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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Checkpoint

6.1

12 i j 12 i j

6.3

Rotational

A visual representation of a rotational vector field in a coordinate plane. The arrows circle the origin in a counterclockwise manner.
6.4

6565 m/sec

6.5

No.

6.7

1.49063 × 10 −18 , 4.96876 × 10 −19 , 9.93752 × 10 −19 N 1.49063 × 10 −18 , 4.96876 × 10 −19 , 9.93752 × 10 −19 N

6.9

No

6.10

f = v f = v

6.11

P y = x Q x = −2 x y P y = x Q x = −2 x y

6.12

No

6.13

2 2

6.14

2 10 π + 2 10 π 2 2 10 π + 2 10 π 2

6.15

Both line integrals equal 1000303.1000303.

6.16

4 17 4 17

6.17

C F · T d s C F · T d s

6.18

−26 −26

6.19

0

6.20

182π2182π2 kg

6.21

3/2

6.22

2 π 2 π

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 C1C1 and C2C2 represent the two curves, then C1F.drC2F.dr.C1F.drC2F.dr.

6.28

f ( x , y ) = e x y 3 + x y f ( x , y ) = e x y 3 + x y

6.29

f ( x , y , z ) = 4 x 3 + sin y cos z + z f ( x , y , z ) = 4 x 3 + sin y cos z + z

6.30

f ( x , y , z ) = G x 2 + y 2 + z 2 f ( x , y , z ) = G x 2 + y 2 + z 2

6.31

It is conservative.

6.32

−10 π −10 π

6.33

Negative

6.34

45 2 45 2

6.35

4 3 4 3

6.36

3 π 2 3 π 2

6.37

g ( x , y ) = x cos y g ( x , y ) = x cos y

6.38

No

6.39

105 π 105 π

6.40

y z 2 y z 2

6.41

Yes

6.42

All points on line y=1.y=1.

6.43

i i

6.44

curl v = 0 curl v = 0

6.45

No

6.46

Yes

6.47

Cylinder x2+y2=4x2+y2=4

6.48

Cone x2+y2=z2x2+y2=z2

6.49

r(u,v)=ucosv,usinv,u,r(u,v)=ucosv,usinv,u, 0<u<,0v<π20<u<,0v<π2

6.50

Yes

6.51

43.02 43.02

6.52

With the standard parameterization of a cylinder, Equation 6.18 shows that the surface area is 2πrh.2πrh.

6.53

2 π ( 2 + sinh −1 ( 1 ) ) 2 π ( 2 + sinh −1 ( 1 ) )

6.54

24

6.55

0

6.56

38.401 π 120.640 38.401 π 120.640

6.57

N ( x , y ) = y 1 + x 2 + y 2 , x 1 + x 2 + y 2 , 1 1 + x 2 + y 2 N ( x , y ) = y 1 + x 2 + y 2 , x 1 + x 2 + y 2 , 1 1 + x 2 + y 2

6.58

0

6.59

400 kg/sec/m

6.60

440 π 3 440 π 3

6.61

Both integrals give 13645.13645.

6.62

π π

6.63

3 2 3 2

6.64

curl E = x , y , −2 z curl E = x , y , −2 z

6.65

Both integrals equal 6π.6π.

6.66

30

6.67

9 ln ( 16 ) 9 ln ( 16 )

6.68

6.777 × 10 9 6.777 × 10 9

Section 6.1 Exercises

1 .

Vectors

3 .

False

5 .


A visual representation of a vector field in two dimensions. The arrows are larger the further they are from the origin. They stretch away from the origin in a radial pattern.
7 .


A visual representation of a vector field in two dimensions. The arrows are larger the further they are from the origin and the further to the left and right they are from the y axis. The arrows asymptotically curve down and to the right in quadrant 1, down and to the left in quadrant 2, up and to the left in quadrant 3, and up and to the right in quadrant four.
9 .


A visual representation of a vector field in two dimensions. The arrows are larger the further away from the origin they are and, even more so, the further away from the y axis they are. They stretch out away from the origin in a radial manner.
11 .


A visual representation of a vector field in two dimensions. The arrows are larger the further away they are from the x axis. The arrows form two radial patterns, one on each side of the y axis. The patterns are clockwise.
13 .


A visual representation of a vector field in three dimensions. The arrows seem to get smaller as both the z component gest close to zero and the x component gets larger, and as both the y and z components get larger. The arrows seem to converge in both of those directions as well.
15 .

F ( x , y ) = sin ( y ) i + ( x cos y sin y ) j F ( x , y ) = sin ( y ) i + ( x cos y sin y ) j

17 .

