Thomas Calculus 13th [Solutions]

Section 16.8 The Divergence Theorem and a Unified Theory 1215
14. Let
2 2 2 y
2
x y z . Then x , , z x 1 x 1 x . Similarly,
x y z x 2 x
3
y
2 2
2 2 2
y 1 y y
and 1 z
3 x y z
F
2
3 z
3 3
2
2 2
2 2
2 2
Flux = dV sin d d d 3sin d d 6 d 12
0 0 1 0 0 0
D
15.
3 2 2 2 3 y 2 y 3 y 2 y
5x 12xy 15x 12 y , y e sin z 3y e sin z, 5z e cos z 15z e sin z
x y z
2 2 2 2 2 2 2 2 2
F 15x 15y 15z 15 Flux 15 dV 15 sin d d d
0 0 1
D
2 2
12 2 3 sin d d 24 2 6 d 48 2 12
0 0 0
16.
1
2x 2z y 2z x
2z
2 2
y
2 2 2
1
x
2 2 1 2 2 2 2
ln x y , tan , z x y x y
x x y y x x x x y z
2 2 2 2
F 2x
2z
x y Flux 2x
2z
x y dz dy dx
2 2 2 2 2 2 2 2
x y x y x y x y
D
2 2 2
2r
cos 2z
2 2
3 2
r dz r dr d 6cos 3r dr d
0 1 1
2 2
r r
0 1
r
2
6 2 1 cos 3 ln 2 2 2 1 d 2 3 ln 2 2 2 1
0
2
17. (a) ( x) 1, ( y) 1, ( z) 1 F 3 Flux 3 dV 3 dV 3 (Volume of the solid)
x y z
D
D
(b) If F is orthogonal to n at every point of S, then F n 0 everywhere Flux F n d 0. But the
S
flux is 3 (Volume of the solid) 0, so F is not orthogonal to n at every point.
18. Yes, the outward flux through the top is 5. The reason is this: Since F ( xi 2 yj ( z 3) k
1 2 1 0, the outward flux across the closed cubelike surface is 0 by the Divergence Theorem. The flux
across the top is therefore the negative of the flux across the sides and base. Routine calculations show that the
sum of these latter fluxes is 5. (The flux across the sides that lie in the xz -plane and the yz -plane are 0,
while the flux across the xy -plane is 3. ) Therefore the flux across the top is 5.
2 2
19. For the field F ( y cos2 x) i ( y sin 2 x) j ( x y x) k,
F 2 y sin 2x 2 y sin 2x
1 1. If F were the curl
of a field A whose component functions have continuous second partial derivatives, then we would have
div F div(curl A) ( A ) 0. Since div F 1, F is not the curl of such a field.
20. From the Divergence Theorem,
2 2 2 1 2 2 2
f x y z x y z x y z
2
2 2 2
f f f
.
2 2 2
n Now,
f d f dV dV
x y z
S D D
f x f y f z
x x y z y x y z z x y z
( , , ) ln ln , ,
2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
f x y z f x y z f x y z f f f x y z
, , ,
1
x x y z y x y z z x y z x y z x y z x y z
2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
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