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

CHAPTER 16 REVIEW 0 337
27. JI 5
curlF · dS =III Ediv(curlF) dV = 0 by Theorem 16.5. 11.
29. ff 5
(f'Vg) · n dS = JJJ E div(f'Vg) dV = fff E(f'V 2 g + 'Vg · 'V f) dV by Exercise 16.5.25.
, 31. If c = c1 i + c2 j + C3 k is an arbitrary constant vector, we define F = fc = fc1 i + fc2 j + fc3 k. Then
. . af af af . · .
0
c and the Divergence Theorem says .ffs F dS = f f f E dJV F dV =>
dtv F = dtv fc = ax Ct + f)y C2 + f)z C3 = 'V f
0
Ifs F · n dS = IIJ E 'V f · cdV. In particular, ifc = i then Jfs f i · n dS = JJJ E 'Vf · i dV =>
Jfs fn1 d~ = jj L :~ dV (where n ~ n 1 i + n2 j + n3 k). Similarly, ifc =j we have Jfs fn2 dS = JJL :~ dV,
and c = k gives /Is f na dS = j J L :~ dV. Then
.fJ~ f n dS = (ff 8 /n1 dS) i + (Jf 8 fn2 dS) j + (ff 5 jn3 dS) k
= (/I L :~ dV) i + (/I L ~; dV) j + (/I L ~~ dV) k = I I L ( :~ i + ~ j + ~~ k) dV
= JJJ E 'V f dV as desired.
16 Review
CONCEPT CHECK
1. See Definitions 1 and 2 in Section 16.1. A vector field can represent, for example, the wind velocity at any location in space,
the speed and direction of the ocean current at any location, or the force vectors of Earth's gravitational field at a location in
space.
2. (a) A conservative vector field F Is a vector field which is the gradient of some scalar function f.
(b) The function f in part (a) is called a potential function for F , that is, F = 'V f .
3. (a) See Definition 16.2.2.
(b) We normally evaluate the line integral using Formula 16:2.3.
(c) The mass ism = fc p (x, y) ds, and the center of mass is (x, Y) where x = ~ fc xp (x, y) ds, y = ~ fc yp (x, y) ds.
(d) See (5) and (6) in Section 16.2 for plane curves; we have similar definitions when 0 is a space curve
· [see the equation preceding (10) in Section 16.2).
(e) For plane curves, see Equations '16.2.7. We have similar results for space curves
[see the equation preceding (10) in Section 16.2 ).
4. (a) See Definition 16.2. 13.
(b) If F is a force field, J~ F · dr represents ti1e work done by Fin moving a particle along the curve 0 . .
(c) Ic F ·dr=J 0
Pdx+Qdy+Rdz
5. See Theorem 16.3.2.
, 0
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