F ( x , y , z ) = ( 2 x y + y ) i + ( x 2 + x + 2 y z ) j + y 2 k F ( x , y , z ) = ( 2 x y + y ) i + ( x 2 + x + 2 y z ) j + y 2 k

19 .

F ( x , y ) = ( 2 x 1 + x 2 + 2 y 2 ) i + ( 4 y 1 + x 2 + 2 y 2 ) j F ( x , y ) = ( 2 x 1 + x 2 + 2 y 2 ) i + ( 4 y 1 + x 2 + 2 y 2 ) j

21 .

F ( x , y ) = ( 1 x ) i y j ( 1 x ) 2 + y 2 F ( x , y ) = ( 1 x ) i y j ( 1 x ) 2 + y 2

23 .

F ( x , y ) = - x i y j x 2 + y 2 F ( x , y ) = - x i y j x 2 + y 2

25 .

F ( x , y ) = y i x j F ( x , y ) = y i x j

27 .

F ( x , y ) = −10 ( x 2 + y 2 ) 3 / 2 ( x i + y j ) F ( x , y ) = −10 ( x 2 + y 2 ) 3 / 2 ( x i + y j )

29 .

E = c | r | 2 r = c | r | r | r | E = c | r | 2 r = c | r | r | r |

31 .

c ( t ) = ( cos t , sin t , e t ) = F ( c ( t ) ) c ( t ) = ( cos t , sin t , e t ) = F ( c ( t ) )

33 .

H

35 .

d. F+GF+G

37 .

a. F+GF+G

Section 6.2 Exercises

39 .

True

41 .

False

43 .

False

45 .

C ( x y ) d s = 10 C ( x y ) d s = 10

47 .

C x y 4 d s = 8192 5 C x y 4 d s = 8192 5

49 .

W = 8 W = 8

51 .

W = 3 π 4 W = 3 π 4

53 .

W = π W = π

55 .

C F · d r = 4 C F · d r = 4

57 .

C y z d x + x z d y + x y d z = −1 C y z d x + x z d y + x y d z = −1

59 .

C ( y 2 ) d x + ( x ) d y = 245 6 C ( y 2 ) d x + ( x ) d y = 245 6

61 .

C x y d x + y d y = 190 3 C x y d x + y d y = 190 3

63 .

C y 2 x 2 y 2 d s = 2 ln 5 C y 2 x 2 y 2 d s = 2 ln 5

65 .

W = −66 W = −66

67 .

W = −10 π 2 W = −10 π 2

69 .

W = 2 W = 2

71 .

a. W=11;W=11; b. W=394;W=394; c. No

73 .

W = 2 π W = 2 π

75 .

C F · d r = 25 5 + 1 120 C F · d r = 25 5 + 1 120

77 .

C y 2 d x + ( x y x 2 ) d y = 6.15 C y 2 d x + ( x y x 2 ) d y = 6.15

79 .

γ x e y d s 7.157 γ x e y d s 7.157

81 .

γ ( y 2 x y ) d x −1.379 γ ( y 2 x y ) d x −1.379

83 .

C F · d r −1.133 C F · d r −1.133

85 .

C F · d r 2 2 . 8 5 7 C F · d r 2 2 . 8 5 7

87 .

flux = 1 3 flux = 1 3

89 .

flux = −20 flux = −20

91 .

flux = 0 flux = 0

93 .

m = 4 π ρ 5 m = 4 π ρ 5

95 .

W = 0 W = 0

97 .

W = k 2 W = k 2

Section 6.3 Exercises

99 .

True

101 .

True

103 .

C F · d r = 24 C F · d r = 24

105 .

C F · d r = e 3 π 2 C F · d r = e 3 π 2

107 .

Not conservative

109 .

Conservative, f(x,y)=3x2+5xy+2y2f(x,y)=3x2+5xy+2y2

111 .

Conservative, f(x,y)=yex+xsin(y)f(x,y)=yex+xsin(y)

113 .

C ( 2 y d x + 2 x d y ) = 32 C ( 2 y d x + 2 x d y ) = 32

115 .

F ( x , y ) = ( 10 x + 3 y ) i + ( 3 x + 20 y ) j F ( x , y ) = ( 10 x + 3 y ) i + ( 3 x + 20 y ) j

117 .

F is not conservative.

119 .

F is conservative and a potential function is f(x,y,z)=xyez.f(x,y,z)=xyez.

121 .

F is conservative and a potential function is f(x,y,z)=z2zxy.f(x,y,z)=z2zxy.

123 .

F is conservative and a potential function is f(x,y,z)=x2y+y2z.f(x,y,z)=x2y+y2z.

