An Illustrative Guide to Multivariable and Vector Calculus

Description

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 S Title Title: An Illustrative Guide to Multivariable and Vector Calculus Copyright © Springer Nature Switzerland AG 2020 Dedication Preface Acknowledgements Contents Important Formulæ Multivariable calculus Vector calculus Chapter 1 Vectors and functions 1.A Some vector algebra essentials 1.B Introduction to sets 1.C Real-valued functions 1.D Coordinate systems 1.E Drawing or visualizing surfaces in $mathbbR3$ 1.F Level sets 1.G Supplementary problems Chapter 2 Differentiation of multivariable functions 2.A The derivative 2.B Limits and continuity 2.C Partial derivatives 2.D Differentiability of f:mathbbRn-3murightarrow mathbbR 2.E Directional derivatives and the gradient 2.F Higher-order derivatives 2.G Composite functions and the chain rule 2.H Implicit functions 2.I Taylor's formula and Taylor series 2.J Supplementary problems Chapter 3 Applications of the differential calculus 3.A Extreme values of f:mathbbRn-3murightarrowmathbbR 3.B Extreme points: The complete story 3.C Differentials and error analysis 3.D Method of least squares 3.E Partial derivatives in equations: Partial differential equations 3.F Supplementary problems Chapter 4 Integration of multivariable functions 4.A Multiple integrals 4.B Iterated integration in $mathbbR2$ 4.C Integration over complex domains 4.D Generalized (improper) integrals in $mathbbR2$ 4.E Change of variables in $mathbbR2$ 4.F Triple integrals 4.G Iterated integration in $mathbbR3$ 4.H Change of variables in $mathbbR3$ 4.I $n$-tuple integrals 4.J Epilogue: Some practical tips for evaluating integrals 4.K Supplementary problems Chapter 5 Vector calculus 5.A Vector-valued functions 5.B Vector fields 5.C Line integrals 5.D Surface integrals 5.E Gauss's theorem 5.F Green's and Stokes's theorems 5.G Supplementary problems Glossary of symbols Bibliography Index