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836 Answers to Selected Problems

Section 12.4 Potential Theory
3
1. Conservative, ϕ(x, y) = xy − 4y
3
2
3. Conservative, ϕ(x, y) = 8x + 2y − y /3
2
2
5. Conservative, ϕ(x, y) = ln(x + y )
7. ϕ(x, y, z) = x − 2y + z
9. Not conservative
11. −27
13. 5 + ln(3/2)
15. −5
17. −403
19. 2e −2
21. Write
m

E(t) = total energy = R (t) · R (t) − ϕ(x(t), y(t), z(t)).
2

Use the fact that mR =∇ϕ (by one of Newton’s laws of motion) to show that E (t) = 0.
Section 12.5 Surface Integrals
√ √
1. 125 2 3. π(29 3/2 − 27)/6 5. 28π 2/3
√ √ √
7. (9/8) ln 4 + 17 + 4 17 9. −10 3
Section 12.6 Applications of Surface Integrals
√
1. 49/12,(12/35,33/35,24/35) 3. 9π K 2,(0,0,2)
5. 78π,(0,0,27/13) 7. 128/3

Section 12.7 Lifting Green’s Theorem to R 3
1. Apply Green’s theorem to the line integral

∂ψ ∂ψ
−ϕ dx + ϕ dy.
∂y ∂x
C
3. Apply Green’s theorem to
∂ϕ ∂ϕ

− dx + dy.
∂y ∂x
C
Section 12.8 The Divergence Theorem of Gauss
1. 256π/3 3. 0 5. 8π/3
7. 2π 9. 0 because ∇· (∇× F) = 0

Section 12.9 Stokes’s Theorem
1. −8π 3. −16π 5. −32/3 7. −108

Section 12.10 Curvilinear Coordinates
2.

2
2
2
2
h 1 = h 2 = a sinh (u)cos (v) + cosh (u)sin (v),h 3 = 1
1 ∂ f 1 ∂ f ∂ f
∇ f (u,v, z) = u u + u v + u z ,
h 1 ∂u h 1 ∂v ∂z
where
a
u u = (sinh(u)cos(v)i + cosh(u)sin(v)j),
h 1

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October 14, 2010 17:50 THM/NEIL Page-836 27410_25_Ans_p801-866