125 .

F is conservative and a potential function is f(x,y)=ex2yf(x,y)=ex2y

127 .

C F · d r = e 2 + 1 C F · d r = e 2 + 1

129 .

C F · d r = -2 C F · d r = -2

131 .

C 1 G · d r = −8 π C 1 G · d r = −8 π

133 .

C 2 F · d r = 7 C 2 F · d r = 7

135 .

C F · d r = 150 C F · d r = 150

137 .

C F · d r = −1 C F · d r = −1

139 .

4 × 10 31 erg 4 × 10 31 erg

141 .

C F · d s = 0.4687 C F · d s = 0.4687

143 .

circulation = π a 2 and flux = 0 circulation = π a 2 and flux = 0

Section 6.4 Exercises

147 .

C 2 x y d x + ( x + y ) d y = 32 3 C 2 x y d x + ( x + y ) d y = 32 3

149 .

C sin x cos y d x + ( x y + cos x sin y ) d y = 1 12 C sin x cos y d x + ( x y + cos x sin y ) d y = 1 12

151 .

C ( y d x + x d y ) = π C ( y d x + x d y ) = π

153 .

C x e −2 x d x + ( x 4 + 2 x 2 y 2 ) d y = 0 C x e −2 x d x + ( x 4 + 2 x 2 y 2 ) d y = 0

155 .

Cy3dxx3ydy=−20πCy3dxx3ydy=−20π

157 .

C x 2 y d x + x y 2 d y = 8 π C x 2 y d x + x y 2 d y = 8 π

159 .

C ( x 2 + y 2 ) d x + 2 x y d y = 0 C ( x 2 + y 2 ) d x + 2 x y d y = 0

161 .

A = 19 π A = 19 π

163 .

A = 3 8 π A = 3 8 π

165 .

C + ( y 2 + x 3 ) d x + x 4 d y = 0 C + ( y 2 + x 3 ) d x + x 4 d y = 0

167 .

A = 9 π 8 A = 9 π 8

169 .

A = 8 3 5 A = 8 3 5

171 .

C ( x 2 y 2 x y + y 2 ) d s = - 5 6 C ( x 2 y 2 x y + y 2 ) d s = - 5 6

173 .

C x d x + y d y x 2 + y 2 = 2 π C x d x + y d y x 2 + y 2 = 2 π

175 .

W = 225 2 W = 225 2

177 .

W = 12 π W = 12 π

179 .

W = 2 π W = 2 π

181 .

C y 2 d x + x 2 d y = 1 3 C y 2 d x + x 2 d y = 1 3

183 .

C 1 + x 3 d x + 2 x y d y = -3 C 1 + x 3 d x + 2 x y d y = -3

185 .

C ( 3 y e sin x ) d x + ( 7 x + y 4 + 1 ) d y = 36 π C ( 3 y e sin x ) d x + ( 7 x + y 4 + 1 ) d y = 36 π

187 .

C F · d r = 2 C F · d r = 2

189 .

C ( y + x ) d x + ( x + sin y ) d y = 0 C ( y + x ) d x + ( x + sin y ) d y = 0

191 .

C x y d x + x 3 y 3 d y = 22 21 C x y d x + x 3 y 3 d y = 22 21

193 .

C F · d r = 15 π 4 C F · d r = 15 π 4

195 .

C sin ( x + y ) d x + cos ( x + y ) d y = 4 C sin ( x + y ) d x + cos ( x + y ) d y = 4

197 .

C F · d r = π C F · d r = π

199 .

C F · n ^ d s = 4 C F · n ^ d s = 4

201 .

C F · n d s = 0 C F · n d s = 0

203 .

C [ y 3 + sin ( x y ) + x y cos ( x y ) ] d x + [ x 3 + x 2 cos ( x y ) ] d y = 4.7124 C [ y 3 + sin ( x y ) + x y cos ( x y ) ] d x + [ x 3 + x 2 cos ( x y ) ] d y = 4.7124

205 .

C ( y + e x ) d x + ( 2 x + cos ( y 2 ) ) d y = 1 3 C ( y + e x ) d x + ( 2 x + cos ( y 2 ) ) d y = 1 3

Section 6.5 Exercises

207 .

False

209 .

True

211 .

