Thomas Calculus 13th [Solutions]

1204 Chapter 16 Integrals and Vector Fields
2 2
8x
8y
2 2 2 2 2 2 2
Flux 4 x y 1 dA 8x 8y 2 dA; z 1 and x y z
2 2
R
4 x y 1
R
2 2 2 1 2
x y 1 Flux = 8r 2 r dr d 2
0 0
39.
x x 2x
e i j
2e 2 y
g( x, y, z) y e 0 g e i j | g | e 1 n F n ;
2x
2x
e 1 e 1
2x x
2x x x
x e 1 2e
2 y e 1
p i | g p | e d dA Flux
dA 2e 2e
dA
x 2x
x x
e e 1 e e
R
R
1 2
4 dA 4 dy dz 4
0 1
R
x
x
40.
1
2
x
1
i j
x i xj
2 x
1 x
2
1
2
x
x
g( x, y, z) y ln x 0 g 1 i j | g | 1 1 since 1 x e n
x
x
2 2
2xy
1 x
2xy
1 x
F n ; p j | g p | 1 d dA Flux
dA
2 2
1 x
x
1 x
x
R
1 e 1
0 1 2 e
1 0 2 ln e
1
2 ln 2 ln e
y dx dz x dz dx x dx x x x
1
2( e e ) 2(0 1) 2
41. On the face z a : g( x, y, z) z g k | g | 1; n k F n 2xz 2ax since z a;
d dx dy
4
Flux 2 a a
ax dx dy 2 ax dx dy a .
0 0
R
On the face z 0 : g( x, y, z) z g k | g | 1; n k F n 2xz 0 since z 0;
d dx dy Flux 0 dx dy 0.
R
On the face x a: g( x, y, z) x g i | g | 1; n i F n 2xy 2ay since x a;
a a
4
d dy dz Flux 2 ay dy dz a .
0 0
On the face x 0: g( x, y, z) x g i | g | 1; n i F n 2xy 0 since x 0 Flux 0.
On the face y a: g( x, y, z) y g j | g | 1; n j F n 2yz 2az since y a;
a a
4.
d dz dx Flux 2az dz dx a
0 0
On the face y 0: g( x, y, z) y g j | g | 1; n j F n 2yz 0 since y 0 Flux 0.
Therefore, Total Flux
4
3 a .
42. Across the cap:
2 2 2 2 2 2
g( x, y, z) x y z 25 g 2xi 2yj 2 zk
| g | 4x 4y 4z
10
2 2
g xi yj zk
y z
n F n x z z ; p k | g p | 2z since z 0 d 10 dA
| g| 5 5 5 5 2z
2 2
2 4
y z 5
2 2 2
Fluxcap F n d x z z dA x y 1 dx dy r 1 r dr d
5 5 5 z
0 0
cap
R
R
2
0
72 d 144 .
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