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Mathematical Methods for Physicists: A concise introduction

TAI L. CHOW

CAMBRIDGE UNIVERSITY PRESS

Mathematical Methods for Physicists A concise introduction This text is designed for an intermediate-level, two-semester undergraduate course in mathematical physics. It provides an accessible account of most of the current, important mathematical tools required in physics these days. Itis assumed that the reader has an adequate preparation in general physics and calculus. The book bridges the gap between an introductory physics course and more advanced courses in classical mechanics, electricity and magnetism, quantum mechanics, and thermal and statistical physics. The text contains a large number of worked examples to illustrate the mathematical techniques developed and to showtheir relevance to physics. The book is designed primarily for undergraduate physics majors, but could also be used by students in other subjects, such as engineering, astronomy and mathematics.
T A I L. C H O W was born and raised in China. He received a BS degree in physics from the National Taiwan University, a Masters degree in physics from Case Western Reserve University, and a PhD inphysics from the University of Rochester. Since 1969, Dr Chow has been in the Department of Physics at California State University, Stanislaus, and served as department chairman for 17 years, until 1992. He served as Visiting Professor of Physics at University of California (at Davis and Berkeley) during his sabbatical years. He also worked as Summer Faculty Research Fellow at Stanford University and atNASA. Dr Chow has published more than 35 articles in physics journals and is the author of two textbooks and a solutions manual.

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PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia http://www.cambridge.org © Cambridge University Press 2000 This edition © Cambridge University Press (Virtual Publishing) 2003 First published in printed format 2000

A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 65227 8 hardbackOriginal ISBN 0 521 65544 7 paperback

ISBN 0 511 01022 2 virtual (netLibrary Edition)

Mathematical Methods for Physicists
A concise introduction

T A I L. C H O W
California State University

Contents

Preface 1

xv

Vector and tensor analysis 1 Vectors and scalars 1 Direction angles and direction cosines 3 Vector algebra 4 Equality of vectors 4 Vector addition 4 Multiplicationby a scalar 4 The scalar product 5 The vector (cross or outer) product 7 The triple scalar product A Á …B  C† 10 The triple vector product 11 Change of coordinate system 11 The linear vector space Vn 13 Vector di€erentiation 15 Space curves 16 Motion in a plane 17 A vector treatment of classical orbit theory 18 Vector di€erential of a scalar ®eld and the gradient Conservative vector ®eld 21 Thevector di€erential operator r 22 Vector di€erentiation of a vector ®eld 22 The divergence of a vector 22 The operator r2 , the Laplacian 24 The curl of a vector 24 Formulas involving r 27 Orthogonal curvilinear coordinates 27
v

20

CON TEN TS

Special orthogonal coordinate systems 32 Cylindrical coordinates …; ; z† 32 Spherical coordinates (r; ; † 34 Vector integration and integraltheorems 35 Gauss' theorem (the divergence theorem) 37 Continuity equation 39 Stokes' theorem 40 Green's theorem 43 Green's theorem in the plane 44 Helmholtz's theorem 44 Some useful integral relations 45 Tensor analysis 47 Contravariant and covariant vectors 48 Tensors of second rank 48 Basic operations with tensors 49 Quotient law 50 The line element and metric tensor 51 Associated tensors 53...