C1
Sp f: R ⊆ �2 → �3
Ď•(u, v) = (x(u, v), y(u, v), z(u, v))
x(u, v), y(u, v), z(u, v)
(u0 , v0 ) ∈ Sp
u
Tu =
∂x ∂y ∂z (u , v )đ??˘ + (u0 , v0 )đ??Ł + (u0 , v0 )đ??¤ ∂u 0 0 ∂u ∂u v
Tv =
1
C1
∂x ∂y ∂z (u0 , v0 )đ??˘ + (u0 , v0 )đ??Ł + (u0 , v0 )đ??¤ ∂v ∂v ∂v
Los campos vectoriales son las funciones vectoriales que estudiaste en la unidad 1.
đ??˘ đ??Ł ∂x ∂y | |Tu Ă— Tv | = ∂u ∂u | ∂x ∂y ∂v ∂v
đ??¤ ∂z | ∂u| = ∂z ∂v
∂z ∂y ∂y ∂z ∂z ∂x ∂x ∂z ∂y ∂x ∂x ∂y = ( ( ) − ( )) đ??˘ − ( ( ) − ( )) đ??Ł + ( ( ) − ( )) đ??¤ ∂v ∂u ∂v ∂u ∂v ∂u ∂v ∂u ∂v ∂u ∂v ∂u
Sp
Tu Ă— Tv = 0
Tu Ă— Tv
Sp = r(u, v) f: R ⊆ �3 → �
đ?‘†đ?‘?
f(x, y, z) ∈ � f
đ?‘†đ?‘?
âˆŹ f(x, y, z)dS = âˆŹ f(Ď•(u, v))|Tu Ă— Tv |dudv = Sp
Sp
∂(x, y) 2 ∂(y, z) 2 ∂(z, x) 2 = âˆŹ f(x(u, v), y(u, v), z(u, v))√[ ] +[ ] +[ ] dudv ∂(u, v) ∂(u, v) ∂(u, v) R
âˆŹđ?‘† 3đ?‘Ľ 2 đ?‘‘đ?‘† đ?‘?
x = senφcosθ y = senφsenθ
z = cosφ
0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π
âˆŹS fdS p
−senθsenφ cosθcosφ =| | = −senφcosφ cosθsenφ senθcosφ ∂(z, x) = −sen2 φsenθ ∂(θ, φ) ∂(x,y) ∂(θ,φ)
∂(y,z) ∂(θ,φ)
= −sen2 φcosθ
|Tu Ă— Tv | = √[−senφcosφ]2 + [−sen2 φcosθ]2 + [−sen2 φsenθ]2 = |senφ|
2Ď€
Ď€
2Ď€
âˆŤ0 [âˆŤ0 3cos 2 θsen2 φsenφdφ]dθ = âˆŤ0 4cos2 θdθ = 4 Ď€
âˆŹS 3x 2 dS = 4Ď€ p
r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k R
Sp
Sp A(Sp ) = âˆŹ |ru Ă— rv |dS R
C1
f
f(x, y) = z
(x, y) ∈ R |ru Ă— rv | = −
x = x y = y z = f(x, y)
A(Sp ) = âˆŹ √1 + [ R
∂f ∂f i − ∂y j ∂x
∂z 2 ∂z 2 ] + [ ] dS ∂x ∂y Ď•(r, θ) = (rcosθ, rsenθ, θ)
R = {0 ≤ r ≤ 1
0 ≤ θ ≤ π}
∂(x,y) ∂(x,y) ∂(x,y) , , ∂(r,θ) ∂(r,θ) ∂(r,θ) ∂(x,y) ∂(r,θ)
=r
∂(y,z) ∂(r,θ)
= senθ
∂(z,x) ∂(r,θ)
= cosθ
|Ď•r Ă— ϕθ | = √r 2 + 1
Ď€ 1 đ?œ‹ đ?œ‹ 1 1 âˆŤ [âˆŤ √r 2 + 1 dr] dθ = âˆŤ [ √r 2 + 1 + log(r + √r 2 + 1)]│0 dθ = Ď€ (√2 + log(1 + √2)) 2 0 0 0 2 1 Ď€(√2 + 2
log(1 + √2))
F
Sp =
r(u, v)
F
Sp
∬F ∙ dS = ∬ F ∙ (ru × rv ) dudv r
ru × rv = n
Sp
