Multivariable Calculus

489 Pages·076.468 MB·English

by  Ron Larson, Bruce Edwards

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Index of Applications
Engineering and Physical Electrical resistance,185 Mass,1055,1061
Sciences Electricity,155,303 on the surface of Earth,486
Electromagnetic theory,577 Maximum area,219,220,221,222,224,
Acceleration,124,128,156,158,176, Emptying a tank of oil,481 240,242,949
253,906 Error Maximum cross-sectional area of an
Air pressure,431 in volume of a ball bearing,233 irrigation canal,223
Air traffic control,154,745,650,850 in volume and surface area of a cube, Maximum volume,221,222,223
Aircraft glide path,193 236 of a box,215,216,220,222,944,949,
Angle of elevation,151,155,156 Explorer 18,694,741 958
Angular rate of change,374 Explorer 55,694 of a can buoy,959
Architecture,694 Falling object,34,315,426,429 of a package,222
Area,116,126,153,256,603,674 Ferris wheel,866 Minimum length,218,221,222,240
Asteroid Apollo,738 Flow rate,286,355,1105 Minimum surface area,222
Atmospheric pressure and altitude,327, Fluid force,541 Minimum time,222,230
353,951 on a circular plate,502 Motion
Automobile aerodynamics,30 of gasoline,501,502 of a liquid,1118,1119
Average speed,40,89 on a stern of a boat,502 of a particle,712
Average temperature,984,1034 in a swimming pool,504,506 pendulum,1155
Average velocity,112 on a tank wall,501,502 spring,1154
Beam deflection,693 of water,501 Moving ladder,154
Beam strength,35,222 Force,289,501,771 Moving shadow,156,158,160
Billiard balls and normal lines,927 Force field,1130 Muzzle velocity,756,757
Boiling temperature,35 Free-falling object,69,82,91 Navigation,695,757
Boyle’s Law,485,504 Frictional force,858,862 Newton’s Law of Gravitation,1041
Braking load,774 Gauss’s Law,1103 Orbit of Earth,708
Breaking strength of a steel cable,364 Gravitational fields,1041 Orbital speed,850
Bridge design,694 Gravitational force,577 Parabolic reflector,684
Building design,445,556,1008,1035, Halley’s comet,694,737 Parachute jump,1148
1064 Harmonic motion,36,38,138,353 Particle motion,128,287,290,823,831,
Buoyant force,501 Heat flux,1123 833,839,840,849,850,861
Cable tension,757,765 Heat transfer,336 Path
Capillary action,1008 Heat-seeking particle,921 of a ball,838
Car performance,35 Heat-seeking path,926 of a baseball,837,838,860
Carbon dating,413 Height of a bomb,839,865
Center of mass,of glass,496 of a baseball,29 of a football,839
Center of pressure on a sail,1001 of a basketball,32 of a projectile,182,712,838,839,964
Centripetal acceleration,850 Highway design,169,193,866 of a shot-put throw,839
Centripetal force,850 Honeycomb,169 Pendulum,138,237,906,1155
Centroid,494,495,502,519 Horizontal motion,355 Planetary motion,741
Chemical mixture problem,427,429 Hyperbolic detection system,691 Planetary orbits,687
Chemical reaction,391,422,550,962 Hyperbolic mirror,695 Planimeter,1122
Circular motion,840,848 Ideal Gas Law,879,898,914 Power,169,906
Comet Hale-Bopp,741 Illumination,222,241 Projectile motion,237,675,705,757,
Construction,154,765 Inflating balloon,150 836,838,839,847,849,850,860,
Cycloidal motion,839,849 Kepler’s Laws,737,738,862 865,913
Depth Kinetic and potential energy,1071,1074 Radioactive decay,356,409,413,421,
of gasoline in a tank,503 Law of Conservation of Energy,1071 431
of water in a swimming pool,153 Lawn sprinkler,169 Refraction of light,959
of water in a vase,29 Length,603 Refrigeration,158
Distance,241 of a catenary,473,503 Resultant force,754,756
Einstein’s Special Theory of Relativity of pursuit,476 Ripples in a pond,149
and Newton’s First Law of Motion, of a stream,475 Rolling a ball bearing,185
204 Linear and angular velocity,158 Satellite antenna,742
Electric circuit,406,426,429 Linear vs. angular speed,156 Satellite orbit,694,866
Electric force,485 Load supported by a beam,1155 Satellites,127
Electric force fields,1041 Load supports,765 Sending a space module into orbit,480,
Electric potential,878 Lunar gravity,253 571
Electrical charge,1105 Magnetic field of Earth,1050 Solar collector,693
Electrical circuits,1147 Map of the ocean floor,926 Sound intensity,40,327,414
(continued on back inside cover)
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9781285060293_FES.qxp 10/18/12 7:39 AM Page 3
DERIVATIVES AND INTEGRALS
s.
