Multivariable Calculus with MATLAB

Transcription

1 Multivariable Calculus with MATLAB

2 Ronald L. Lipsman Jonathan M. Rosenberg Multivariable Calculus with MATLAB With Applications to Geometry and Physics

3 Ronald L. Lipsman Department of Mathematics University of Maryland College Park, MD, USA Jonathan M. Rosenberg Department of Mathematics University of Maryland College Park, MD, USA ISBN DOI / ISBN (ebook) Library of Congress Control Number: Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

4 z Preface y x 0 The preface of a book gives the authors their best chance to answer an extremely important question: What makes this book special? This book is a reworking and updating for MATLAB of our previous book ( joint with Kevin R. Coombes) Multivariable Calculus with Mathematica, Springer, It represents our attempt to enrich and enliven the teaching of multivariable calculus and mathematical methods courses for scientists and engineers. Most books in these subjects are not substantially different from those of fifty years ago. (Well, they may include fancier graphics and omit several topics, but those are minor changes.) This book is different. We do touch on most of the classical topics; however, we have made a particular effort to illustrate each point with a significant example. More importantly, we have tried to bring fundamental physical applications Kepler s laws, electromagnetism, fluid flow, energy estimation back to a prominent position in the subject. From one perspective, the subject of multivariable calculus only exists because it can be applied to important problems in science. In addition, we have included a discussion of the geometric invariants of curves and surfaces, providing, in effect, a brief introduction to differential geometry. This material provides a natural extension to the traditional syllabus. We believe that we have succeeded in resurrecting material that used to be in the course while introducing new material. A major reason for that success is that we use the computational power of the mathematical software system MATLAB to carry a large share of the load. MATLAB is tightly integrated into every portion of this book. We use its graphical capabilities to draw pictures of curves and surfaces; we use its symbolical capabilities to compute curvature and torsion; we use its numerical capabilities to tackle problems that are well beyond the typical mundane examples of textbooks that treat the subject without using a computer. Finally, and this is something not done in any other books at this level, we give a serious yet elementary explanation of how various numerical algorithms work, and what their advantages and disadvantages are. Again, this is something that could not be accomplished without a software package such as MATLAB. v

5 vi Preface As an additional benefit from introducing MATLAB, we are able to improve students understanding of important elements of the traditional syllabus. Our students are better able to visualize regions in the plane and in space. They develop a better feel for the geometric meaning of the gradient; for the method of steepest descent; for the orthogonality of level curves and gradient flows. Because they have tools for visualizing cross sections of solids, they are better able to find the limits of integration in multiple integrals. To summarize, we think this book is special because, by using it: students obtain a better understanding of the traditional material; students see the deep connections between mathematics and science; students learn more about the intrinsic geometry of curves and surfaces; students acquire skill using MATLAB, a powerful piece of modern mathematical software; instructors can choose from a more exciting variety of problems than in standard textbooks; and both students and instructors are exposed to a more holistic approach to the subject one that embraces not only algebraic/calculus-based solutions to problems, but also numerical, graphical/geometric and qualitative approaches to the subject and its problems. Conventions Throughout the book, MATLAB commands, such as solve, are printed in typewriter boldface. Theorems and general principles, such as: derivatives measure change, are printed in a slanted font. When new terms, such as torsion, are introduced, they are printed in an italic font. File names and URLs (web addresses) are printed in typewriter font. Everything else is printed in a standard font. At the start of each chapter, below the title, is a small illustration. Each is a graphic generated by a MATLAB command. Most are taken from the MATLAB solution to one of the problems in the accompanying problem set. A few are taken from the chapter itself. Finally, in this Preface, the graphic represents a more eclectic choice. We leave it to the industrious reader to identify the source of these graphics, as well as to reproduce the figure. Acknowledgments We above all want to thank our former collaborators for their contributions to this project. Kevin Coombes (now at the Department of Biomedical Informatics at Ohio State University) was a co-author of Multivariable Calculus with Mathematica and kindly agreed to let us adapt that book for MATLAB. Brian Hunt was a co-author of A Guide to MATLAB and taught us many useful MATLAB tricks and

