Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

338 0 CHAPTER 16 VECTOR CALCULUS
6. (a) fc F · dr is independent of path if the line integral has the same value for any two curves that have the same initial and
tenninal points.
(b) See Theorem 16.3.4.
7. See the statement of Green's Theorem on page.ll08 [ET 1084].
8. See Equations 16.4.5.
8R 8Q) . ( 8P 8R) . ( 8Q 8P) .
( 8y 8z 8z 8x ax 8y .
9. (a)curlF= --- •+ - - - J + - - - k=V x F
(c) For curl F , see the discussion accompanying Figure 1 on page 111 8 [ET 1094] as well as Figure 6 and the accompanying
discussion on page 1150 [ET 1126]. For div F, see the discussion following Example 5 on page 1119 [ET 1095] as well as
the discussion preceding (8) on page 1157 [ET 1133].
10. See Theorem 16.3.6; see Theorem 16.5.4.
11. (a) See (I) and (2) and the accompanying discussion in Section 16.6; See Figure 4 and· the accompanying discussion on
page 1124 [ET 1100].
(b) See Definition 16.6.6.
(c) See Equation 16.6.9.
12. (a) See (1) in Section 16.7.
(b) We normally evaluate the surface integral using Fonnula 16.7.2.
(c) See Formula 16.7.4.
(d) The mass ism = Jf 5
p(x,y,z)dS and the center of mass is (x,y,z) wherex = ~ JJ 8
xp(x,y,z) dS,
y = ~ Ifs yp(x, y, z ) dS, z = ~ .ffs zp(x, y , z) dS.
13. (a) See Figures 6 and 7 and the accompanying discussion in Section 16.7. A Mtibius strip is a nonorientable surface; see
Figures 4 and 5 and the accompanying discussion on page 1139 [ET 111 5].
(b) See Definition 16.7.8.
(c) See Formula 16.7.9.
(d) SeeFonnula 16.7.10.
14. See the statement of Stokes' Theorem on page 1146 [ET I J 22].
15. See the statement of the Divergence Theorem on page 11 53 [ET J 129].
16. In each theorem, we have an integral of a "derivative" over a region on the left side, while the right side involves the values of
the original function only on the boundary of the region.
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