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This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/9/10 4:30 PM Page i MULTIVARIABLE CA L C U L U S SEVENTH EDITION JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/9/10 4:30 PM Page ii Multivariable Calculus, Seventh Edition James Stewart Executive Editor: Liz Covello Assistant Editor: Liza Neustaetter Editorial 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1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be e-mailed to permissionrequest@cengage.com Library of Congress Control Number: 2010936601 Text Designer: TECH· arts Photo Researcher: Terri Wright, www.terriwright.com Copy Editor: Kathi Townes ISBN-13: 978-0-538-49787-9 ISBN-10: 0-538-49787-4 Cover Designer: Irene Morris Cover Illustration: Irene Morris Compositor: Stephanie Kuhns, TECH· arts Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with oﬃce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local oﬃce at www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd To learn more about Brooks/Cole, visit www.cengage.com/brookscole Trademarks ExamView ® and ExamViewPro ® are 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right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/9/10 4:30 PM Page iii Contents Preface 10 vii Parametric Equations and Polar Coordinates 659 10.1 Curves Defined by Parametric Equations Laboratory Project 10.2 N Polar Coordinates Laboratory Project Bézier Curves 677 678 N Families of Polar Curves Areas and Lengths in Polar Coordinates 10.5 Conic Sections 10.6 Conic Sections in Polar Coordinates Problems Plus 668 669 10.4 Review 11 Running Circles around Circles Calculus with Parametric Curves Laboratory Project 10.3 N 660 688 689 694 702 709 712 Infinite Sequences and Series 713 11.1 Sequences 714 Laboratory Project N Logistic Sequences 727 11.2 Series 727 11.3 The Integral Test and Estimates of Sums 11.4 The Comparison Tests 11.5 Alternating Series 11.6 Absolute Convergence and the Ratio and Root Tests 11.7 Strategy for Testing Series 11.8 Power Series 11.9 Representations of Functions as Power Series 738 746 751 756 763 765 770 iii Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/9/10 4:30 PM Page iv iv CONTENTS 11.10 Taylor and Maclaurin Series Laboratory Project Writing Project 11.11 Review Problems Plus 791 How Newton Discovered the Binomial Series N 791 792 Radiation from the Stars 801 802 805 Vectors and the Geometry of Space 809 12.1 Three-Dimensional Coordinate Systems 12.2 Vectors 12.3 The Dot Product 12.4 The Cross Product 12.5 824 832 The Geometry of a Tetrahedron N Equations of Lines and Planes Laboratory Project 12.6 N Problems Plus 840 840 Putting 3D in Perspective Cylinders and Quadric Surfaces Review 810 815 Discovery Project 13 An Elusive Limit Applications of Taylor Polynomials Applied Project 12 N N 777 850 851 858 861 Vector Functions 863 13.1 Vector Functions and Space Curves 13.2 Derivatives and Integrals of Vector Functions 13.3 Arc Length and Curvature 13.4 Motion in Space: Velocity and Acceleration Applied Project Review Problems Plus N 864 871 877 Kepler’s Laws 886 896 897 900 Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/9/10 4:30 PM Page v CONTENTS 14 Partial Derivatives 901 14.1 Functions of Several Variables 14.2 Limits and Continuity 14.3 Partial Derivatives 14.4 Tangent Planes and Linear Approximations 14.5 The Chain Rule 14.6 Directional Derivatives and the Gradient Vector 14.7 Maximum and Minimum Values N Discovery Project 14.8 902 916 924 N 980 Quadratic Approximations and Critical Points 980 981 Applied Project N Rocket Science Applied Project N Hydro-Turbine Optimization Problems Plus 957 970 Designing a Dumpster Lagrange Multipliers Review 939 948 Applied Project 15 v 988 990 991 995 Multiple Integrals 997 15.1 Double Integrals over Rectangles 15.2 Iterated Integrals 15.3 Double Integrals over General Regions 15.4 Double Integrals in Polar Coordinates 15.5 Applications of Double Integrals 15.6 Surface Area 15.7 Triple Integrals 1006 1027 N Volumes of Hyperspheres 1051 Triple Integrals in Cylindrical Coordinates 1051 N The Intersection of Three Cylinders Triple Integrals in Spherical Coordinates Applied Project 15.10 1021 1041 Discovery Project 15.9 1012 1037 Discovery Project 15.8 998 N Roller Derby Problems Plus 1057 1063 Change of Variables in Multiple Integrals Review 1056 1064 1073 1077 Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/11/10 10:31 AM Page vi vi CONTENTS 16 Vector Calculus 1079 16.1 Vector Fields 1080 16.2 Line Integrals 1087 16.3 The Fundamental Theorem for Line Integrals 16.4 Green’s Theorem 16.5 Curl and Divergence 16.6 Parametric Surfaces and Their Areas 16.7 Surface Integrals 1134 16.8 Stokes’ Theorem 1146 Writing Project 1108 1115 The Divergence Theorem 16.10 Summary Problems Plus 1123 Three Men and Two Theorems 16.9 Review 17 N 1099 1152 1152 1159 1160 1163 Second-Order Differential Equations 1165 17.1 Second-Order Linear Equations 17.2 Nonhomogeneous Linear Equations 17.3 Applications of Second-Order Differential Equations 17.4 Series Solutions Review 1166 1172 1180 1188 1193 Appendixes A1 F Proofs of Theorems A2 G Complex Numbers H Answers to Odd-Numbered Exercises A5 A13 Index A43 Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/9/10 4:30 PM Page vii Preface A great discovery solves a great problem but there is a grain of discovery in the solution of any problem Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery GEORGE POLYA The art of teaching, Mark Van Doren said, is the art of assisting discovery I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty In this edition, as in the first six editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject Newton undoubtedly experienced a sense of triumph when he made his great discoveries I want students to share some of that excitement The emphasis is on understanding concepts I think that nearly everybody agrees that this should be the primary goal of calculus instruction