Multivariable Calculus 7th Edition By James Stewart PDF

www.EngineeringEBooksPdf.com  This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial...

www.EngineeringEBooksPdf.com  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). www.EngineeringEBooksPdf.com 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). www.EngineeringEBooksPdf.com 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 © 2012, 2008 Brooks/Cole, Cengage Learning James Stewart ALL RIGHTS RESERVED. No part of this work covered by the copy- right herein may be reproduced, transmitted, stored, or used in any Executive Editor: Liz Covello form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, Assistant Editor: Liza Neustaetter taping, Web distribution, information networks, or information stor- Editorial Assistant: Jennifer Staller age and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior writ- Media Editor : Maureen Ross ten permission of the publisher. Marketing Manager: Jennifer Jones Marketing Coordinator: Michael Ledesma For product information and technology assistance, contact us at Marketing Communications Manager: Mary Anne Payumo Cengage Learning Customer & Sales Support, 1-800-354-9706. 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Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Brooks/Cole, visit www.cengage.com/brookscole. Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com. Trademarks ExamView ® and ExamViewPro ® are registered trademarks of FSCreations, Inc. Windows is a registered trademark of the Microsoft Corporation and used herein under license. Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc. Used herein under license. Derive is a registered trademark of Soft Warehouse, Inc. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. K10T10 Tools for Enriching is a trademark used herein under license. Printed in the United States of America 1 2 3 4 5 6 7 1 4 1 3 1 2 11 1 0 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). www.EngineeringEBooksPdf.com 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 iii Contents Preface vii 10 Parametric Equations and Polar Coordinates 659 10.1 Curves Defined by Parametric Equations 660 Laboratory Project N Running Circles around Circles 668 10.2 Calculus with Parametric Curves 669 Laboratory Project N Bézier Curves 677 10.3 Polar Coordinates 678 Laboratory Project N Families of Polar Curves 688 10.4 Areas and Lengths in Polar Coordinates 689 10.5 Conic Sections 694 10.6 Conic Sections in Polar Coordinates 702 Review 709 Problems Plus 712 11 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 738 11.4 The Comparison Tests 746 11.5 Alternating Series 751 11.6 Absolute Convergence and the Ratio and Root Tests 756 11.7 Strategy for Testing Series 763 11.8 Power Series 765 11.9 Representations of Functions as Power Series 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). www.EngineeringEBooksPdf.com 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 777 Laboratory Project N An Elusive Limit 791 Writing Project N How Newton Discovered the Binomial Series 791 11.11 Applications of Taylor Polynomials 792 Applied Project N Radiation from the Stars 801 Review 802 Problems Plus 805 12 Vectors and the Geometry of Space 809 12.1 Three-Dimensional Coordinate Systems 810 12.2 Vectors 815 12.3 The Dot Product 824 12.4 The Cross Product 832 Discovery Project N The Geometry of a Tetrahedron 840 12.5 Equations of Lines and Planes 840 Laboratory Project N Putting 3D in Perspective 850 12.6 Cylinders and Quadric Surfaces 851 Review 858 Problems Plus 861 13 Vector Functions 863 13.1 Vector Functions and Space Curves 864 13.2 Derivatives and Integrals of Vector Functions 871 13.3 Arc Length and Curvature 877 13.4 Motion in Space: Velocity and Acceleration 886 Applied Project N Kepler’s Laws 896 Review 897 Problems Plus 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). www.EngineeringEBooksPdf.com 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 v 14 Partial Derivatives 901 14.1 Functions of Several Variables 902 14.2 Limits and Continuity 916 14.3 Partial Derivatives 924 14.4 Tangent Planes and Linear Approximations 939 14.5 The Chain Rule 948 14.6 Directional Derivatives and the Gradient Vector 957 14.7 Maximum and Minimum Values 970 Applied Project N Designing a Dumpster 980 Discovery Project N Quadratic Approximations and Critical Points 980 14.8 Lagrange Multipliers 981 Applied Project N Rocket Science 988 Applied Project N Hydro-Turbine Optimization 990 Review 991 Problems Plus 995 15 Multiple Integrals 997 15.1 Double Integrals over Rectangles 998 15.2 Iterated Integrals 1006 15.3 Double Integrals over General Regions 1012 15.4 Double Integrals in Polar Coordinates 1021 15.5 Applications of Double Integrals 