Vector Calculus

Normal 0 false false false Vector Calculus, Fourth Edition , uses the language and notation of vectors and matrices to teach multivariable calculus. It is ideal for students with a solid background in single-variable calculus who are capable of thinking in more general terms about the topics in the course. This text is distinguished from others by its readable narrative, numerous figures, thoughtfully selected examples, and carefully crafted exercise sets. Colley includes not only basic and advanced exercises, but also mid-level exercises that form a necessary bridge between the two. Front Cover 1 Title Page 5 Copyright Page 6 Dedication Page 7 About the Author 8 CONTENTS 9 Preface 11 Acknowledgments 13 To the Student: Some Preliminary Notation 17 1 Vectors 23 1.1 Vectors in Two and Three Dimensions 23 1.2 More About Vectors 30 1.3 The Dot Product 40 1.4 The Cross Product 49 1.5 Equations for Planes Distance Problems 62 1.6 Some n-dimensional Geometry 70 1.7 New Coordinate Systems 84 True/False Exercises for Chapter 1 97 Miscellaneous Exercises for Chapter 1 97 2 Differentiation in Several Variables 104 2.1 Functions of Several Variables Graphing Surfaces 104 2.2 Limits 119 2.3 The Derivative 138 2.4 Properties Higher-order Partial Derivatives 156 2.5 The Chain Rule 164 2.6 Directional Derivatives and the Gradient 180 2.7 Newton’s Method (optional) 198 True/False Exercises for Chapter 2 204 Miscellaneous Exercises for Chapter 2 205 3 Vector-Valued Functions 211 3.1 Parametrized Curves and Kepler’s Laws 211 3.2 Arclength and Differential Geometry 224 3.3 Vector Fields: An Introduction 243 3.4 Gradient, Divergence, Curl, and the Del Operator 249 True/False Exercises for Chapter 3 259 Miscellaneous Exercises for Chapter 3 259 4 Maxima and Minima in Several Variables 266 4.1 Differentials and Taylor’s Theorem 266 4.2 Extrema of Functions 285 4.3 Lagrange Multipliers 300 4.4 Some Applications of Extrema 315 True/False Exercises for Chapter 4 327 Miscellaneous Exercises for Chapter 4 328 5 Multiple Integration 332 5.1 Introduction: Areas and Volumes 332 5.2 Double Integrals 336 5.3 Changing the Order of Integration 356 5.4 Triple Integrals 359 5.5 Change of Variables 371 5.6 Applications of Integration 395 5.7 Numerical Approximations of Multiple Integrals (optional) 410 True/False Exercises for Chapter 5 423 Miscellaneous Exercises for Chapter 5 425 6 Line Integrals 430 6.1 Scalar and Vector Line Integrals 430 6.2 Green’s Theorem 451 6.3 Conservative Vector Fields 461 True/False Exercises for Chapter 6 472 Miscellaneous Exercises for Chapter 6 473 7 Surface Integrals and Vector Analysis 477 7.1 Parametrized Surfaces 477 7.2 Surface Integrals 491 7.3 Stokes’s and Gauss’s Theorems 512 7.4 Further Vector Analysis Maxwell’s Equations 532 True/False Exercises for Chapter 7 544 Miscellaneous Exercises for Chapter 7 545 8 Vector Analysis in Higher Dimensions 552 8.1 An Introduction to Differential Forms 552 8.2 Manifolds and Integrals of k-forms 558 8.3 The Generalized Stokes’s Theorem 575 True/False Exercises for Chapter 8 583 Miscellaneous Exercises for Chapter 8 583 Suggestions for Further Reading 585 Answers to Selected Exercises 587 INDEX 621