Chapter 6

> 来源: OpenStax《Calculus Volume 3》| 原页: https://openstax.org/books/calculus-volume-3/pages/chapter-6

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Chapter 6

Calculus Volume 3Chapter 6

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Chapter 6

Checkpoint

6.1

$12\mathbf{\text{i}} - \mathbf{\text{j}}$

6.2

6.3

Rotational

6.4

$\sqrt{65}$ m/sec

6.5

No.

6.6

6.7

$–1.49063\ \times \ 10^{-18},4.96876\ \times \ 10^{-19},–9.93752\ \times \ 10^{-19}\text{N}$

6.8

6.9

No

6.10

$\text{∇}f = \mathbf{\text{v}}$

6.11

$P_{y} = x \neq Q_{x} = -2xy$

6.12

No

6.13

$\sqrt{2}$

6.14

$2\sqrt{10}\pi + 2\sqrt{10}\pi^{2}$

6.15

Both line integrals equal $- \frac{1000\sqrt{30}}{3}.$

6.16

$4\sqrt{17}$

6.17

${\int_{C}{\mathbf{\text{F}} \cdot \mathbf{\text{T}}}}ds$

6.18

$-26$

6.19

0

6.20

$18\sqrt{2}\pi^{2}$ kg

6.21

3/2

6.22

$2\pi$

6.23

0

6.24

Yes

6.25

The region in the figure is connected. The region in the figure is not simply connected.

6.26

2

6.27

If $C_{1}$ and $C_{2}$ represent the two curves, then ${\int_{C_{1}}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} \neq {\int_{C_{2}}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}.}}$

6.28

$f(x,y) = e^{x}y^{3} + xy$

6.29

$f\left( {x,y,z} \right) = 4x^{3} + \text{sin}\mspace{2mu} y\ \text{cos}\ z + z$

6.30

$f(x,y,z) = \frac{G}{\sqrt{x^{2} + y^{2} + z^{2}}}$

6.31

It is conservative.

6.32

$-10\pi$

6.33

Negative

6.34

$\frac{45}{2}$

6.35

$\frac{2}{3}$

6.36

$\frac{3\pi}{2}$

6.37

$g\left( {x,y} \right) = \text{−}x\ \text{cos}\ y$

6.38

No

6.39

$105\pi$

6.40

$y - z^{2}$

6.41

Yes

6.42

All points on line $y = 1.$

6.43

$\text{−}\mathbf{\text{i}}$

6.44

$\text{curl}\ \mathbf{\text{v}} = \mathbf{0}$

6.45

No

6.46

Yes

6.47

Cylinder $x^{2} + y^{2} = 4$

6.48

Cone $x^{2} + y^{2} = z^{2}$

6.49

$\mathbf{\text{r}}(u,v) = \left\langle {u\ \text{cos}\ v,u\ \text{sin}\ v,u} \right\rangle,$ $0 < u < \infty,0 \leq v < \frac{\pi}{2}$

6.50

Yes

6.51

$\approx 43.02$

6.52

With the standard parameterization of a cylinder, Equation 6.18 shows that the surface area is $2\pi rh.$

6.53

$2\pi\left( {\sqrt{2} + \text{sinh}^{-1}(1)} \right)$

6.54

24

6.55

0

6.56

$38.401\pi \approx 120.640$

6.57

$\mathbf{\text{N}}(x,y) = \left\langle {\frac{\text{−}y}{\sqrt{1 + x^{2} + y^{2}}},\frac{\text{−}x}{\sqrt{1 + x^{2} + y^{2}}},\frac{1}{\sqrt{1 + x^{2} + y^{2}}}} \right\rangle$

6.58

0

6.59

400 kg/sec/m

6.60

$- \frac{440\pi}{3}$

6.61

Both integrals give $0$

6.62

$\text{−}\pi$

6.63

$\frac{3}{2}$

6.64

$\text{curl}\ \mathbf{\text{E}} = \left\langle {x,y,-2z} \right\rangle$

6.65

Both integrals equal $6\pi.$

6.66

30

6.67

$9\ \text{ln}(16)$

6.68

$\approx 6.777\ \times \ 10^{9}$

Section 6.1 Exercises

1.

Vectors

3.

False

5.

7.

9.

11.