1027 15.6 Surface Area 1037 15.7 Triple Integrals 1041 Discovery Project N Volumes of Hyperspheres 1051 15.8 Triple Integrals in Cylindrical Coordinates 1051 Discovery Project N The Intersection of Three Cylinders 1056 15.9 Triple Integrals in Spherical Coordinates 1057 Applied Project N Roller Derby 1063 15.10 Change of Variables in Multiple Integrals 1064 Review 1073 Problems Plus 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). www.EngineeringEBooksPdf.com 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 1099 16.4 Green’s Theorem 1108 16.5 Curl and Divergence 1115 16.6 Parametric Surfaces and Their Areas 1123 16.7 Surface Integrals 1134 16.8 Stokes’ Theorem 1146 Writing Project N Three Men and Two Theorems 1152 16.9 The Divergence Theorem 1152 16.10 Summary 1159 Review 1160 Problems Plus 1163 17 Second-Order Differential Equations 1165 17.1 Second-Order Linear Equations 1166 17.2 Nonhomogeneous Linear Equations 1172 17.3 Applications of Second-Order Differential Equations 1180 17.4 Series Solutions 1188 Review 1193 Appendixes A1 F Proofs of Theorems A2 G Complex Numbers A5 H Answers to Odd-Numbered Exercises 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). www.EngineeringEBooksPdf.com 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 stu- dent 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 exper- imentation, and other approaches have changed how we teach conceptual reasoning in fun- damental 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 con- ceptual 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 instruc- tors. 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 cov- ered 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). www.EngineeringEBooksPdf.com 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 Cal- culus: 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 3. Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understand- ing 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 prod- uct on page 832. New examples have been added (see Example 4 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 sub- stantial 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 8 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). www.EngineeringEBooksPdf.com 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 ix PREFACE ix Technology Enhancements The media and technology to support the text have been enhanced to give professors greater control over their course, to provide extra help to deal with the varying levels of student preparedness for the calculus course, and to improve support for conceptual understanding. New Enhanced WebAssign features including a customizable Cengage YouBook, Just in Time review, Show Your Work, Answer Evaluator, Personalized Study Plan, Master Its, solution videos, lecture video clips (with associated questions), and Visualizing Calculus (TEC animations with associated questions) have been developed to facilitate improved student learning and flexible classroom teaching. Tools for Enriching Calculus (TEC) has been completely redesigned and is accessible in Enhanced WebAssign, CourseMate, and PowerLecture. Selected Visuals and Modules are available at www.stewartcalculus.com. Features CONCEPTUAL EXERCISES The most important way to foster conceptual understanding is through the problems that we assign. To that end I have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first few exercises in Sections 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a True-False Quiz. Other exercises test concep- tual understanding through graphs or tables (see Exercises 10.1.24–27, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–42, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.5–10, 16.1.11–18, 16.2.17–18, and 16.3.1–2). Another type of exercise uses verbal description to test conceptual understanding. I par- ticularly value problems that combine and compare graphical, numerical, and algebraic approaches. GRADED EXERCISE SETS Each exercise set is carefully graded, progressing from basic conceptual exercises and skill- development problems to more challenging problems involving applications and proofs. REAL-WORLD DATA My assistants and I spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting real-world data to intro- duce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. Functions of two variables are illustrated by a table of values of the wind-chill index as a function of air temperature and wind speed (Example 2 in Section 14.1). Partial derivatives are intro- duced in Section 14.3 by examining a column in a table of values of the heat index (per- ceived air temperature) as a function of the actual temperature and the relative humidity. This example is pursued further in connection with linear approximations (Example 3 in Section 14.4). Directional derivatives are introduced in Section 14.6 by using a tempera- ture contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas. Double integrals are used to estimate the average snowfall in Colorado on December 20–21, 2006 (Example 4 in Section 15.1). Vector fields are introduced in Sec- tion 16.