

factor

waves of

(A x

elliptic

Aj)

e <(mt

~ nif)

harmonic vibra

The

vibrations take place in the plane of Aj and A2 , in an ellipse of which A x and A% are conjugate semi-diam tions.

The displacement of the vibrating point from the eters. center of the ellipse is given by the real part of the factor. The velocity of the point after it has been divided by nj is given by the pure imaginary part. The wave advances

The other factors in the expres ""* is a The factor damper in the direction m 2 As the wave proceeds in the direction m^ it dies away. The factor e*** is a damper in time. If na is negative the wave dies away as time goes on. If n2 is posi The tive the wave increases in energy as time increases.

in the direction

sion

are dampers. .

presence (for unlimited time) of any such factor in an ex pression which represents an actual vibration is clearly inad missible.

It contradicts the

law of conservation of energy.

In any physical vibration of a conservative system na

cessarily negative or zero.

The general

expression (9) therefore represents a train of plane waves of elliptic harmonic vibrations damped in a definite direction and in time. Two such waves may be com

pounded by adding the bivectors which represent them. If n t is the same for both the resulting the exponent m r train of waves advances in the same direction and has the

VECTOR ANALYSIS

436

same period and wave-length

as the individual waves.

vibrations, however, take place in a different ellipse.

The If the

waves are

the resultant

By combining two directions but

- m*-

Ae-^- e-""

<mi

**.

waves which advance in opposite

in other respects equal a system of

obtained. (mi

The theory of

trains of

which are

stationary waves

- lr

- n0

+ A e~ m

+ $-"i") = 2Acos

bivectors

and

(n^

e- m*

e^ int

their applications will not be

The object in entering at all upon this very short and condensed discussion of bivectors was first to show carried further.

the reader

how

the simple idea of a direction has to give way to the more complicated but no less useful idea of a directional

when

the generalization from real to imaginary vectors made, and second to set forth the manner in which a single

ellipse

bivector

D may

waves of

elliptic

vectors

be employed to represent a train of plane harmonic vibrations. This application of bi

may be used to give

the Theory of Light a wonderfully

1 simple and elegant treatment. 1 Such use of bivectors is made by Professor Gibbs in his course of lectures on " The Electromagnetic Theory of Light" delivered biannually at Yale University. Bivectors were not used in the second part of this chapter, because in the opinion

of the present author they possess no essential advantage over real vectors until the more advanced parts of the theory, rotation of the plane of polarization by

magnets and

crystals, total

and metallic

reflection, etc., are reached.

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