13.

15.

$\mathbf{\text{F}}(x,y) = \text{sin}(y)\mathbf{\text{i}} + (x\ \text{cos}\ y - \text{sin}\ y)\mathbf{\text{j}}$

17.

$\mathbf{\text{F}}(x,y,z) = (2xy + y)\mathbf{\text{i}} + (x^{2} + x + 2yz)\mathbf{\text{j}} + y^{2}\mathbf{\text{k}}$

19.

$\mathbf{\text{F}}(x,y) = \left( \frac{2x}{1 + x^{2} + 2y^{2}} \right)\mathbf{\text{i}} + \left( \frac{4y}{1 + x^{2} + 2y^{2}} \right)\mathbf{\text{j}}$

21.

$\mathbf{\text{F}}(x,y) = \frac{(1 - x)\mathbf{\text{i}} - y\mathbf{\text{j}}}{\sqrt{{(1 - x)}^{2} + y^{2}}}$

23.

$\mathbf{\text{F}}(x,y) = \frac{- x\mathbf{\text{i}} - y\mathbf{\text{j}}}{\sqrt{x^{2} + y^{2}}}$

25.

$\mathbf{\text{F}}(x,y) = y\mathbf{\text{i}} - x\mathbf{\text{j}}$

27.

$\mathbf{\text{F}}(x,y) = \frac{-10}{\left( x^{2} + y^{2} \right)^{3\text{/}2}}\left( {x\mathbf{\text{i}} + y\mathbf{\text{j}}} \right)$

29.

$\left. ||\mathbf{E} \right.|| = \frac{c}{x^{2} + y^{2}}\sqrt{x^{2} + y^{2}} = \frac{c}{\sqrt{x^{2} + y^{2}}} = \frac{c}{r}$

31.

$\mathbf{\text{c}}\text{'}(t) = \left( {\text{cos}\ t,\text{−}\text{sin}\ t,e^{\text{−}t}} \right) = \mathbf{\text{F}}\left( {\mathbf{\text{c}}(t)} \right)$

33.

H

35.

d\. $\text{−}\mathbf{\text{F}} + \mathbf{\text{G}}$

37.

a\. $\mathbf{\text{F}} + \mathbf{\text{G}}$

Section 6.2 Exercises

39.

True

41.

False

43.

False

45.

${\int_{C}^{}{(x - y)ds}} = 10$

47.

${\int_{C}^{}{xy^{4}ds}} = \frac{8192}{5}$

49.

$W = 8$

51.

$W = \frac{3\pi}{4}$

53.

$W = \pi$

55.

${\int_{C}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = 4$

57.

${\int_{C}^{}{yzdx + xzdy + xydz}} = -1$

59.

${\int_{C}^{}{\left( y^{2} \right)dx + (x)dy}} = \frac{245}{6}$

61.

$\int_{C}^{}{xydx + ydy = \frac{190}{3}}$

63.

$\int_{C}{\frac{y}{2x^{2} - y^{2}}ds = \sqrt{2}\ \text{ln}\ 5}$

65.

$W = -66$

67.

$W = -10\pi^{2}$

69.

$W = 2$

71.

a\. $W = 11;$ b. $W = \frac{39}{4};$ c. No

73.

$W = 2\pi$

75.

$\int_{C}^{}xy~ds = \frac{25\sqrt{5} + 1}{120}$

77.

${\int_{C}^{}{y^{2}dx + \left( {xy - x^{2}} \right)dy}} = 6.15$

79.

$\int_{\gamma}^{}{xe^{y}ds \approx 7.157}$

81.

$\int_{\gamma}^{}{\left( {y^{2} - xy} \right)dx \approx -1.379}$

83.

${\int_{C}^{}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} \approx -1.133$

85.

$\int_{C}^{}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}} \approx 22.857}$

87.

$\text{flux} = - \frac{1}{3}$

89.

$\text{flux} = -20$

91.

$\text{flux} = 0$

93.

$m = 4\pi\rho\sqrt{5}$

95.

$W = 0$

97.

$W = \frac{k}{2}$

Section 6.3 Exercises

99.

True

101.

True

103.

${\int_{C}^{}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = 24$

105.

${\int_{C}^{}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = e - \frac{3\pi}{2}$

107.

