An illustrative guide to multivariable and vector calculus

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. Preface......Page 6 Contents......Page 9 Multivariable calculus......Page 11 Vector calculus......Page 13 1 Vectors and functions......Page 15 1.A Some vector algebra essentials......Page 16 1.B Introduction to sets......Page 23 1.C Real-valued functions......Page 31 1.D Coordinate systems......Page 39 1.E Drawing or visualizing surfaces in $mathbbR3$......Page 41 1.F Level sets......Page 52 1.G Supplementary problems......Page 57 2.A The derivative......Page 62 2.B Limits and continuity......Page 66 2.C Partial derivatives......Page 75 2.D Differentiability of f:mathbbRn-3murightarrow mathbbR......Page 80 2.E Directional derivatives and the gradient......Page 87 2.F Higher-order derivatives......Page 93 2.G Composite functions and the chain rule......Page 97 2.H Implicit functions......Page 114 2.I Taylor's formula and Taylor series......Page 126 2.J Supplementary problems......Page 132 3.A Extreme values of f:mathbbRn-3murightarrowmathbbR......Page 137 3.B Extreme points: The complete story......Page 145 3.C Differentials and error analysis......Page 157 3.D Method of least squares......Page 158 3.E Partial derivatives in equations: Partial differential equations......Page 164 3.F Supplementary problems......Page 183 4.A Multiple integrals......Page 188 4.B Iterated integration in $mathbbR2$......Page 195 4.C Integration over complex domains......Page 198 4.D Generalized (improper) integrals in $mathbbR2$......Page 204 4.E Change of variables in $mathbbR2$......Page 209 4.F Triple integrals......Page 215 4.G Iterated integration in $mathbbR3$......Page 218 4.H Change of variables in $mathbbR3$......Page 222 4.I $n$-tuple integrals......Page 224 4.J Epilogue: Some practical tips for evaluating integrals......Page 226 4.K Supplementary problems......Page 228 5.A Vector-valued functions......Page 234 5.B Vector fields......Page 249 5.C Line integrals......Page 257 5.D Surface integrals......Page 271 5.E Gauss's theorem......Page 284 5.F Green's and Stokes's theorems......Page 292 5.G Supplementary problems......Page 304 Glossary of symbols......Page 312 Bibliography......Page 315 Index......Page 317