Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, 3ed PDF
Preview Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, 3ed
div orad an . url informal text and o® vector all calculus that third edition h.m.schey div, grad, an informal text h. m. schey curl, and all that on vector calculus third edition Dect W- W- NORTON & COMPANY New York - London Copyright © 1997, 1992, 1973 by W W Norton & Company, Inc Alll rights reserved Printed in the United States of America The text of this book is composed in Times Roman with the display set in Optima Composition by University Graphics. Inc Library of Congress Cataloging-in-Publication Data Schey. H. M. (Harry Montz), 1930- Div, grad, curl, and all that an informal text on vector calculus /H M Schey —3rd ed Poem Includes bibliographical references (pp 160-61 and index 1 Vector analysis I Title QA433 S28 1996 515’ 63—de20 96-4942 ISBN 0-393-96997-5 (pbk ) W W Norton & Company. Inc . 500 Fifth Avenue, New York, NY 10110 http //web wwnorton com W W Norton & Company, Ltd , 10 Coptic Street, London WCIA IPU 1234567890 Zusammengestohlen aus Verschiedenem diesem und jenem Ludwig van Beethoven Contents Preface Chapter! Introduction, Vector Functions, and Electrostatics Introduction Vector Functions Electrostatics Problems Chapter II Surface Integrals and the Divergence Gauss’ Law The Unit Normal Vector Definition of Surface Integrals Evaluating Surface Integrals Flux Using Gauss’ Law to Find the Field The Divergence The Divergence in Cylindrical and Spherical Coordinates The Del Notation The Divergence Theorem Two Simple Applications of the Divergence Theorem Problems Chapter II Line Integrals and the Curl Work and Line Integrals eune— Contents Line Integrals Involving Vector Functions Path Independence The Curl The Curl in Cylindrical and Spherical Coordinates The Meaning of the Curl Differential Form of the Circulation Law Stokes’ Theorem An Application of Stokes’ Theorem Stokes’ Theorem and Simply Connected Regions Path Independence and the Curl Problems Chapter IV The Gradient Line Integrals and the Gradient Finding the Electrostatic Field Using Laplace’s Equation Directional Derivatives and the Gradient Geometric Significance of the Gradient The Gradient in Cylindrical and Spherical Coordinates Problems Solutions to Problems Bibliography Index 100 101 102 114 114 121 123 130 137 140 143 156 160 162 Preface to the Third Edition Uf it ain't broke, don't fix it Anonymous This new edition constitutes a fine-tuning of its predecessor. Sev- eral new problems have been added, two other problems awk- wardly worded in the earlier editions have been revised, and a diagram has been corrected. The major change involves replacing the operators div, grad, and curl by the appropriate expressions using the V operator, to bring the text into conformity with mod- ern notational practice. I have, however, resisted retitling the book V- , V, V X, and All That. I wish to express my gratitude to Richard Liu, Stephen Nettel, and Sally Seidel for their useful reviews of the previous edition. I take particular pleasure in thanking those of my readers who over the years have been good enough to send me comments, criticisms, and suggestions which have contributed significantly to the quality of the text. Chapter | Introduction, Vector Functions, and Electrostatics One lesson, Nature, let me learn of thee. Matthew Arnold Introduction In this text the subject of vector calculus is presented in the con- text of simple electrostatics. We follow this procedure for two reasons. First, much of vector calculus was invented for use in electromagnetic theory and is ideally suited to it. This presenta- tion will therefore show what vector calculus is and at the same time give you an idea of what it’s for. Second, we have a deep- seated conviction that mathematics—in any case some mathe- matics—is best discussed in a context which is not exclusively 1 mathematical. Thus, we will soft pedal mathematical rigor, which Introduction, Vector Functions, and Electrostatics we think is an obstacle to learning this subject on a first exposure to it, and appeal as much as possible to physical and geometric intuition. Now, if you want to learn vector calculus but know little or nothing about electrostatics, you needn’t be put off by our approach; no very great knowledge of physics is required to read and understand this text. Only the simplest features of electro- statics are involved, and these are presented in a few pages near the beginning. It should not be an impediment to anyone. In fact, the entire discussion is based on a search for a convenient method of finding the electrostatic field given the distribution of electric charges which produce it. This is the thread which runs through, and unifies, our presentation, so that as a bare minimum all you really need do is take our word for the fact that the electric field is an important enough quantity to warrant spending some time and effort in setting up a general method for calculating it. In the process, we hope you will learn the elements of vector calculus. Having said what you do nor need to know, we must now say what you do need to know. To begin with, you should, of course, be fluent in elementary calculus. Beyond that you should know how to work with functions of several variables, partial deriva- tives, and multiple (double and triple) integrals.' Finally, you must know something about vectors. This, however, is a subject of which too many writers and teachers have made heavy weather. What you should know about it can be listed quickly~ definition of vector, addition and subtraction of vectors, multi- plication of vectors by scalars, dot and cross products, and finally, resolution of vectors into components. An hour's time with any reasonable text on the subject should provide you with all you need to know of it to follow this text. Vector Functions One of the most important quantitites we deal with in the study of electricity is the electric field, and much of our presentation will make use of this quantity. Since the electric field is an exam- ple of what we call a vector function, we begin our discussion with a brief resumé of the function concept. A function of one variable, generally written y = f(x), is a rule ' Differential equations are used in one section of this text The section is not essential and may be omitted if the mathematics is too frightening
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