Not conservative

109.

Conservative, $f(x,y) = 3x^{2} + 5xy + 2y^{2}$

111.

Conservative, $f(x,y) = ye^{x} + x\ \text{sin}(y)$

113.

${\int_{C}{(2ydx + 2xdy)}} = 32$

115.

$\mathbf{\text{F}}(x,y) = (10x + 3y)\mathbf{\text{i}} + (3x + 20y)\mathbf{\text{j}}$

117.

F is not conservative.

119.

F is conservative and a potential function is $f(x,y\text{,}\ z) = xye^{z}.$

121.

F is conservative and a potential function is $f(x,y,z) = z^{2}–z–\frac{x}{y}.$

123.

F is conservative and a potential function is $f(x,y\text{,}\ z) = x^{2}y + y^{2}z.$

125.

F is conservative and a potential function is $f\left( {x,y} \right) = e^{x^{2}y}$

127.

${\int_{C}{\mathbf{\text{F}} \cdot dr}} = e^{2} + 1$

129.

${\int_{C}{\mathbf{\text{F}} \cdot dr}} = -2$

131.

${\int_{C_{1}}{\mathbf{\text{G}} \cdot d\mathbf{\text{r}}}} = -8\pi$

133.

${\int_{C_{2}}{\mathbf{\text{G}} \cdot d\mathbf{\text{r}}}} = 7$

135.

${\int_{C}\mathbf{\text{F}}} \cdot d\mathbf{\text{r}} = 159$

137.

${\int_{C}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = -1$

139.

$4\ \times \ 10^{29}\text{erg}$

141.

$\int_{C}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}} \approx 2.9923}$

143.

$\text{circulation} = \pi a^{2}\ \text{and flux} = 0$

Section 6.4 Exercises

147.

${\int_{C}^{}{2xydx + (x + y)dy}} = \frac{32}{3}$

149.

${\int_{C}^{}{\text{sin}\ x\ \text{cos}\ ydx + (xy + \text{cos}\ x\ \text{sin}\ y)dy}} = \frac{1}{12}$

151.

${\int_{C}{(\text{−}ydx + xdy)}} = \pi$

153.

$\int_{C}{xe^{-2x}dx + \left( {x^{4} + 2x^{2}y^{2}} \right)dy = 0}$

155.

$\int_{C}{y^{3}dx - x^{3}ydy = -20\pi}$

157.

${\int_{C}{\text{−}x^{2}ydx + xy^{2}dy}} = 8\pi$

159.

${\int_{C}{\left( {x^{2} + y^{2}} \right)dx + 2xydy}} = 0$

161.

$A = 19\pi$

163.

$A = \frac{3\pi}{8}$

165.

$\int_{C +}{\left( {y^{2} + x^{3}} \right)dx + x^{4}dy = 0}$

167.

$A = \frac{9\pi}{8}$

169.

$A = \frac{8\sqrt{3}}{5}$

171.

$\int_{C}{\left( {x^{2}y - 2xy + y^{2}} \right)ds = \frac{1}{2}}$

173.

${\int_{C}^{}\frac{xdx + ydy}{x^{2} + y^{2}}} = 0$

175.

$W = \frac{225}{2}$

177.

$W = 12\pi$

179.

$W = 2\pi$

181.

$\int_{C}{y^{2}dx + x^{2}dy = \frac{1}{3}}$

183.

${\int_{C}^{}{\sqrt{1 + x^{3}}dx + 2xydy}} = -3$

185.

${\int_{C}^{}{\left( {3y - e^{\text{sin}\ x}} \right)dx}} + \left( {7x + \sqrt{y^{4} + 1}} \right)dy = 36\pi$

187.

${\int_{C}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = 2$

189.

${\int_{C}{(y + x)dx + (x + \text{sin}\ y)dy}} = 0$

191.

$\int_{C}{xydx + x^{3}y^{3}dy = \frac{22}{21}}$

193.

${\int_{C}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = \frac{15\pi}{4}$

195.

${\int_{C}^{}{\text{sin}(x + y)dx + \text{cos}(x + y)dy}} = 4$

197.

${\int_{C}^{}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = \pi$

199.

$\int_{C}{\mathbf{\text{F}} \cdot \mathbf{\hat{N}}ds = 4}$

201.