6 Preface vii tips. Paul Green helped develop MATLAB exercises for multivariable calculus that eventually worked their way into this book. Jonathan Rosenberg thanks the National Science Foundation for its support under grant DMS Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. College Park, MD, USA October 1, 2017 Ronald L. Lipsman Jonathan M. Rosenberg

7 Contents Preface... v 1 Introduction Benefits of Mathematical Software What s in This Book Chapter Descriptions What s Not in This Book How to Use This Book The MATLAB Interface A Word on Terminology Software Versions... 8 Problem Set A. Review of One-Variable Calculus... 9 Glossary of MATLAB Commands Options to MATLAB Commands References Vectors and Graphics Vectors Applications of Vectors Parametric Curves Graphing Surfaces Parametric Surfaces Problem Set B. Vectors and Graphics Glossary of MATLAB Commands Options to MATLAB Commands Geometry of Curves Parametric Curves Geometric Invariants Arclength The Frenet Frame ix

8 x Contents Curvature and Torsion Differential Geometry of Curves The Osculating Circle Plane Curves Spherical Curves Helical Curves Congruence Two More Examples Problem Set C. Curves Glossary of MATLAB Commands Kinematics Newton s Laws of Motion Kepler s Laws of Planetary Motion Studying Equations of Motion with MATLAB Problem Set D. Kinematics Glossary of MATLAB Commands Reference Directional Derivatives Visualizing Functions of Two Variables Three-Dimensional Graphs Graphing Level Curves The Gradient of a Function of Two Variables Partial Derivatives and the Gradient Directional Derivatives Functions of Three or More Variables Problem Set E. Directional Derivatives Glossary of MATLAB Commands and Options Options to MATLAB Commands Geometry of Surfaces The Concept of a Surface Basic Examples The Implicit Function Theorem Geometric Invariants Curvature Calculations with MATLAB Problem Set F. Surfaces Glossary of MATLAB Commands and Options Options to MATLAB Commands References

9 Contents xi 7 Optimization in Several Variables The One-Variable Case Analytic Methods Numerical Methods Newton s Method Functions of Two Variables Second Derivative Test Steepest Descent Multivariable Newton s Method Three or More Variables Constrained Optimization and Lagrange Multipliers Problem Set G. Optimization Glossary of MATLAB Commands Multiple Integrals Automation and Integration Regions in the Plane Viewing Simple Regions Polar Regions Algorithms for Numerical Integration Algorithms for Numerical Integration in a Single Variable Algorithms for Numerical Multiple Integration Viewing Solid Regions A More Complicated Example Cylindrical Coordinates More General Changes of Coordinates Problem Set H. Multiple Integrals Glossary of MATLAB Commands Multidimensional Calculus The Fundamental Theorem of Line Integrals Green s Theorem Stokes Theorem The Divergence Theorem Vector Calculus and Physics Problem Set I. Multivariable Calculus Glossary of MATLAB Commands Options to MATLAB Commands References

10 xii Contents 10 Physical Applications of Vector Calculus Motion in a Central Force Field Newtonian Gravitation Electricity and Magnetism Fluid Flow Heat and Wave Equations The Heat Equation The Wave Equation Problem Set J. Physical Applications Glossary of MATLAB Commands and Options Options to MATLAB Commands Appendix: Energy Minimization and Laplace s Equation References MATLAB Tips Sample Solutions Problem Set A: Problem Problem Set B: Problem Problem Set C: Problem Problem Set D: Problem Problem Set E: Problem 5, Parts (b) and (c) Problem Set F: Problem 10, Part (e) Problem Set G: Problem 10, Part (b) Problem Set H: Problem 2, Parts (c) and (d) Problem Set I: Problems 4(b) and 5(a) Problem Set J: Problem 3, Parts (a) and (b), but only subpart (ii) Index