s Basic Differentiation Rules
e
c
Suc 1. d (cid:2)cu(cid:3)(cid:3)cu(cid:4) 2. d (cid:2)u ± v(cid:3)(cid:3)u(cid:4)± v(cid:4) 3. d (cid:2)uv(cid:3)(cid:3) uv(cid:4)(cid:5) vu(cid:4)
k dx dx dx
or d (cid:8)u(cid:9) vu(cid:4)(cid:2)uv(cid:4) d d
w 4. (cid:3) 5. (cid:2)c(cid:3)(cid:3)0 6. (cid:2)un(cid:3)(cid:3) nun(cid:2)1u(cid:4)
e dx v v2 dx dx
m
Ho 7. d (cid:2)x(cid:3)(cid:3)1 8. d (cid:2)(cid:4)u(cid:4)(cid:3)(cid:3) (cid:4)u(cid:4)(cid:6)u(cid:4)(cid:7), u(cid:6)0 9. d (cid:2)ln u(cid:3)(cid:3) u(cid:4)
r dx dx u dx u
o
ds f 10. d (cid:2)eu(cid:3)(cid:3)euu(cid:4) 11. d (cid:2)log u(cid:3)(cid:3) u(cid:4) 12. d (cid:2)au(cid:3)(cid:3)(cid:6)ln a(cid:7)auu(cid:4)
ar dx dx a (cid:6)ln a(cid:7)u dx
C
a 13. d (cid:2)sin u(cid:3)(cid:3)(cid:6)cos u(cid:7)u(cid:4) 14. d (cid:2)cos u(cid:3)(cid:3)(cid:2)(cid:6)sin u(cid:7)u(cid:4) 15. d (cid:2)tan u(cid:3)(cid:3)(cid:6)sec2 u(cid:7)u(cid:4)
ul dx dx dx
m
r d d d
Fo 16. (cid:2)cot u(cid:3)(cid:3)(cid:2)(cid:6)csc2 u(cid:7)u(cid:4) 17. (cid:2)sec u(cid:3)(cid:3)(cid:6)sec u tan u(cid:7)u(cid:4) 18. (cid:2)csc u(cid:3)(cid:3)(cid:2)(cid:6)csc u cot u(cid:7)u(cid:4)
ut dx dx dx
o d u(cid:4) d (cid:2)u(cid:4) d u(cid:4)
r 19. (cid:2)arcsin u(cid:3)(cid:3) 20. (cid:2)arccos u(cid:3)(cid:3) 21. (cid:2)arctan u(cid:3)(cid:3)
ea dx (cid:5)1(cid:2)u2 dx (cid:5)1(cid:2)u2 dx 1(cid:5)u2
T
d (cid:2)u(cid:4) d u(cid:4) d (cid:2)u(cid:4)
22. dx(cid:2)arccot u(cid:3)(cid:3) 1(cid:5)u2 23. dx(cid:2)arcsec u(cid:3)(cid:3) (cid:4)u(cid:4)(cid:5)u2(cid:2)1 24. dx(cid:2)arccsc u(cid:3)(cid:3) (cid:4)u(cid:4)(cid:5)u2(cid:2)1
d d d
25. (cid:2)sinh u(cid:3)(cid:3)(cid:6)cosh u(cid:7)u(cid:4) 26. (cid:2)cosh u(cid:3)(cid:3)(cid:6)sinh u(cid:7)u(cid:4) 27. (cid:2)tanh u(cid:3)(cid:3)(cid:6)sech2 u(cid:7)u(cid:4)
dx dx dx
d d d
28. (cid:2)coth u(cid:3)(cid:3)(cid:2)(cid:6)csch2 u(cid:7)u(cid:4) 29. (cid:2)sech u(cid:3)(cid:3)(cid:2)(cid:6)sech u tanh u(cid:7)u(cid:4) 30. (cid:2)csch u(cid:3)(cid:3)(cid:2)(cid:6)csch u coth u(cid:7)u(cid:4)
dx dx dx
d u(cid:4) d u(cid:4) d u(cid:4)
31. (cid:2)sinh(cid:2)1 u(cid:3)(cid:3) 32. (cid:2)cosh(cid:2)1 u(cid:3)(cid:3) 33. (cid:2)tanh(cid:2)1 u(cid:3)(cid:3)
dx (cid:5)u2(cid:5)1 dx (cid:5)u2(cid:2)1 dx 1(cid:2)u2
d u(cid:4) d (cid:2)u(cid:4) d (cid:2)u(cid:4)
34. dx(cid:2)coth(cid:2)1 u(cid:3)(cid:3) 1(cid:2)u2 35. dx(cid:2)sech(cid:2)1 u(cid:3)(cid:3) u(cid:5)1(cid:2)u2 36. dx(cid:2)csch(cid:2)1 u(cid:3)(cid:3) (cid:4)u(cid:4)(cid:5)1(cid:5)u2
Basic Integration Formulas
(cid:10) (cid:10) (cid:10) (cid:10) (cid:10)