${\int_{C}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}ds}} = 0$

203.

${\int_{C}{\left\lbrack {\text{−}y^{3} + \text{sin}\left( {xy} \right) + xy\ \text{cos}\left( {xy} \right)} \right\rbrack dx + \left\lbrack {x^{3} + x^{2}\text{cos}\left( {xy} \right)} \right\rbrack dy}} = 4.7124$

205.

$\int_{C}{\left( {y + e^{\sqrt{x}}} \right)dx + \left( {2x + \text{cos}\left( y^{2} \right)} \right)dy = \frac{1}{3}}$

Section 6.5 Exercises

207.

False

209.

True

211.

True

213.

$\text{curl}\ \mathbf{\text{F}} = \mathbf{\text{i}} + x^{2}\mathbf{\text{j}} + y^{2}\mathbf{\text{k}}$

215.

$\text{curl}\ \mathbf{\text{F}} = \left( {xz^{2} - xy^{2}} \right)\mathbf{\text{i}} + \left( {x^{2}y - yz^{2}} \right)\mathbf{\text{j}} + \left( {y^{2}z - x^{2}z} \right)\mathbf{\text{k}}$

217.

$\text{curl}\ \mathbf{\text{F}} = \mathbf{\text{i}} + \mathbf{\text{j}} + \mathbf{\text{k}}$

219.

$\text{curl}\ \mathbf{\text{F}} = \text{−}y\mathbf{\text{i}} - z\mathbf{\text{j}} - x\mathbf{\text{k}}$

221.

$\text{curl}\ \mathbf{\text{F}} = 0$

223.

$\text{div}\ \mathbf{\text{F}} = 3yz^{2} + 2y\ \text{sin}\mspace{2mu} z + 2xe^{2z}$

225.

$\text{div}\ \mathbf{\text{F}} = 2(x + y + z)$

227.

$\text{div}\ \mathbf{\text{F}} = \frac{1}{\sqrt{x^{2} + y^{2}}}$

229.

$\text{div}\ \mathbf{\text{F}} = a + b$

231.

$\text{div}\ \mathbf{\text{F}} = x + y + z$

233.

Harmonic

235.

$\text{div}\ (\mathbf{\text{F}}\ \times \ \mathbf{\text{G}}) = 2z + 3x$

237.

$\text{div}\ \mathbf{\text{F}} = 2\left( x^{2}~ + ~y^{2}~ + ~z^{2} \right)$

239.

$\text{curl}\ \mathbf{\text{r}} = 0$

241.

$\text{curl}\ \frac{\mathbf{\text{r}}}{r^{3}} = 0$

243.

$\text{curl}\ \mathbf{\text{F}} = \frac{2x}{x^{2} + y^{2}}\mathbf{\text{k}}$

245.

$\text{div}\ \mathbf{\text{F}} = 0$

247.

$\text{div}\ \mathbf{\text{F}} = 2 - 2e^{-6}$

249.

$\text{div}\ \mathbf{\text{F}} = 0$

251.

$\text{curl}\ \mathbf{\text{F}} = \mathbf{\text{j}} - 3\mathbf{\text{k}}$

253.

$\text{curl}\ \mathbf{\text{F}} = 2\mathbf{\text{j}} - \mathbf{\text{k}}$

255.

$a = 3$

257.

F is conservative.

259.

$\text{div}\ \mathbf{\text{F}} = \text{cosh}\ x + \text{sinh}\ y - xy$

261.

$(bz - cy)\mathbf{\text{i}} + (cx - az)\mathbf{\text{j}} + (ay - bx)\mathbf{\text{k}}$

263.

$\text{curl}\ \mathbf{\text{F}} = 2\omega$

265.

$\mathbf{\text{F}}\ \times \ \mathbf{\text{G}}$ does not have zero divergence.

267.

$\nabla \cdot \mathbf{\text{F}} = -200k\left\lbrack {1 + 2\left( {x^{2} + y^{2} + z^{2}} \right)} \right\rbrack e^{\text{−}x^{2} + y^{2} + z^{2}}$

Section 6.6 Exercises

269.

True

271.

True

273.

