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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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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

12ij12ij

6.2


A visual representation of the given radial field in a coordinate plane. The magnitudes increase further from the origin. The arrow seem to be stretching away from the origin in a rectangular shape.
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.

6.6


A visual representation of the given vector field in three dimensions. The x and z components are always 2 and 1, respectively. The y component is z/2. The closer z comes to zero, the smaller the y component is, and the further away z is from zero, the larger the y component is.
6.7

1.49063×10−18,4.96876×10−19,9.93752×10−19N1.49063×10−18,4.96876×10−19,9.93752×10−19N

6.8


A visual representation of the given given vector in two dimensions. The arrows seem to be forming several ovals. The first is around the origin, where the arrows curve to the right above and below the x axis. The closer the arrows are to the x axis, the flatter they are. There appear to be six other ovals, three on either side of the central one. The vectors get longer as they get farther from the origin, and then they start to get shorter again.
6.9

No

6.10

f=vf=v

6.11

Py=xQx=−2xyPy=xQx=−2xy

6.12

No

6.13

22

6.14

13+26+3π413+26+3π4

6.15

Both line integrals equal 1000303.1000303.

6.16

417417

6.17

CF·TdsCF·Tds

6.18

−26−26

6.19

0

6.20

182π2182π2 kg

6.21

3/2

6.22

2π2π

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 C1FdrC2Fdr.C1FdrC2Fdr.

6.28

f(x,y)=exy3+xyf(x,y)=exy3+xy

6.29

f(x,y,z)=4x3+sinycosz+zf(x,y,z)=4x3+sinycosz+z

6.30

f(x,y,z)=Gx2+y2+z2f(x,y,z)=Gx2+y2+z2

6.31

It is conservative.

6.32

−10π−10π

6.33

Negative

6.34

452452

6.35

4343

6.36

3π23π2

6.37

g(x,y)=xcosyg(x,y)=xcosy

6.38

No

6.39

105π105π

6.40

yz2yz2

6.41

Yes

6.42

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

6.43

ii

6.44

curlv=0curlv=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

43.0243.02

6.52

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

6.53

2π(2+sinh−1(1))2π(2+sinh−1(1))

6.54

24

6.55

0

6.56

38.401π120.64038.401π120.640

6.57

N(x,y)=y1+x2+y2,x1+x2+y2,11+x2+y2N(x,y)=y1+x2+y2,x1+x2+y2,11+x2+y2

6.58

0

6.59

400 kg/sec/m

6.60

440π3440π3

6.61

Both integrals give 13645.13645.

6.62

ππ

6.63

3232

6.64

curlE=x,y,−2zcurlE=x,y,−2z

6.65

Both integrals equal 6π.6π.

6.66

30

6.67

9ln(16)9ln(16)

6.68

6.777×1096.777×109

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+(xcosysiny)jF(x,y)=sin(y)i+(xcosysiny)j

17.

F(x,y,z)=(2xy+y)i+(x2+x+2yz)j+y2kF(x,y,z)=(2xy+y)i+(x2+x+2yz)j+y2k

19.

F(x,y)=(2x1+x2+2y2)i+(4y1+x2+2y2)jF(x,y)=(2x1+x2+2y2)i+(4y1+x2+2y2)j

21.

F(x,y)=(1x)iyj(1x)2+y2F(x,y)=(1x)iyj(1x)2+y2

23.

F(x,y)=(yixj)x2+y2F(x,y)=(yixj)x2+y2

25.

F(x,y)=yixjF(x,y)=yixj

27.

F(x,y)=−10(x2+y2)3/2(xi+yj)F(x,y)=−10(x2+y2)3/2(xi+yj)

29.

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

31.

c(t)=(cost,sint,et)=F(c(t))c(t)=(cost,sint,et)=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(xy)ds=10C(xy)ds=10

47.

Cxy4ds=81925Cxy4ds=81925

49.

W=8W=8

51.

W=3π4W=3π4

53.

W=πW=π

55.

CF·dr=4CF·dr=4

57.

Cyzdx+xzdy+xydz=−1Cyzdx+xzdy+xydz=−1

59.

C(y2)dx+(x)dy=2456C(y2)dx+(x)dy=2456

61.

Cxydx+ydy=1903Cxydx+ydy=1903

63.

Cy2x2y2ds=2ln5Cy2x2y2ds=2ln5

65.

W=−66W=−66

67.

W=−10π2W=−10π2

69.

W=2W=2

71.

a. W=11;W=11; b. W=11;W=11; c. Yes

73.

W=2πW=2π

75.

CF·dr=255+1120CF·dr=255+1120

77.

Cy2dx+(xyx2)dy=6.15Cy2dx+(xyx2)dy=6.15

79.

γxeyds7.157γxeyds7.157

81.

γ(y2xy)dx−1.379γ(y2xy)dx−1.379

83.

CF·dr−1.133CF·dr−1.133

85.