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns. PROJECTS One way of involving students and making them active learners is to have them work (per- haps in groups) on extended projects that give a feeling of substantial accomplishment when completed. I have included four kinds of projects: Applied Projects involve applica- tions that are designed to appeal to the imagination of students. The project after Section 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). www.EngineeringEBooksPdf.com 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 x x PREFACE 14.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired velocity. Labo- ratory Projects involve technology; the one following Section 10.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer. Discovery Projects explore aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.7), and intersections of three cylinders (after Section 15.8). The Writing Project after Section 17.8 explores the historical and physical origins of Green’s Theorem and Stokes’ Theorem and the interactions of the three men involved. Many additional projects can be found in the Instructor’s Guide. TOOLS FOR TEC is a companion to the text and is intended to enrich and complement its contents. (It ENRICHING™ CALCULUS is now accessible in Enhanced WebAssign, CourseMate, and PowerLecture. Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC modules that provide a laboratory environment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises. Instruc- tors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating addi- tional exercises, labs, and projects that make use of the Visuals and Modules. HOMEWORK HINTS Homework Hints presented in the form of questions try to imitate an effective teaching assistant by functioning as a silent tutor. Hints for representative exercises (usually odd- numbered) are included in every section of the text, indicated by printing the exercise number in red. They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress, and are available to students at stewartcalculus.com and in CourseMate and Enhanced WebAssign. ENHANCED W E B A S S I G N Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. With the seventh edition we have been working with the calculus community and WebAssign to develop a more robust online homework sys- tem. Up to 70% of the exercises in each section are assignable as online homework, includ- ing free response, multiple choice, and multi-part formats. The system also includes Active Examples, in which students are guided in step-by-step tutorials through text examples, with links to the textbook and to video solutions. New enhancements to the system include a customizable eBook, a Show Your Work feature, Just in Time review of precalculus prerequisites, an improved Assignment Editor, and an Answer Evaluator that accepts more mathematically equivalent answers and allows for homework grading in much the same way that an instructor grades. www.stewartcalculus.com This site includes the following. Homework Hints Algebra Review Lies My Calculator and Computer Told Me History of Mathematics, with links to the better historical websites Additional Topics (complete with exercise sets): Fourier Series, Formulas for the Remainder Term in Taylor Series, Rotation of Axes Archived Problems (Drill exercises that appeared in previous editions, together with their solutions) 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). www.EngineeringEBooksPdf.com 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 xi PREFACE xi Challenge Problems (some from the Problems Plus sections from prior editions) Links, for particular topics, to outside web resources Selected Tools for Enriching Calculus (TEC) Modules and Visuals Content 10 Parametric Equations This chapter introduces parametric and polar curves and applies the methods of calculus and Polar Coordinates to them. Parametric curves are well suited to laboratory projects; the three presented here involve families of curves and Bézier curves. A brief treatment of conic sections in polar coordinates prepares the way for Kepler’s Laws in Chapter 13. 11 Infinite Sequences and Series The convergence tests have intuitive justifications (see page 738) as well as formal proofs. Numerical estimates of sums of series are based on which test was used to prove conver- gence. The emphasis is on Taylor series and polynomials and their applications to physics. Error estimates include those from graphing devices. 