1. kf(cid:6)u(cid:7) du(cid:3)k f(cid:6)u(cid:7) du 2. (cid:2)f(cid:6)u(cid:7) ± g(cid:6)u(cid:7)(cid:3) du(cid:3) f(cid:6)u(cid:7) du ± g(cid:6)u(cid:7) du
(cid:10) (cid:10)
un(cid:5)1
3. du(cid:3)u(cid:5)C 4. un du(cid:3) (cid:5)C, n(cid:6)(cid:2)1
(cid:10) (cid:10) n(cid:5)1
du (cid:4) (cid:4)
5. (cid:3)ln u (cid:5)C 6. eu du(cid:3)eu(cid:5)C
(cid:10)u (cid:10)
(cid:11) 1 (cid:12)
7. au du(cid:3) au(cid:5)C 8. sin u du (cid:3)(cid:2)cos u (cid:5)C
(cid:10) ln a (cid:10)
(cid:4) (cid:4)
9. cos u du (cid:3)sin u (cid:5)C 10. tan u du (cid:3)(cid:2)ln cos u (cid:5) C
(cid:10) (cid:10)
(cid:4) (cid:4) (cid:4) (cid:4)
11. cot u du (cid:3)ln sin u (cid:5)C 12. sec u du (cid:3)ln sec u (cid:5)tan u (cid:5) C
g (cid:10) (cid:10)
n
ni (cid:4) (cid:4)
ar 13. csc u du (cid:3)(cid:2)ln csc u (cid:5)cot u (cid:5)C 14. sec2 u du(cid:3)tan u (cid:5)C
e
L (cid:10) (cid:10)
e
g
ga 15. csc2 u du(cid:3)(cid:2)cot u (cid:5)C 16. sec u tan u du(cid:3)sec u (cid:5)C
en (cid:10) (cid:10)
C
Cole, 17. (cid:10)csc u cot u du(cid:3)(cid:2)csc u (cid:5)C 18. (cid:10)(cid:5)a2du(cid:2)u2 (cid:3)arcsin au (cid:5) C
/ (cid:4) (cid:4)
ks du 1 u du 1 u
o 19. (cid:3) arctan (cid:5)C 20. (cid:3) arcsec (cid:5) C
ro a2(cid:5)u2 a a u(cid:5)u2(cid:2)a2 a a
B
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TRIGONOMETRY
Definition of the Six Trigonometric Functions
Right triangle definitions,where 0 < (cid:7)< (cid:8)(cid:13)2. y
HAθy pdojtaecneunste Opposite c t saoinns (cid:7)(cid:7)(cid:7)(cid:3)(cid:3)(cid:3) hhooayyppdppppj scc esocct (cid:7)(cid:7)(cid:7)(cid:3)(cid:3)(cid:3) ohhaapyyddpppjj (−(−232,2,12)(−2212),56π23341π)51023°3π5°120°π2 9(00,° 610)°4π35°3(0π124°,π62(3)22(,2322, )21)
adj opp
0° 0
Circular function definitions,where (cid:7)is any angle. (−1, 0) π 180° 360° 2π (1, 0) x
y y r
(x, y) r = x2 + y2 sin (cid:7)(cid:3) xr csc (cid:7)(cid:3) yr (− 23, −12) 76π52π1202°5°240° 300°31353°70π°116π ( 23, −21)
y xr θ x c taons (cid:7)(cid:7)(cid:3)(cid:3) yr sc eoct (cid:7)(cid:7)(cid:3)(cid:3) xx (− 22,−(−212,)− 34) 43π 270° 3(20π, −1)53π(41, −( 322), − 22)
x y 2 2 2 2
Reciprocal Identities Double-Angle Formulas
1 1 1 sin 2u (cid:3)2 sin u cos u
sin x(cid:3) sec x(cid:3) tan x(cid:3)
csc x cos x cot x cos 2u (cid:3)cos2 u(cid:2)sin2 u(cid:3)2 cos2 u(cid:2) 1(cid:3) 1(cid:2) 2 sin2 u
1 1 1 2 tan u
csc x(cid:3) cos x(cid:3) cot x (cid:3) tan 2u (cid:3)
sin x sec x tan x 1(cid:2)tan2 u
Quotient Identities Power-Reducing Formulas
tan x (cid:3) sin x cot x (cid:3) cos x sin2 u(cid:3) 1(cid:2)cos 2u
cos x sin x 2
1(cid:5)cos 2u
Pythagorean Identities cos2 u(cid:3)
2
sin2 x(cid:5) cos2 x(cid:3) 1 1(cid:2)cos 2u