CF·dr22.857CF·dr22.857

87.

flux=13flux=13

89.

flux=−20flux=−20

91.

flux=0flux=0

93.

m=4πρ5m=4πρ5

95.

W=0W=0

97.

W=k2W=k2

Section 6.3 Exercises

99.

True

101.

True

103.

CF·dr=24CF·dr=24

105.

CF·dr=e3π2CF·dr=e3π2

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(2ydx+2xdy)=32C(2ydx+2xdy)=32

115.

F(x,y)=(10x+3y)i+(3x+10y)jF(x,y)=(10x+3y)i+(3x+10y)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.

CF·dr=e2+1CF·dr=e2+1

129.

CF·dr=41CF·dr=41

131.

C1G·dr=−8πC1G·dr=−8π

133.

C2F·dr=7C2F·dr=7

135.

CF·dr=150CF·dr=150

137.

CF·dr=−1CF·dr=−1

139.

4×1031erg4×1031erg

141.

CF·ds=0.4687CF·ds=0.4687

143.

circulation=πa2and flux=0circulation=πa2and flux=0

Section 6.4 Exercises

147.

C2xydx+(x+y)dy=323C2xydx+(x+y)dy=323

149.

Csinxcosydx+(xy+cosxsiny)dy=112Csinxcosydx+(xy+cosxsiny)dy=112

151.

C(ydx+xdy)=πC(ydx+xdy)=π

153.

Cxe−2xdx+(x4+2x2y2)dy=0Cxe−2xdx+(x4+2x2y2)dy=0

155.

Cy3dxx3dy=−24πCy3dxx3dy=−24π

157.

Cx2ydx+xy2dy=8πCx2ydx+xy2dy=8π

159.

C(x2+y2)dx+2xydy=0C(x2+y2)dx+2xydy=0

161.

A=19πA=19π

163.

A=38πA=38π

165.

C+(y2+x3)dx+x4dy=0C+(y2+x3)dx+x4dy=0

167.

A=9π8A=9π8

169.

A=835A=835

171.

C(x2y2xy+y2)ds=3C(x2y2xy+y2)ds=3

173.

Cxdx+ydyx2+y2=2πCxdx+ydyx2+y2=2π

175.

W=2252W=2252

177.

W=12πW=12π

179.

W=2πW=2π

181.

Cy2dx+x2dy=13Cy2dx+x2dy=13

183.

C1+x3dx+2xydy=3C1+x3dx+2xydy=3

185.

C(3yesinx)dx+(7x+y4+1)dy=36πC(3yesinx)dx+(7x+y4+1)dy=36π

187.

CF·dr=2CF·dr=2

189.

C(y+x)dx+(x+siny)dy=0C(y+x)dx+(x+siny)dy=0

191.

Cxydx+x3y3dy=2221Cxydx+x3y3dy=2221

193.

CF·dr=15π4CF·dr=15π4

195.

Csin(x+y)dx+cos(x+y)dy=4Csin(x+y)dx+cos(x+y)dy=4

197.

CF·dr=πCF·dr=π

199.

CF·n^ds=4CF·n^ds=4

201.

CF·nds=32πCF·nds=32π

203.

C[y3+sin(xy)+xycos(xy)]dx+[x3+x2cos(xy)]dy=4.7124C[y3+sin(xy)+xycos(xy)]dx+[x3+x2cos(xy)]dy=4.7124

205.

C(y+ex)dx+(2x+cos(y2))dy=13C(y+ex)dx+(2x+cos(y2))dy=13

Section 6.5 Exercises

207.

False

209.

True

211.

True

213.

curlF=i+x2j+y2kcurlF=i+x2j+y2k

215.

curlF=(xz2xy2)i+(x2yyz2)j+(y2zx2z)kcurlF=(xz2xy2)i+(x2yyz2)j+(y2zx2z)k

217.

curlF=i+j+kcurlF=i+j+k

219.

curlF=yizjxkcurlF=yizjxk

221.

curlF=0curlF=0

223.

divF=3yz2+2ysinz+2xe2zdivF=3yz2+2ysinz+2xe2z

225.

divF=2(x+y+z)divF=2(x+y+z)

227.

divF=1x2+y2divF=1x2+y2

229.

divF=a+bdivF=a+b

231.

divF=x+y+zdivF=x+y+z

233.

Harmonic

235.

div(F×G)=2z+3xdiv(F×G)=2z+3x

237.

divF=2r2divF=2r2

239.

curlr=0curlr=0

241.

curlrr3=0curlrr3=0

243.

curlF=2xx2+y2kcurlF=2xx2+y2k

245.

divF=0divF=0

247.

divF=22e−6divF=22e−6

249.

divF=0divF=0

251.

curlF=j3kcurlF=j3k

253.

curlF=2jkcurlF=2jk

255.

a=3a=3

257.

F is conservative.

259.

divF=coshx+sinhyxydivF=coshx+sinhyxy

261.