12 Vectors and The material on three-dimensional analytic geometry and vectors is divided into two chap- The Geometry of Space ters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces. 13 Vector Functions This chapter covers vector-valued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culminating in Kepler’s laws. 14 Partial Derivatives Functions of two or more variables are studied from verbal, numerical, visual, and alge- braic points of view. In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. 15 Multiple Integrals Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions. Double and triple integrals are used to compute probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of three cylinders. Cylindrical and spherical coordinates are introduced in the context of eval- uating triple integrals. 16 Vector Calculus Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized. 17 Second-Order Since first-order differential equations are covered in Chapter 9, this final chapter deals Differential Equations with second-order linear differential equations, their application to vibrating springs and electric circuits, and series solutions. Ancillaries Multivariable Calculus, Seventh Edition, is supported by a complete set of ancillaries developed under my direction. Each piece has been designed to enhance student under- standing and to facilitate creative instruction. With this edition, new media and technolo- gies have been developed that help students to visualize calculus and instructors to customize content to better align with the way they teach their course. The tables on pages xiii–xiv describe each of these ancillaries. 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). www.EngineeringEBooksPdf.com 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 xii xii PREFACE 0 Acknowledgments The preparation of this and previous editions has involved much time spent reading the reasoned (but sometimes contradictory) advice from a large number of astute reviewers. I greatly appreciate the time they spent to understand my motivation for the approach taken. I have learned something from each of them. SEVENTH EDITION REVIEWERS Amy Austin, Texas A&M University Richard Millspaugh, University of North Dakota Anthony J. Bevelacqua, University of North Dakota Lon H. Mitchell, Virginia Commonwealth University Zhen-Qing Chen, University of Washington—Seattle Ho Kuen Ng, San Jose State University Jenna Carpenter, Louisiana Tech University Norma Ortiz-Robinson, Virginia Commonwealth University Le Baron O. Ferguson, University of California—Riverside Qin Sheng, Baylor University Shari Harris, John Wood Community College Magdalena Toda, Texas Tech University Amer Iqbal, University of Washington—Seattle Ruth Trygstad, Salt Lake Community College Akhtar Khan, Rochester Institute of Technology Klaus Volpert, Villanova University Marianne Korten, Kansas State University Peiyong Wang, Wayne State University Joyce Longman, Villanova University In addition, I would like to thank Jordan Bell, George Bergman, Leon Gerber, Mary Pugh, and Simon Smith for their suggestions; Al Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; George Bergman, David Bleecker, Dan Clegg, Victor Kaftal, Anthony Lam, Jamie Law- son, Ira Rosenholtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exercises; Dan Drucker for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, Philip Straffin, and Klaus Volpert for ideas for projects; Dan Ander- son, Dan Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises and suggesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in proofreading; and Jeff Cole and Dan Clegg for their careful preparation and proofreading of the answer manuscript. In addition, I thank those who have contributed to past editions: Ed Barbeau, Fred Brauer, Andy Bulman-Fleming, Bob Burton, David Cusick, Tom DiCiccio, Garret Etgen, Chris Fisher, Stuart Goldenberg, Arnold Good, Gene Hecht, Harvey Keynes, E.L. Koh, Zdislav Kovarik, Kevin Kreider, Emile LeBlanc, David Leep, Gerald Leibowitz, Larry Peterson, Lothar Redlin, Carl Riehm, John Ringland, Peter Rosenthal, Doug Shaw, Dan Silver, Norton Starr, Saleem Watson, Alan Weinstein, and Gail Wolkowicz. I also thank Kathi Townes and Stephanie Kuhns of TECHarts for their production serv- ices and the following Brooks/Cole staff: Cheryll Linthicum, content project manager; Liza Neustaetter, assistant editor; Maureen Ross, media editor; Sam Subity, managing media editor; Jennifer Jones, marketing manager; and Vernon Boes, art director. They have all done an outstanding job. I have been very fortunate to have worked with some of the best