$\mathbf{\text{r}}\left( {u,v} \right) = \left\langle {u,v,2 - 3u + 2v} \right\rangle$ for $\text{−}\infty \leq u < \infty$ and $\text{−}\infty \leq v < \infty.$

275.

$\mathbf{\text{r}}(u,v) = \left\langle {u,v,\frac{1}{3}\left( {16 - 2u + 4v} \right)} \right\rangle$ for $|u| < \infty$ and $|v| < \infty.$

277.

$\mathbf{\text{r}}(u,v) = \left\langle {3\ \text{cos}\ u,3\ \text{sin}\ u,v} \right\rangle$ for $0 \leq u \leq \frac{\pi}{2},0 \leq v \leq 3$

279.

$A = 28\pi = 87.9646$

281.

${\iint_{S}{zdS}} = 8\pi$

283.

${\iint_{S}{\left( {x^{2} + y^{2}} \right)zdS}} = 16\pi$

285.

$\iint_{S}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}dS = \frac{4\pi}{3}}$

287.

$m \approx 13.0639$

289.

$m \approx 228.5313$

291.

${\iint_{S}{gdS}} = 3\sqrt{14}$

293.

${\iint_{S}\left( {x - y^{2} + z} \right)}dS \approx 0.9617$

295.

${\iint_{S}{\left( {x^{2} + y^{2}} \right)dS}} = \frac{4\pi}{3}$

297.

${\iint_{S}{x^{2}zdS}} = \frac{1023\pi\sqrt{2}}{5}$

299.

${\iint_{S}{\left( {z + y} \right)dS}} \approx 10.1$

301.

$m = \pi a^{3}$

303.

$\iint_{S}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}dS = \frac{13}{24}}$

305.

$\iint_{S}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}dS = \frac{3}{4}}$

307.

$\int\limits_{0}^{8}{\int\limits_{0}^{6}{\left( {4 - 3y + \frac{1}{16}y^{2} + z} \right)\left( {\frac{1}{4}\sqrt{17}} \right)dzdy}}$

309.

$\int\limits_{0}^{2}{\int\limits_{0}^{6}{\left\lbrack {x^{2} - 2(8 - 4x) + z} \right\rbrack\sqrt{17}dzdx}}$

311.

${\iint_{S}{\left( {x^{2}z + y^{2}z} \right)dS}} = \frac{\pi a^{5}}{2}$

313.

${\iint_{S}{x^{2}yzdS}} = 171\sqrt{14}$

315.

${\iint_{S}{yzdS}} = \frac{\sqrt{2}\pi}{4}$

317.

${\iint_{S}{(x\mathbf{\text{i}} + y\mathbf{\text{j}}) \cdot dS}} = 16\pi$

319.

$m = \frac{\pi a^{7}}{192}$

321.

$F \approx 4.57\ \text{lb}.$

323.

$8\pi a$

325.

The net flux is zero.

Section 6.7 Exercises

327.

${\iint_{S}{(\text{curl}\ \mathbf{\text{F}} \cdot \mathbf{\text{N}})dS}} = \pi a^{2}$

329.

${\iint_{S}{(\text{curl}\ \mathbf{\text{F}} \cdot \mathbf{\text{N}})dS}} = 18\pi$

331.

${\iint_{S}{(\text{curl}\ \mathbf{\text{F}} \cdot \mathbf{\text{N}})dS}} = -8\pi$

333.

${\iint_{S}{(\text{curl}\ \mathbf{\text{F}} \cdot \mathbf{\text{N}})dS}} = 0$

335.

$\int_{C}\textbf{F} \cdot d\mathbf{r} = 0$

337.

${\int_{s}\textbf{F} \cdot d\mathbf{r} = - 3\pi \approx - 9}.4248$

339.

$\iint\limits_{S}{\text{curl}\ \mathbf{\text{F}} \cdot d\mathbf{\text{S}} = 0}$

341.

${\iint_{S}{\text{curl}\ \mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 2.6667$

343.

${\iint_{S}{(\text{curl}\ \mathbf{\text{F}} \cdot \mathbf{\text{N}})dS}} = - \frac{1}{6}$

345.

${\int\limits_{C}\left( {\frac{1}{2}y^{2}dx + zdy + xdz} \right)} = - \frac{\pi}{4}$

347.

