January 27, 2005 11:56
L24-CH16
Sheet number 1 Page number 693
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CHAPTER 16
Topics in Vector Calculus EXERCISE SET 16.1 1. (a) III because the vector field is independent of y and the direction is that of the negative x-axis for negative x, and positive for positive (b) IV, because the y-component is constant, and the x-component varies periodically with x 2. (a) I, since the vector field is constant (b) II, since the vector field points away from the origin 3. (a) true
(b) true
(c) true
4. (a) false, the lengths are equal to 1 (c) false, the x-component is then zero 5.
y
6.
(b) false, the y-component is then zero
y
7.
y
x
x
x
8.
y
9.
y
y
x
x
11. (a) ∇φ = φx i + φy j =
10.
x
y x i+ j = F, so F is conservative for all x, y 1 + x2 y 2 1 + x2 y 2
(b) ∇φ = φx i + φy j = 2xi − 6yj + 8zk = F so F is conservative for all x, y 12. (a) ∇φ = φx i + φy j = (6xy − y 3 )i + (4y + 3x2 − 3xy 2 )j = F, so F is conservative for all x, y (b) ∇φ = φx i + φy j + φz k = (sin z + y cos x)i + (sin x + z cos y)j + (x cos z + sin y)k = F, so F is conservative for all x, y 13. div F = 2x + y, curl F = zi 14. div F = z 3 + 8y 3 x2 + 10zy, curl F = 5z 2 i + 3xz 2 j + 4xy 4 k 15. div F = 0, curl F = (40x2 z 4 − 12xy 3 )i + (14y 3 z + 3y 4 )j − (16xz 5 + 21y 2 z 2 )k 16. div F = yexy + sin y + 2 sin z cos z, curl F = −xexy k 693