Thomas Calculus 13th [Solutions]

1234 Chapter 16 Integrals and Vector Fields
15. Assume that S is a surface to which Stokes Theorem applies. Then E dr ( E)
n d
C
S
B n d B n d . Thus the voltage around a loop equals the negative of the rate of change of
t
t
S
S
magnetic flux through the loop.
16. According to Gausss Law, F n d 4 GmM for any surface enclosing the origin. But if F H then
S
the integral over such a closed surface would have to be 0 by the Divergence Theorem since div F 0.
17.
C
S
S
S
f g dr f g n d
(Stokes Theorem)
S
f g f g n d
(Section 16.8,Exercise 19b)
( f )( 0)
f g n d
(Section 16.7, Equation 8)
f g n d
18. F1 F2 F2 F1 0 F2 F 1 is conservative F2 F1 f ; also, F1 F2
2
F2 F 1 0 f 0 (so f is harmonic). Finally, on the surface S,
f n F2 F1
n
2
F2 n F1 n 0. Now, f f f f f f so the Divergence Theorem gives
2 2
f dV f f dV f f dV f f n d 0, and since
D D D S
2 2
f dV 0 0 F2 F1
dV 0 f f n d 0, as claimed.
D D S
2
f 0 we have
i j k
19. False; let F yi xj 0 F ( y) x
( x) y
0 and F
x y z
0i 0j 0k 0
x y 0
20.
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
ru rv ru rv sin ru rv 1 cos ru rv ru rv cos ru rv ru rv
2 2 2
ru rv EG F d ru rv
du dv EG F du dv
21. r xi yj zk r 1 1 1 3 r dV 3 dV 3 V V 1 rdV 1 r n d , by
3 3
D D D S
the Divergence Theorem
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