Solução_Calculo_Stewart_6e

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F. TX.10 CHAPTER 17 REVIEW ET CHAPTER 16 ¤ 301 21. From the graph it appears that for points above the x-axis, vectors starting near a particular point are longer than vectors ending there, so divergence is positive. The opposite is true at points below the x-axis, where divergence is negative. F (x, y) = xy, x + y 2 ⇒ div F = ∂ (xy)+ ∂ ∂x ∂y x + y 2 = y +2y =3y. Thus div F > 0 for y>0,anddiv F < 0 for y
F. 302 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16 3. (a) See Definition 17.2.2 [ ET 16.2.2]. (b) We normally evaluate the line integral using Formula 17.2.3 [ ET 16.2.3]. (c) The mass is m = C ρ (x, y) ds,andthecenterofmassis(x, y) where x = 1 m C xρ (x, y) ds, y = 1 m (d) See (5) and (6) in Section 17.2 [ ET 16.2] for plane curves; we have similar definitions when C is a space curve (see the equation preceding (10) in Section 17.2 [ ET 16.2]). (e) For plane curves, see Equations 17.2.7 [ ET 16.2.7]. We have similar results for space curves (see the equation preceding (10) in Section 17.2 [ ET 16.2]). 4. (a) See Definition 17.2.13 [ ET 16.2.13]. (b) If F is a force field, F · dr represents the work done by F in moving a particle along the curve C. C (c) F · dr = Pdx+ Qdy+ Rdz C C 5. See Theorem 17.3.2 [ ET 16.3.2]. C yρ (x, y) ds. 6. (a) F · dr is independent of path if the line integral has the same value for any two curves that have the same initial and C terminal points. (b) See Theorem 17.3.4 [ ET 16.3.4]. 7. See the statement of Green’s Theorem on page 1091 [ ET 1055]. 8. See Equations 17.4.5 [ ET 16.4.5]. ∂R 9. (a) curl F = ∂y − ∂Q ∂P i + ∂z ∂z − ∂R ∂Q j + ∂x ∂x − ∂P k = ∇ × F ∂y (b) div F = ∂P ∂x + ∂Q ∂y + ∂R ∂z = ∇ · F (c) For curl F, see the discussion accompanying Figure 1 on page 1100 [ ET 1064] as well as Figure 6 and the accompanying discussion on page 1132 [ ET 1096]. For div F, see the discussion following Example 5 on page 1102 [ ET 1066] as well as the discussion preceding (8) on page 1139 [ ET 1103]. 10. See Theorem 17.3.6 [ ET 16.3.6]; see Theorem 17.5.4 [ ET 16.5.4]. 11. (a) See (1) and (2) and the accompanying discussion in Section 17.6 [ ET 16.6] ; See Figure 4 and the accompanying discussion on page 1107 [ ET 1071] . (b) See Definition 17.6.6 [ ET 16.6.6 ]. (c) See Equation 17.6.9 [ ET 16.6.9]. 12. (a)See(1)inSection17.7[ET16.7]. (b) We normally evaluate the surface integral using Formula 17.7.2 [ ET 16.7.2]. (c) See Formula 17.7.4 [ ET 16.7.4]. TX.10 (d) The mass is m = ρ(x, y, z) dS and the center of mass is (x, y, z) where x = 1 S m y = 1 yρ(x, y, z) dS, z = 1 m S m zρ(x, y, z) dS. S S xρ(x, y, z) dS,
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