${\iint_{S}{(\text{curl}\ \mathbf{\text{F}} \cdot \mathbf{\text{N}})dS}} = 3\pi$

349.

${\int_{C}^{}{(c\mathbf{\text{k}}\ \times \ \mathbf{\text{R}}) \cdot d\text{r}}} = 2\pi c$

351.

${\iint_{S}{\text{curl}\ \mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 0$

353.

$\int_{C}\mathbf{F} \cdot d\mathbf{r} = -4$

355.

${\iint_{S}{\text{curl}\ \mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 0$

357.

${\iint_{S}{\text{curl}\ \mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = -36\pi$

359.

$\iint_{S}\text{curl}\ \textbf{F} \cdot d\textbf{S} = 0$

361.

$\int_{C}\mathbf{F} \cdot d\mathbf{r} = 0$

363.

$\iint_{S}{\text{curl}(\mathbf{\text{F}}) \cdot d\mathbf{\text{S}} = 84.8230}$

365.

$A = \iint_{S}\text{curl}~\textbf{F} \cdot d\mathbf{S} = 0$

367.

$\iint_{S}\text{curl}~\textbf{F} \cdot d\mathbf{S} = 2\pi$

369.

$C = \pi\left( {\text{cos}\ \varphi - \text{sin}\ \varphi} \right)$

371.

${\int_{C}{\mathbf{\text{F}} \cdot d\mathbf{\text{r}}}} = 48\pi$

373.

$\iint_{S}\text{curl}~\textbf{F} \cdot d\mathbf{S} = 0$

375.

0

Section 6.8 Exercises

377.

${\int_{S}^{}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}ds}} = 24\pi~ \approx 75.3982$

379.

${\int_{S}^{}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}ds}} = \frac{243\pi}{2} \approx 381.704$

381.

${\int_{S}^{}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}ds}} = 12\pi~ \approx 37.6991$

383.

${\int_{S}^{}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}ds}} = \frac{9\pi a^{4}}{2}$

385.

${\iint_{S}{\mathbf{\text{F}} \cdot}}d\mathbf{\text{S}} = \frac{4\pi}{3}$

387.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 0$

389.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = \frac{384\pi}{5} \approx 241.2743$

391.

${\iint_{D}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} =$ Net flux $= 0$; flux through the paraboloid $= –\pi$

393.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = \frac{2\pi}{3}$

395.

$16\sqrt{6}\pi$

397.

$- \frac{128\pi}{3}$

399.

$–224\pi~ \approx -703.7168$

401.

20

403.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 8$

405.

${\iint_{S}{\mathbf{\text{F}} \cdot \mathbf{\text{N}}dS}} = \frac{1}{8}$

407.

$\iint_{S}\left\| \textbf{R} \right\|\textbf{R} \cdot \mathbf{n}dS = 4\pi a^{4}$

409.

${\iiint_{R}{z^{2}dV}} = \frac{4\pi}{15}$

411.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 3e - 2 + \frac{1}{2}\text{sin}1 \approx 6.5759$

413.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 21$

415.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = 72$

417.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = - \frac{32\pi}{3} \approx -33.5103$

419.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = \pi a^{4}b^{2}$

421.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = \frac{5\pi}{2}$

423.

${\iint_{S}{\mathbf{\text{F}} \cdot d\mathbf{\text{S}}}} = \frac{21\pi}{2}$

425.

$e^{- 1} - 1$

Review Exercises

427.

False

429.

False

431.

433.

Conservative, $f(x,y) = xy - 2e^{y}$

435.

Conservative, $f(x,y,z) = x^{2}y + y^{2}z + z^{2}x$

437.

$- \frac{16}{3}$

439.

$\frac{32\sqrt{2}}{9}\left( {3\sqrt{3} - 1} \right)$

441.

Divergence: $e^{x} + xe^{xy} + xye^{xyz},$ curl: $xze^{xyz}\mathbf{\text{i}} - yze^{xyz}\mathbf{\text{j}} + ye^{xy}\mathbf{\text{k}}$

443.

$-18\pi$

445.

$\text{−}\pi$

447.

$24\pi$

449.

${\sqrt{2}\left( {2\sqrt{2} + \pi} \right)} = 4 + \pi\sqrt{2}$

451.

${8\pi}\text{/}3$

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