n F(x, y, z) = (x, y, z)
Sp
r(φ, θ) = (senφcosθ senφsenθ, cosφ) 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π
Sp F
rφ × rθ F(r(φ, θ))
F(r(φ, θ)) ∙ (ru × rv )
rφ × rθ = (sen2 φcosθ, sen2 φsenθ, senφcosφ) F(r(φ, θ)) = (cosθsenφ, senθsenφ, cosφ) F(r(φ, θ)) ∙ (ru × rv ) = cosφsen2 φcosθ + sen3 φsen2 θ + sen2 φcosφcosθ
π 2π 4 ∫ ∫ (cosφsen2 φcosθ + sen3 φsen2 θ + sen2 φcosφcosθ)dθdφ = = π 3 0 0
So (๐ ฅ, ๐ ฆ, ๐ ง) โ So n1
n2 n1 = โ n2
So
So โ ฌS F โ dS = โ ฌR F โ (ru ร rv )dS o
So r1 , r2
F C
1
r1 โ ฌ F โ dS = โ ฌ F โ dS r1
r2
Y si r2 la invierte serรก โ ฌ F โ dS = โ โ ฌ F โ dS r1
r2
F = (F1 , F2 , F3 ) C1
F3 (x, y, z)
F1 (x, y, z) F2 (x, y, z) z = f(x, y)
R
∂f ∂f − F2 + F3 ) dA ∂x ∂y
∬ F ∙ dS = ∬ (−F1 So
R
F(x, y, z) = (0, x, z) 𝑧= 0
∂f
∂f
∬S F ∙ dS = ∬R (−F1 ∂x − F2 ∂y + F3 ) dA o
F1 = 0 F2 = z = −𝑥 2 − 𝑦 2 + 1
F3 = y R
2
2
f(x, y) = z = −𝑥 − 𝑦 + 1
f(x, y) = z = 0
f(x, y) = z = −x 2 − y 2 + 1 ∂f ∂x
= −2x
∂f ∂y
= −2y
∬S F ∙ dS = ∬S F ∙ dS + ∬S F ∙ dS o
−x 2 − y 2 + 1
2
S1 =
S2 = 0
∬ (−F1 R
1
∂f ∂f − F2 + F3 ) dA = ∬ [−(0)(−2x) − (x)(−2y) + (−x 2 − y 2 + 1)] dA ∂x ∂y R ∬ [−2xy − x 2 − y 2 + 1] dA R
ϕ(r, θ) ϕ(r, θ) = (rcosθ, rsenθ) 2π
∂(x,y) ∂(r,θ)
=đ?‘&#x;
0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π
1
2Ď€
âˆŹ F ∙ dS = âˆŤ [âˆŤ (−2r 2 senθcosθ − r 2 + 1)dr]dθ = âˆŤ S1
0
0
0
1 4 (2 − 2senθcosθ)dθ = Ď€ 3 3
z=0
âˆŹ F ∙ dS = − âˆŹ zdA = − âˆŹ (0)dA = 0 S2
R
R 4
4
âˆŹS F ∙ dS = âˆŹS F ∙ dS + âˆŹS F ∙ dS = 3 Ď€ + 0 = 3 Ď€ o
1
2
r R ∈ ℝ2 C
1
P, Q
r ∮ Pdx + Qdy = ∬ ( r
R
∫R 2xydx + xydy
∂Q ∂P − ) dxdy ∂x ∂y
x2 + y2 = 9
R
x 2 + y 2 = 25
R
R
xy
P(x, y) = 2xy, Q(x, y) = xy ∂Q ∂P − ∂y ∂x
5
𝜋
∫ [∫ (𝑟𝑠𝑒𝑛𝜃 − 2𝑟𝑐𝑜𝑠𝜃)𝑟𝑑𝑟]𝑑𝜃 = ∫ − 0
3
=y
∂P ∂y
= 2x
= y − 2x x = rcos𝜃 y = rsen𝜃
𝜋
∂Q ∂x
0
∂(x,y) ∂(r,θ)
=r
98 196 (2𝑐𝑜𝑠𝜃 − 𝑠𝑒𝑛𝜃) 𝑑𝜃 = 3 3
∫R 2xydx + xydy =
R⊆ℝ ∂R
196 3
F(x, y) = P𝐢 + Q𝐣 r
2
R
rotF ∫ F ∙ dS = ∬ (rotF) ∙ 𝐤dA ∂R
R
divF
n=
