Last Bear

This textbook focuses on one of the most valuable skills in multivariable and vector calculus: visualization. With over one hundred carefully drawn color images, students who have long struggled picturing, for example, level sets or vector fields will find these abstract concepts rendered with clarity and ingenuity. This illustrative approach to the material covered in standard multivariable and vector calculus textbooks will serve as a much-needed and highly useful companion. Emphasizing portability, this book is an ideal complement to other references in the area. It begins by exploring preliminary ideas such as vector algebra, sets, and coordinate systems, before moving into the core areas of multivariable differentiation and integration, and vector calculus. Sections on the chain rule for second derivatives, implicit functions, PDEs, and the method of least squares offer additional depth; ample illustrations are woven throughout. Mastery Checks engage students in material on the spot, while longer exercise sets at the end of each chapter reinforce techniques. An Illustrative Guide to Multivariable and Vector Calculus will appeal to multivariable and vector calculus students and instructors around the world who seek an accessible, visual approach to this subject. Higher-level students, called upon to apply these concepts across science and engineering, will also find this a valuable and concise resource. Cover 1 S Title 2 Title: An Illustrative Guide to Multivariable and Vector Calculus 3 Copyright © Springer Nature Switzerland AG 2020 4 Dedication 5 Preface 6 Acknowledgements 8 Contents 9 Important Formulæ 11 Multivariable calculus 11 Vector calculus 13 Chapter 1 Vectors and functions 15 1.A Some vector algebra essentials 16 1.B Introduction to sets 23 1.C Real-valued functions 31 1.D Coordinate systems 39 1.E Drawing or visualizing surfaces in $mathbbR3$ 41 1.F Level sets 52 1.G Supplementary problems 57 Chapter 2 Differentiation of multivariable functions 62 2.A The derivative 62 2.B Limits and continuity 66 2.C Partial derivatives 75 2.D Differentiability of f:mathbbRn-3murightarrow mathbbR 80 2.E Directional derivatives and the gradient 87 2.F Higher-order derivatives 93 2.G Composite functions and the chain rule 97 2.H Implicit functions 114 2.I Taylor's formula and Taylor series 126 2.J Supplementary problems 132 Chapter 3 Applications of the differential calculus 137 3.A Extreme values of f:mathbbRn-3murightarrowmathbbR 137 3.B Extreme points: The complete story 145 3.C Differentials and error analysis 157 3.D Method of least squares 158 3.E Partial derivatives in equations: Partial differential equations 164 3.F Supplementary problems 183 Chapter 4 Integration of multivariable functions 188 4.A Multiple integrals 188 4.B Iterated integration in $mathbbR2$ 195 4.C Integration over complex domains 198 4.D Generalized (improper) integrals in $mathbbR2$ 204 4.E Change of variables in $mathbbR2$ 209 4.F Triple integrals 215 4.G Iterated integration in $mathbbR3$ 218 4.H Change of variables in $mathbbR3$ 222 4.I $n$-tuple integrals 224 4.J Epilogue: Some practical tips for evaluating integrals 226 4.K Supplementary problems 228 Chapter 5 Vector calculus 234 5.A Vector-valued functions 234 5.B Vector fields 249 5.C Line integrals 257 5.D Surface integrals 271 5.E Gauss's theorem 284 5.F Green's and Stokes's theorems 292 5.G Supplementary problems 304 Glossary of symbols 312 Bibliography 315 Index 317