mathematics editors in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, Richard Stratton, and now Liz Covello. All of them have contributed greatly to the success of this book. JAMES STEWART 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). www.EngineeringEBooksPdf.com 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 xiii Ancillaries for Instructors Ancillaries for Instructors and Students PowerLecture Stewart Website ISBN 0-8400-5414-9 www.stewartcalculus.com This comprehensive DVD contains all art from the text in both Contents: Homework Hints Algebra Review Additional jpeg and PowerPoint formats, key equations and tables from the Topics Drill exercises Challenge Problems Web Links text, complete pre-built PowerPoint lectures, an electronic ver- History of Mathematics Tools for Enriching Calculus (TEC) sion of the Instructor’s Guide, Solution Builder, ExamView test- ing software, Tools for Enriching Calculus, video instruction, TEC Tools for Enriching™ Calculus and JoinIn on TurningPoint clicker content. By James Stewart, Harvey Keynes, Dan Clegg, and developer Hu Hohn Instructor’s Guide Tools for Enriching Calculus (TEC) functions as both a power- by Douglas Shaw ful tool for instructors, as well as a tutorial environment in ISBN 0-8400-5407-6 which students can explore and review selected topics. The Each section of the text is discussed from several viewpoints. Flash simulation modules in TEC include instructions, writ- The Instructor’s Guide contains suggested time to allot, points ten and audio explanations of the concepts, and exercises. to stress, text discussion topics, core materials for lecture, work- TEC is accessible in CourseMate, WebAssign, and Power- shop/discussion suggestions, group work exercises in a form Lecture. Selected Visuals and Modules are available at suitable for handout, and suggested homework assignments. An www.stewartcalculus.com. electronic version of the Instructor’s Guide is available on the PowerLecture DVD. Enhanced WebAssign Complete Solutions Manual www.webassign.net WebAssign’s homework delivery system lets instructors deliver, Multivariable collect, grade, and record assignments via the web. Enhanced By Dan Clegg and Barbara Frank WebAssign for Stewart’s Calculus now includes opportunities ISBN 0-8400-4947-1 for students to review prerequisite skills and content both at the Includes worked-out solutions to all exercises in the text. start of the course and at the beginning of each section. In addi- tion, for selected problems, students can get extra help in the Solution Builder form of “enhanced feedback” (rejoinders) and video solutions. www.cengage.com /solutionbuilder Other key features include: thousands of problems from Stew- This online instructor database offers complete worked out solu- art’s Calculus, a customizable Cengage YouBook, Personal tions to all exercises in the text. Solution Builder allows you to Study Plans, Show Your Work, Just in Time Review, Answer create customized, secure solutions printouts (in PDF format) Evaluator, Visualizing Calculus animations and modules, matched exactly to the problems you assign in class. quizzes, lecture videos (with associated questions), and more! Printed Test Bank Cengage Customizable YouBook By William Steven Harmon YouBook is a Flash-based eBook that is interactive and cus- ISBN 0-8400-5408-4 tomizable! Containing all the content from Stewart’s Calculus, Contains text-specific multiple-choice and free response test YouBook features a text edit tool that allows instructors to mod- items. ify the textbook narrative as needed. With YouBook, instructors can quickly re-order entire sections and chapters or hide any ExamView Testing content they don’t teach to create an eBook that perfectly Create, deliver, and customize tests in print and online formats matches their syllabus. Instructors can further customize the with ExamView, an easy-to-use assessment and tutorial software. text by adding instructor-created or YouTube video links. ExamView contains hundreds of multiple-choice and free Additional media assets include: animated figures, video clips, response test items. ExamView testing is available on the Power- highlighting, notes, and more! YouBook is available in Lecture DVD. Enhanced WebAssign. Electronic items Printed items (Table continues on page xiv.) xiii 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). www.EngineeringEBooksPdf.com 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 xiv CourseMate Study Guide www.cengagebrain.com Multivariable CourseMate is a perfect self-study tool for students, and By Richard St. Andre requires no set up from instructors. CourseMate brings course ISBN 0-8400-5410-6 concepts to life with interactive learning, study, and