True

213 .

curl F = i + x 2 j + y 2 k curl F = i + x 2 j + y 2 k

215 .

curl F = ( x z 2 x y 2 ) i + ( x 2 y y z 2 ) j + ( y 2 z x 2 z ) k curl F = ( x z 2 x y 2 ) i + ( x 2 y y z 2 ) j + ( y 2 z x 2 z ) k

217 .

curl F = i + j + k curl F = i + j + k

219 .

curl F = y i z j x k curl F = y i z j x k

221 .

curl F = 0 curl F = 0

223 .

div F = 3 y z 2 + 2 y sin z + 2 x e 2 z div F = 3 y z 2 + 2 y sin z + 2 x e 2 z

225 .

div F = 2 ( x + y + z ) div F = 2 ( x + y + z )

227 .

div F = 1 x 2 + y 2 div F = 1 x 2 + y 2

229 .

div F = a + b div F = a + b

231 .

div F = x + y + z div F = x + y + z

233 .

Harmonic

235 .

div ( F × G ) = 2 z + 3 x div ( F × G ) = 2 z + 3 x

237 .

div F = 2 r 2 div F = 2 r 2

239 .

curl r = 0 curl r = 0

241 .

curl r r 3 = 0 curl r r 3 = 0

243 .

curl F = 2 x x 2 + y 2 k curl F = 2 x x 2 + y 2 k

245 .

div F = 0 div F = 0

247 .

div F = 2 2 e −6 div F = 2 2 e −6

249 .

div F = 0 div F = 0

251 .

curl F = j 3 k curl F = j 3 k

253 .

curl F = 2 j k curl F = 2 j k

255 .

a = 3 a = 3

257 .

F is conservative.

259 .

div F = cosh x + sinh y x y div F = cosh x + sinh y x y

261 .

( b z c y ) i ( c x a z ) j + ( a y b x ) k ( b z c y ) i ( c x a z ) j + ( a y b x ) k

263 .

curl F = 2 ω curl F = 2 ω

265 .

F×GF×G does not have zero divergence.

267 .

· F = −200 k [ 1 + 2 ( x 2 + y 2 + z 2 ) ] e x 2 + y 2 + z 2 · F = −200 k [ 1 + 2 ( x 2 + y 2 + z 2 ) ] e x 2 + y 2 + z 2

Section 6.6 Exercises

269 .

True

271 .

True

273 .

r(u,v)=u,v,23u+2vr(u,v)=u,v,23u+2v for u<u< and v<.v<.

275 .

r(u,v)=u,v,13(162u+4v)r(u,v)=u,v,13(162u+4v) for |u|<|u|< and |v|<.|v|<.

277 .

r(u,v)=3cosu,3sinu,vr(u,v)=3cosu,3sinu,v for 0uπ2,0v30uπ2,0v3

279 .

A = 87.9646 A = 87.9646

281 .

S z d S = 8 π S z d S = 8 π

283 .

S ( x 2 + y 2 ) z d S = 16 π S ( x 2 + y 2 ) z d S = 16 π

285 .

S F · N d S = 4 π 3 S F · N d S = 4 π 3

287 .

m 13.0639 m 13.0639

289 .

m 228.5313 m 228.5313

291 .

S g d S = 3 14 S g d S = 3 14

293 .

S ( x 2 + y z ) d S 0.9617 S ( x 2 + y z ) d S 0.9617

295 .

S ( x 2 + y 2 ) d S = 4 π 3 S ( x 2 + y 2 ) d S = 4 π 3

297 .

S x 2 z d S = 1023 2 π 5 S x 2 z d S = 1023 2 π 5

299 .

S ( z + y ) d S 10.1 S ( z + y ) d S 10.1

301 .

m = π a 3 m = π a 3

303 .

S F · N d S = 13 24 S F · N d S = 13 24

305 .

S F · N d S = 3 4 S F · N d S = 3 4

307 .

0 8 0 6 ( 4 3 y + 1 16 y 2 + z ) ( 1 4 17 ) d z d y 0 8 0 6 ( 4 3 y + 1 16 y 2 + z ) ( 1 4 17 ) d z d y

309 .

0 2 0 6 [ x 2 2 ( 8 4 x ) + z ] 17 d z d x 0 2 0 6 [ x 2 2 ( 8 4 x ) + z ] 17 d z d x

311 .

S ( x 2 z + y 2 z ) d S = π a 5 2 S ( x 2 z + y 2 z ) d S = π a 5 2

313 .

S x 2 y z d S = 171 14 S x 2 y z d S = 171 14

315 .

S y z d S = 2 π 4 S y z d S = 2 π 4

317 .

S ( x i + y j ) · d S = 16 π S ( x i + y j ) · d S = 16 π

319 .

m = π a 7 192 m = π a 7 192

321 .

F 4.57 lb . F 4.57 lb .

323 .

8 π a 8 π a

325 .

The net flux is zero.

Section 6.7 Exercises

327 .

S ( curl F · N ) d S = π a 2 S ( curl F · N ) d S = π a 2

329 .