(x′ (t)− y′(t)) √[x′(t)]2 +[y′(t)]2
𝑟(𝑡) = (𝑥(𝑡), 𝑦(𝑡)) ∫ F ∙ ndS = ∬ (divF)dA ∂R
∬R(rotF) ∙ 𝐤dA x=y
R
F(x, y) = (xy)i + (x − 2y)j x2 = y
x≥0 (rotF) ∙
k
Q(x, y) = xy
P(x, y) = x − 2y (rotF) ∙ 𝐤 =
∂Q ∂P − = y+2 ∂x ∂y x = x2
x=1 x=0 0≤x≤1
x2 ≤ y ≤ x
1 x 1 1 2 ∫ ∫ (y + 2)dydx = ∫ − x(x 3 + 3x − 4) dx = 2 5 0 x2 0
F(x, y) = ey x 2 i + ex j
∬R(divF)dA [0,1] × [0,1]
P(x, y) = ey x 2
Q(x, y) = ex divF =
∂P ∂Q + ∂y ∂x
= 2xey + 0
1
1
1
∬ 2xey dxdy = ∫ [∫ 2xey dx]dy = ∫ ey dy = e − 1 R
0
0
0
r R C
1
P, Q
R ∫ Pdx + Qdy = ∬ ( r
R
∂Q ∂P − ) ∂x ∂y
∫r xdx + 2x 2 ydy (0,0), (1,0), (1,1)
0≤y≤x P(x, y) = x ∂Q ∂P − ∂x ∂y
∂Q ∂x
Q(x, y) = x 2 y
= 4xy 1
x
1
∫ [∫ 4xydy]dx = ∫ 2x 3 dx = 0
0
0
1 2
0≤x≤1 = 4xy,
∂P ∂y
=0
z = f(x, y)
C
2
R (x, y) ∈ R
R F(x, y, z)
∂R ∬ rotF ∙ dS = ∬ (∇ × F) ∙ dS = ∫ F ∙ ds R
R∗
∂R∗ R
∫R −y 3 dx + ydy + zdz x2 + y2 = 1
z=1−x−y
R
∂R
C1
i ∂ ∇ Ă— F = || ∂x −xy 3
j k ∂ ∂ | = 3xy 2 đ??¤ = (0,0,3y 2 ) ∂y ∂z| y z
âˆŹ 3xy 2 ∙ đ?‘‘đ?‘† = âˆŹ 3xy 2 dxdy đ?‘…∗
đ?‘…
� x = rcosθ y = rsenθ
0 ≤ r ≤ 1, 0 ≤ θ ≤ 2
1 2Ď€ 1 3 âˆŤ [âˆŤ (3r 2 sen2 θ)rdθ]dr = âˆŤ 3 Ď€r 3 dr = Ď€ 4 0 0 0
R ∗
2
R∗
φ: R ⊆ â„? → Sp ∂R C
1
R
F
R âˆŹ (∇ Ă— F) ∙ dS = âˆŤ F ∙ dS R
∂R
F(x, y, z) = (2y 2 + x)đ??˘ + zđ??Ł + (x 2 + 4y)đ??¤ z = 9 − √x 2 + y 2
z=0
z = 9 − √x 2 + y 2 Ď•(x, y, z) = (rcosθ, rsenθ, 0)
âˆ‡Ă—F = |
0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π
i
j
k
∂ ∂x 2
∂ ∂y
∂ ∂z
z
x 2 + 4y
2y + x
| = (4 − 1) − (2x − 0) + (0 − 2y) = 3i − 2xj − 2yk
n i j k Ď•r Ă— ϕθ = | cosθ senθ 0| = (0)i − (0)j + (r)k, −rsenθ rcosθ 0 rr Ă— rθ 1 n= = (0,0, r) = (0,0,1) |rr Ă— rθ | r
|ϕr × ϕθ | = r
∇ × F ∙ n = (3, −2x, −2y) ∙ (0,0,1) = −2y = −2senθ
2π 3 2π 3 ∬ (∇ × F) ∙ dS = ∫ [∫ 2senθdr]dθ = ∫ 2 − 2 cos(3) dθ = 8πsen2 ( ) 2 R 0 0 0
ℝ3
F C1
∇×F=0 F
f
∇f = F ℝ2
F(x, y) = P(x, yi) + Q(x, y)j
∂P ∂Q = ∂y ∂x
F(x, y) = P(y 2 cosx + 2)i + Q(2ysenx + 2)j
∂P = 2ycosx ∂y ∂Q = 2ycosx ∂x