(bzcy)i(cxaz)j+(aybx)k(bzcy)i(cxaz)j+(aybx)k

263.

curlF=2ωcurlF=2ω

265.

F×GF×G does not have zero divergence.

267.

·F=−200k[1+2(x2+y2+z2)]ex2+y2+z2·F=−200k[1+2(x2+y2+z2)]ex2+y2+z2

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=87.9646A=87.9646

281.

SzdS=8πSzdS=8π

283.

S(x2+y2)zdS=16πS(x2+y2)zdS=16π

285.

SF·NdS=4π3SF·NdS=4π3

287.

m13.0639m13.0639

289.

m228.5313m228.5313

291.

SgdS=34SgdS=34

293.

S(x2+yz)dS0.9617S(x2+yz)dS0.9617

295.

S(x2+y2)dS=4π3S(x2+y2)dS=4π3

297.

Sx2zdS=10232π5Sx2zdS=10232π5

299.

S(z+y)dS10.1S(z+y)dS10.1

301.

m=πa3m=πa3

303.

SF·NdS=1324SF·NdS=1324

305.

SF·NdS=34SF·NdS=34

307.

0806(43y+116y2+z)(1417)dzdy0806(43y+116y2+z)(1417)dzdy

309.

0206[x22(84x)+z]17dzdx0206[x22(84x)+z]17dzdx

311.

S(x2z+y2z)dS=πa52S(x2z+y2z)dS=πa52

313.

Sx2yzdS=17114Sx2yzdS=17114

315.

SyzdS=2π4SyzdS=2π4

317.

S(xi+yj)·dS=16πS(xi+yj)·dS=16π

319.

m=πa7192m=πa7192

321.

F4.57lb.F4.57lb.

323.

8πa8πa

325.

The net flux is zero.

Section 6.7 Exercises

327.

S(curlF·N)dS=πa2S(curlF·N)dS=πa2

329.

S(curlF·N)dS=18πS(curlF·N)dS=18π

331.

S(curlF·N)dS=−8πS(curlF·N)dS=−8π

333.

S(curlF·N)dS=0S(curlF·N)dS=0

335.

CF·dS=0CF·dS=0

337.

CF·dS=−9.4248CF·dS=−9.4248

339.

ScurlF·dS=0ScurlF·dS=0

341.

ScurlF·dS=2.6667ScurlF·dS=2.6667

343.

S(curlF·N)dS=16S(curlF·N)dS=16

345.

C(12y2dx+zdy+xdz)=π4C(12y2dx+zdy+xdz)=π4

347.

S(curlF·N)dS=−3πS(curlF·N)dS=−3π

349.

C(ck×R)·dS=2πcC(ck×R)·dS=2πc

351.

ScurlF·dS=0ScurlF·dS=0

353.

F·dS=−4F·dS=−4

355.

ScurlF·dS=0ScurlF·dS=0

357.

ScurlF·dS=−36πScurlF·dS=−36π

359.

ScurlF·N=0ScurlF·N=0

361.

CF·dr=0CF·dr=0

363.

Scurl(F)·dS=84.8230Scurl(F)·dS=84.8230

365.

A=S(×F)·ndS=0A=S(×F)·ndS=0

367.

S(×F)·ndS=2πS(×F)·ndS=2π

369.

C=π(cosφsinφ)C=π(cosφsinφ)

371.

CF·dr=48πCF·dr=48π

373.

S(×F)·n=0S(×F)·n=0

375.

0

Section 6.8 Exercises

377.

SF·nds=75.3982SF·nds=75.3982

379.

SF·nds=127.2345SF·nds=127.2345

381.

SF·nds=37.6991SF·nds=37.6991

383.

SF·nds=9πa42SF·nds=9πa42

385.

SF·dS=π3SF·dS=π3

387.

SF·dS=0SF·dS=0

389.

SF·dS=241.2743SF·dS=241.2743

391.

DF·dS=πDF·dS=π

393.

SF·dS=2π3SF·dS=2π3

395.

166π166π

397.

1283π1283π

399.

−703.7168−703.7168

401.

20

403.

SF·dS=8SF·dS=8

405.

SF·NdS=18SF·NdS=18

407.

SRR·nds=4πa4SRR·nds=4πa4

409.

Rz2dV=4π15Rz2dV=4π15

411.

SF·dS=6.5759SF·dS=6.5759

413.

SF·dS=21SF·dS=21

415.

SF·dS=72SF·dS=72

417.

SF·dS=−33.5103SF·dS=−33.5103

419.

SF·dS=πa4b2SF·dS=πa4b2

421.

SF·dS=52πSF·dS=52π

423.

SF·dS=21π2SF·dS=21π2

425.

(1e−1)(1e−1)

Chapter 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.

163163

439.

3229(331)3229(331)

441.

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

443.

−2π−2π

445.

ππ

447.

31π/231π/2

449.

2(2+π)2(2+π)

451.

2π/32π/3

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