S ( curl F · N ) d S = 18 π S ( curl F · N ) d S = 18 π

331 .

S ( curl F · N ) d S = −8 π S ( curl F · N ) d S = −8 π

333 .

S ( curl F · N ) d S = 0 S ( curl F · N ) d S = 0

335 .

C F · d S = 0 C F · d S = 0

337 .

C F · d S = −9.4248 C F · d S = −9.4248

339 .

S curl F · d S = 0 S curl F · d S = 0

341 .

S curl F · d S = 2.6667 S curl F · d S = 2.6667

343 .

S ( curl F · N ) d S = 1 6 S ( curl F · N ) d S = 1 6

345 .

C ( 1 2 y 2 d x + z d y + x d z ) = π 4 C ( 1 2 y 2 d x + z d y + x d z ) = π 4

347 .

S ( curl F · N ) d S = 3 π S ( curl F · N ) d S = 3 π

349 .

C ( c k × R ) · d S = 2 π c C ( c k × R ) · d S = 2 π c

351 .

S curl F · d S = 0 S curl F · d S = 0

353 .

F · d S = −4 F · d S = −4

355 .

S curl F · d S = 0 S curl F · d S = 0

357 .

S curl F · d S = −36 π S curl F · d S = −36 π

359 .

S curl F · N = 0 S curl F · N = 0

361 .

C F · d r = 0 C F · d r = 0

363 .

S curl ( F ) · d S = 84.8230 S curl ( F ) · d S = 84.8230

365 .

A = S ( × F ) · n d S = 0 A = S ( × F ) · n d S = 0

367 .

S ( × F ) · n d S = 2 π S ( × F ) · n d S = 2 π

369 .

C = π ( cos φ sin φ ) C = π ( cos φ sin φ )

371 .

C F · d r = 48 π C F · d r = 48 π

373 .

S ( × F ) · n = 0 S ( × F ) · n = 0

375 .

0

Section 6.8 Exercises

377 .

S F · n d s = 75.3982 S F · n d s = 75.3982

379 .

S F · n d s = 127.2345 S F · n d s = 127.2345

381 .

S F · n d s = 37.6991 S F · n d s = 37.6991

383 .

S F · n d s = 9 π a 4 2 S F · n d s = 9 π a 4 2

385 .

S F · d S = 4 π 3 S F · d S = 4 π 3

387 .

S F · d S = 0 S F · d S = 0

389 .

S F · d S = 241.2743 S F · d S = 241.2743

391 .

D F · d S = π D F · d S = π

393 .

S F · d S = 2 π 3 S F · d S = 2 π 3

395 .

16 6 π 16 6 π

397 .

128 3 π 128 3 π

399 .

−703.7168 −703.7168

401 .

20

403 .

S F · d S = 8 S F · d S = 8

405 .

S F · N d S = 1 8 S F · N d S = 1 8

407 .

S R R · n d s = 4 π a 4 S R R · n d s = 4 π a 4

409 .

R z 2 d V = 4 π 15 R z 2 d V = 4 π 15

411 .

S F · d S = 6.5759 S F · d S = 6.5759

413 .

S F · d S = 21 S F · d S = 21

415 .

S F · d S = 72 S F · d S = 72

417 .

S F · d S = −33.5103 S F · d S = −33.5103

419 .

S F · d S = π a 4 b 2 S F · d S = π a 4 b 2

421 .

S F · d S = 5 2 π S F · d S = 5 2 π

423 .

S F · d S = 531 π 32 S F · d S = 531 π 32

425 .

( 1 e −1 ) ( 1 e −1 )

Review Exercises

427 .

False

429 .

False

431 .


A vector field in two dimensions. All quadrants are shown. The arrows are larger the further from the y axis they become. They point up and to the right for positive x values and down and to the right for negative x values. The further from the y axis they are, the steeper the slope they have.
433 .

Conservative, f(x,y)=xy2eyf(x,y)=xy2ey

435 .

Conservative, f(x,y,z)=x2y+y2z+z2xf(x,y,z)=x2y+y2z+z2x

437 .

16 3 16 3

439 .

32 2 9 ( 3 3 1 ) 32 2 9 ( 3 3 1 )

441 .

Divergence: ex+xexy+xyexyz,ex+xexy+xyexyz, curl: xzexyziyzexyzj+yexykxzexyziyzexyzj+yexyk

443 .

−18 π −18 π

445 .

π π

447 .

24 π 24 π

449 .

2 ( 2 2 + π ) = 4 + π 2 2 ( 2 2 + π ) = 4 + π 2

451 .

8 π / 3 8 π / 3

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