tan2 u(cid:3)
1(cid:5)tan2 x(cid:3) sec2 x 1(cid:5)cot2 x(cid:3)csc2 x 1(cid:5)cos 2u
Cofunction Identities Sum-to-Product Formulas
(cid:11)(cid:8) (cid:12) (cid:11)(cid:8) (cid:12) (cid:11)u(cid:5)v(cid:12) (cid:11)u(cid:2)v(cid:12)
sin (cid:2) x (cid:3) cos x cos (cid:2)x (cid:3)sin x sin u (cid:5)sin v (cid:3)2 sin cos
2 2 2 2
(cid:11)(cid:8) (cid:12) (cid:11)(cid:8) (cid:12) (cid:11)u(cid:5)v(cid:12) (cid:11)u(cid:2)v(cid:12)
csc (cid:2)x (cid:3) sec x tan (cid:2)x (cid:3)cot x sin u (cid:2)sin v (cid:3)2 cos sin
2 2 2 2
sec(cid:11)(cid:8)(cid:2)x(cid:12) (cid:3) csc x cot(cid:11)(cid:8)(cid:2)x(cid:12) (cid:3)tan x cos u (cid:5)cos v (cid:3)2 cos(cid:11)u(cid:5)2 v(cid:12) cos(cid:11)u(cid:2)2 v(cid:12)
2 2
(cid:11)u(cid:5)v(cid:12) (cid:11)u(cid:2)v(cid:12)
Even/Odd Identities cos u (cid:2)cos v (cid:3)(cid:2)2 sin sin
2 2
sin(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)sin x cos(cid:6)(cid:2)x(cid:7)(cid:3)cos x Product-to-Sum Formulas ning
csc(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)csc x tan(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)tan x ar
1 e
sec(cid:6)(cid:2)x(cid:7)(cid:3) sec x cot(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)cot x sin u sin v(cid:3) 2(cid:2)cos(cid:6)u(cid:2)v(cid:7)(cid:2)cos(cid:6)u(cid:5)v(cid:7)(cid:3) ge L
a
Sum and Difference Formulas cos u cos v(cid:3) 1(cid:2)cos(cid:6)u(cid:2)v(cid:7)(cid:5)cos(cid:6)u(cid:5) v(cid:7)(cid:3) eng
sin(cid:6)u ± v(cid:7)(cid:3) sin u cos v ± cos u sin v 2 e, C
cos(cid:6)u ± v(cid:7)(cid:3) cos u cos v (cid:9) sin u sin v sin u cos v(cid:3) 1(cid:2)sin(cid:6)u(cid:5)v(cid:7)(cid:5)sin(cid:6)u(cid:2)v(cid:7)(cid:3) Col
2 /
tan u ± tan v ks
tan(cid:6)u ± v(cid:7)(cid:3) 1 o
1 (cid:9) tan u tan v cos u sin v(cid:3) (cid:2)sin(cid:6)u(cid:5)v(cid:7)(cid:2)sin(cid:6)u(cid:2)v(cid:7)(cid:3) ro
2 B
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Multivariable
Calculus
10e
Ron Larson
The Pennsylvania State University
The Behrend College
Bruce Edwards
University of Florida
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
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Multivariable Calculus © 2014, 2010, 2006 Brooks/Cole, Cengage Learning
Tenth Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright
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Contents
11
Vectors and the Geometry of Space 747
11.1 Vectors in the Plane 748
11.2 Space Coordinates and Vectors in Space 758
11.3 The Dot Product ofTwo Vectors 766
11.4 The Cross Product ofTwo Vectors in Space 775
