Page 855 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 835

Section 11.4 The Gradient Field
1. yzi + xzj + xyk,i + j + k,
√ √
3,− 3
z
z
6
6
3. (2y + e )i + 2xj + xe k,(2 + e )i − 4j − 2e k,
√ √
12
6
6
20 + 4e + 5e ,− 20 + 4e + 5e 12
5. 2y sinh(2xy)i + 2x sinh(2xy)j − cosh(z)k,cosh(1)k,cosh(1),−cosh(1)
√
2
7. 1/ 3 (8y − z + 16xy − x)
√

2
2 3
2
9. 1/ 5 (2x z + 3x yz )
√ √
11. x + y + 2z = 4; x = y = 1 + 2t, z = 2(1 + 2t)
13. x = y; x = 1 + 2t, y = 1 − 2t, z = 0
15. x = 1; x = 1 + 2t, y = π, z = 1
17. Level surfaces are planes x + z = k.
Section 11.5 Divergence and Curl
In Problems 1, 3, and 5, ∇· F is given first, then ∇× F.
y
y
1. 4,O 3. 2y + xe + 2,(e − 2x)k
5. cosh(x) fxz sinh(xyz) − 1,(−1 − xy sinh(xyz))i − j + yz sinh(xyz)k(i, j,k)
In Problems 7 and 9, ∇ϕ is given.
7. i − j + 4zk
3
2
3 2
2
9. −6x yz i − 2x z j − 4x yzk
11. (cos(x + y + z) − x sin(x + y + z))i − x sin(x + y + z)(j + k)
13. ∇· (ϕF) =∇ϕ · F + ϕ(∇· F)
∇× (ϕF)=∇ϕ × F + ϕ(∇× F)
CHAPTER TWELVE VECTOR INTEGRAL CALCULUS
Section 12.1 Line Integrals
√
1. 0 3. 26 2/3 5. sin(3) − 81/2
7. 0 9. −422/5 11. −27/2

Section 12.2 Green’s Theorem
1. −8 3. −12 5. −40 7. 512π
9. 0 11. 95/4
13. By Green’s theorem,

∂u ∂u ∂ ∂u ∂ ∂u

− dx + dy = − − dA
C ∂y ∂x D ∂x ∂x ∂y ∂y

Section 12.3 An Extension of Green’s Theorem
1. 0
3. 2π if C encloses the origin; 0 otherwise
5. 0

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October 14, 2010 17:50 THM/NEIL Page-835 27410_25_Ans_p801-866