exam For each section of the text, the Study Guide provides students preparation tools that support the printed textbook. CourseMate with a brief introduction, a short list of concepts to master, as for Stewart’s Calculus includes: an interactive eBook, Tools well as summary and focus questions with explained answers. for Enriching Calculus, videos, quizzes, flashcards, and more! The Study Guide also contains “Technology Plus” questions, For instructors, CourseMate includes Engagement Tracker, a and multiple-choice “On Your Own” exam-style questions. first-of-its-kind tool that monitors student engagement. Maple CD-ROM CalcLabs with Maple Maple provides an advanced, high performance mathe- matical computation engine with fully integrated numerics Multivariable By Philip B. Yasskin and Robert Lopez & symbolics, all accessible from a WYSIWYG technical docu- ISBN 0-8400-5812-8 ment environment. CalcLabs with Mathematica CengageBrain.com To access additional course materials and companion resources, Multivariable By Selwyn Hollis please visit www.cengagebrain.com. At the CengageBrain.com ISBN 0-8400-5813-6 home page, search for the ISBN of your title (from the back Each of these comprehensive lab manuals will help students cover of your book) using the search box at the top of the page. learn to use the technology tools available to them. CalcLabs This will take you to the product page where free companion contain clearly explained exercises and a variety of labs and resources can be found. projects to accompany the text. Ancillaries for Students Linear Algebra for Calculus by Konrad J. Heuvers, William P. Francis, John H. Kuisti, Student Solutions Manual Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner ISBN 0-534-25248-6 Multivariable By Dan Clegg and Barbara Frank This comprehensive book, designed to supplement the calculus ISBN 0-8400-4945-5 course, provides an introduction to and review of the basic ideas of linear algebra. Provides completely worked-out solutions to all odd-numbered exercises in the text, giving students a chance to check their answers and ensure they took the correct steps to arrive at an answer. Electronic items Printed items xiv 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). www.EngineeringEBooksPdf.com 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. 97817_10_ch10_p659-669.qk_97817_10_ch10_p659-669 11/3/10 4:12 PM Page 659 10 Parametric Equations and Polar Coordinates The Hale-Bopp comet, with its blue ion tail and white dust tail, appeared in the sky in March 1997. In Section 10.6 you will see how polar coordinates © Dean Ketelsen provide a convenient equation for the path of this comet. So far we have described plane curves by giving y as a function of x 关 y 苷 f 共x兲兴 or x as a function of y 关x 苷 t共 y兲兴 or by giving a relation between x and y that defines y implicitly as a function of x 关 f 共x, y兲 苷 0兴. In this chapter we discuss two new methods for describing curves. Some curves, such as the cycloid, are best handled when both x and y are given in terms of a third variable t called a parameter 关x 苷 f 共t兲, y 苷 t共t兲兴. Other curves, such as the cardioid, have their most convenient description when we use a new coordinate system, called the polar coordinate system. 659 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). www.EngineeringEBooksPdf.com 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. 97817_10_ch10_p659-669.qk_97817_10_ch10_p659-669 11/3/10 4:12 PM Page 660 660 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES 10.1 Curves Defined by Parametric Equations y Imagine that a particle moves along the curve C shown in Figure 1. It is impossible to C describe C by an equation of the form y 苷 f 共x兲 because C fails the Vertical Line Test. But (x, y)={ f(t), g(t)} the x- and y-coordinates of the particle are functions of time and so we can write x 苷 f 共t兲 and y 苷 t共t兲. Such a pair of equations is often a convenient way of describing a curve and gives rise to the following definition. Suppose that x and y are both given as functions of a third variable t (called a param- 0 x eter) by the equations FIGURE 1 x 苷 f 共t兲 y 苷 t共t兲 (called parametric equations). Each value of t determines a point 共x, y兲, which we can plot in a coordinate plane. As t varies, the point 共x, y兲 苷 共 f 共t兲, t共t兲兲 varies and traces out a curve C, which we call a parametric curve. The parameter t does not necessarily represent time and, in fact, we could use a letter other than t for the parameter. But in many applications of parametric curves, t does denote time and therefore we can interpret 共x, y兲 苷 共 f 共t兲, t共t兲兲 as the position of a particle at time t. EXAMPLE 1 Sketch and identify the curve defined by the parametric equations x 苷 t 2 ⫺ 2t y苷t⫹1