11.5 Lines and Planes in Space 783
Section Project:Distances in Space 793
11.6 Surfaces in Space 794
11.7 Cylindrical and Spherical Coordinates 804
Review Exercises 811
P.S.Problem Solving 813
12
Vector-Valued Functions 815
12.1 Vector-Valued Functions 816
Section Project: Witch of Agnesi 823
12.2 Differentiation and Integration of Vector-Valued
Functions 824
12.3 Velocity and Acceleration 832
12.4 Tangent Vectors and Normal Vectors 841
12.5 Arc Length and Curvature 851
Review Exercises 863
P.S.Problem Solving 865
13
Functions of Several Variables 867
13.1 Introduction to Functions of Several Variables 868
13.2 Limits and Continuity 880
13.3 Partial Derivatives 890
Section Project: Moiré Fringes 899
13.4 Differentials 900
13.5 Chain Rules for Functions of Several Variables 907
13.6 Directional Derivatives and Gradients 915
13.7 Tangent Planes and Normal Lines 927
Section Project: Wildflowers 935
13.8 Extrema of Functions of Two Variables 936
13.9 Applications of Extrema 944
Section Project: Building a Pipeline 951
13.10 Lagrange Multipliers 952
Review Exercises 960
P.S.Problem Solving 963
iii
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iv Contents
14 Multiple Integration 965
14.1 Iterated Integrals and Area in the Plane 966
14.2 Double Integrals and Volume 974
14.3 Change of Variables: Polar Coordinates 986
14.4 Center of Mass and Moments of Inertia 994
Section Project: Center of Pressure on a Sail 1001
14.5 Surface Area 1002
Section Project: Capillary Action 1008
14.6 Triple Integrals and Applications 1009
14.7 Triple Integrals in Other Coordinates 1020
Section Project: Wrinkled and Bumpy Spheres 1026
14.8 Change of Variables: Jacobians 1027
Review Exercises 1034
P.S.Problem Solving 1037
15 Vector Analysis 1039
15.1 Vector Fields 1040
15.2 Line Integrals 1051
15.3 Conservative Vector Fields and Independence
of Path 1065
15.4 Green’s Theorem 1075
Section Project: Hyperbolic and Trigonometric
Functions 1083
15.5 Parametric Surfaces 1084
15.6 Surface Integrals 1094
Section Project: Hyperboloid of One Sheet 1105
15.7 Divergence Theorem 1106
15.8 Stokes’s Theorem 1114
Review Exercises 1120
Section Project: The Planimeter 1122
P.S.Problem Solving 1123
16
Additional Topics in Differential Equations 1125
16.1 Exact First-Order Equations 1126
16.2 Second-Order Homogeneous Linear Equations 1133
16.3 Second-Order Nonhomogeneous Linear Equations 1141
Section Project: Parachute Jump 1148
16.4 Series Solutions of Differential Equations 1149
Review Exercises 1153
P.S.Problem Solving 1155
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