In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation: Focus on conceptual understanding I have tried to implement this goal through the Rule of Three: “Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways The Rule of Three has been expanded to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as well In writing the seventh edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus The book contains elements of reform, but within the context of a traditional curriculum Alternative Versions I have written several other calculus textbooks that might be preferable for some instructors Most of them also come in single variable and multivariable versions ■ Calculus, Seventh Edition, Hybrid Version, is similar to the present textbook in content and coverage except that all end-of-section exercises are available only in Enhanced WebAssign The printed text includes all end-of-chapter review material ■ Calculus: Early Transcendentals, Seventh Edition, is similar to the present textbook except that the exponential, logarithmic, and inverse trigonometric functions are covered in the first semester vii Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97879_FM7eMV_FM7eMV_pi-xiv.qk_97879_FM7eMV_FM7eMV_pi-xiv 11/11/10 10:33 AM Page viii viii PREFACE ■ Calculus: Early Transcendentals, Seventh Edition, Hybrid Version, is similar to Calculus: Early Transcendentals, Seventh Edition, in content and coverage except that all end-of-section exercises are available only in Enhanced WebAssign The printed text includes all end-of-chapter review material ■ Essential Calculus is a much briefer book (800 pages), though it contains almost all of the topics in Calculus, Seventh Edition The relative brevity is achieved through briefer exposition of some topics and putting some features on the website ■ Essential Calculus: Early Transcendentals resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter ■ Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understanding even more strongly than this book The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters ■ Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book It is suitable for students taking Engineering and Physics courses concurrently with calculus ■ Brief Applied Calculus is intended for students in business, the social sciences, and the life sciences What’s New in the Seventh Edition? The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers Here are some of the many improvements that I’ve incorporated into this edition: ■ Some material has been rewritten for greater clarity or for better motivation See, for instance, the introduction to series on page 727 and the motivation for the cross product on page 832 ■ New examples have been added (see Example on page 1045 for instance), and the solutions to some of the existing examples have been amplified ■ The art program has been revamped: New figures have been incorporated and a substantial percentage of the existing figures have been redrawn ■ The data in examples and exercises have been updated to be more timely ■ One new project has been added: Families of Polar Curves (page 688) exhibits the fascinating shapes of polar curves and how they evolve within a family ■ The section on the surface area of the graph of a function of two variables has been restored as Section 15.6 for the convenience of instructors who like to teach it after double integrals, though the full treatment of surface area remains in Chapter 16 ■ I continue to seek out examples of how calculus applies to so many aspects of the real world On page 933 you will see beautiful images of the earth’s magnetic field strength and its second vertical derivative as calculated from Laplace’s equation I thank Roger Watson for bringing to my attention how this is used in geophysics and mineral exploration ■ More than 25% of the exercises are new Here are some of my favorites: 11.2.49–50, 11.10.71–72, 12.1.44, 12.4.43–44, 12.5.80, 14.6.59–60, 15.8.42, and Problems 4, 5, and on pages 861–62 Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... current editions, and alternate formats, please visit www .cengage. com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2010 Cengage Learning. .. Locate your local oﬃce at www .cengage. com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd To learn more about Brooks/Cole, visit www .cengage. com/brookscole Trademarks... discovering calculus both for its practical power and its surprising beauty In this edition, as in the first six editions, I aim to convey to the student a sense of the utility of calculus and
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Xem thêm: Cengage learning multivariable calculus 7th edition 0538497874 , Cengage learning multivariable calculus 7th edition 0538497874 , 1: Curves Defined by Parametric Equations, 2: Calculus with Parametric Curves, 4: Areas and Lengths in Polar Coordinates, 6: Conic Sections in Polar Coordinates, 3: The Integral Test and Estimates of Sums, 6: Absolute Convergence and the Ratio and Root Tests, 7: Strategy for Testing Series, 9: Representations of Functions as Power Series, 10: Taylor and Maclaurin Series, 11: Applications of Taylor Polynomials, 5: Equations of Lines and Planes, 6: Cylinders and Quadric Surfaces, 1: Vector Functions and Space Curves, 2: Derivatives and Integrals of Vector Functions, 3: Arc Length and Curvature, 4: Motion in Space: Velocity and Acceleration, 1: Functions of Several Variables, 4: Tangent Planes and Linear Approximations, 6: Directional Derivatives and the Gradient Vector, 7: Maximum and Minimum Values, 1: Double Integrals over Rectangles, 3: Double Integrals over General Regions, 4: Double Integrals in Polar Coordinates, 5: Applications of Double Integrals, 8: Triple Integrals in Cylindrical Coordinates, 9: Triple Integrals in Spherical Coordinates, 10: Change of Variables in Multiple Integrals, 3: The Fundamental Theorem for Line Integrals, 6: Parametric Surfaces and Their Areas, 3: Applications of Second-Order Differential Equations