Kusse and Erik A. Westwig Mathematical Physics Applied Mathematics for Scientists and Engineers 2nd Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co.com This Page Intentionally Left Blank www. Kusse and ErikA. Westwig Mathematical Physics www.com Related Titles Vaughn, M.
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Hardcover ISBN 3-527-40548-8 www. Kusse and Erik A. Westwig Mathematical Physics Applied Mathematics for Scientists and Engineers 2nd Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co.com The Authors All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and Bruce R.
Kusse publisher do not warrant the information contained in College of Engineering these books, including this book, to be free of errors. Cornell University Readers are advised to keep in mind that statements, Ithaca, NY data, illustrations, procedural details or other items brk2@cornell.edu may inadvertently be inaccurate. Erik Westwig Library of Congress Card No.: Palisade Corporation applied for Ithaca, NY ewestwig@palisade.com British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library. For a Solution Manual, lecturers should contact the editorial department at physics@wiley-vch.de, stating their Bibliographicinformation published by affiliation and the course in which they wish to use the Die Dentsehe Bibliothek book.
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb. 02006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheirn All rights reserved (including those of translation into other languages). No part of this book may be repro- duced in any form by photoprinting, microfilm, or ~ any other means - nor transmitted or translated into a machine language without written permission from the publishers.
Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printing Strauss GmbH, Morlenbach Binding J. Schaffer Buchbinderei GmbH, Griinstadt Printed in the Federal Republic of Germany Printed on acid-free paper ISBN-13: 978-3-52740672-2 ISBN-10: 3-527-40672-7 www.com This book is the result of a sequence of two courses given in the School of Applied and Engineering Physics at Cornell University.
The intent of these courses has been to cover a number of intermediate and advanced topics in applied mathematics that are needed by science and engineering majors. The courses were originally designed for junior level undergraduates enrolled in Applied Physics, but over the years they have attracted students from the other engineering departments, as well as physics, chemistry, astronomy and biophysics students. Course enrollment has also expanded to include freshman and sophomores with advanced placement and graduate students whose math background has needed some reinforcement. While teaching this course, we discovered a gap in the available textbooks we felt appropriate for Applied Physics undergraduates.
There are many good introductory calculus books. One such example is Calculus andAnalytic Geometry by Thomas and Finney, which we consider to be a prerequisitefor our book. There are also many good textbooks covering advanced topics in mathematical physics such as Mathematical Methods for Physicists by Arfken. Unfortunately,these advanced books are generally aimed at graduate students and do not work well for junior level undergraduates.
It appeared that there was no intermediate book which could help the typical student make the transition between these two levels. Our goal was to create a book to fill this need. The material we cover includes intermediate topics in linear algebra, tensors, curvilinearcoordinatesystems,complex variables, Fourier series, Fourier and Laplace transforms, differential equations, Dirac delta-functions, and solutions to Laplace’s equation. In addition, we introduce the more advanced topics of contravariance and covariance in nonorthogonal systems, multi-valued complex functions described with branch cuts and Riemann sheets, the method of steepest descent, and group theory.
These topics are presented in a unique way, with a generous use of illustrations and V www.com vi PREFACE graphs and an informal writing style, so that students at the junior level can grasp and understand them. Throughout the text we attempt to strike a healthy balance between mathematical completeness and readability by keeping the number of formal proofs and theorems to a minimum. Applications for solving real, physical problems are stressed. There are many examples throughout the text and exercises for the students at the end of each chapter.
Unlike many text books that cover these topics, we have used an organization that is fundamentally pedagogical. We consider the book to be primarily a teaching tool, although we have attempted to also make it acceptable as a reference. Consistent with this intent, the chapters are arranged as they have been taught in our two course sequence, rather than by topic. Consequently, you will find a chapter on tensors and a chapter on complex variables in the first half of the book and two more chapters, covering more advanced details of these same topics, in the second half.
In our first semester course, we cover chapters one through nine, which we consider more important for the early part of the undergraduate curriculum. The last six chapters are taught in the second semester and cover the more advanced material. We would like to thank the many Cornell students who have taken the AEP 3211322 course sequence for their assistance in finding errors in the text, examples, and exercises. would like to thank Ralph Westwig for his research help and the loan of many useful books.
He is also indebted to his wife Karen and their son John for their infinite patience. WESTWIG Ithaca, New York www.com CONTENTS 1 A Review of Vector and Matrix Algebra Using SubscriptlSummationConventions 1 1.2 Vector Operations, 5 2 Differential and Integral Operations on Vector and Scalar Fields 18 2.1 Plotting Scalar and Vector Fields, 18 2.4 Integral Definitions of the Differential Operators, 34 2.5 TheTheorems, 35 3 Curvilinear Coordinate Systems 44 3.1 The Position Vector, 44 3.2 The Cylindrical System, 45 3.3 The Spherical System, 48 3.4 General Curvilinear Systems, 49 3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical Systems, 58 www.com viii CONTENTS 4 Introduction to Tensors 67 4.1 The Conductivity Tensor and Ohm’s Law, 67 4.2 General Tensor Notation and Terminology, 71 4.3 TransformationsBetween Coordinate Systems, 7 1 4.5 Tensor Transformationsin Curvilinear Coordinate Systems, 84 4.6 Pseudo-Objects, 86 5 The Dirac &Function 100 5.1 Examples of Singular Functions in Physics, 100 5.2 Two Definitions of &t), 103 5.3 6-Functions with Complicated Arguments, 108 5.4 Integrals and Derivatives of 6(t), 111 5.5 Singular Density Functions, 114 5.6 The Infinitesimal Electric Dipole, 121 5.7 Riemann Integration and the Dirac &Function, 125 6 Introduction to Complex Variables 135 6.1 A Complex Number Refresher, 135 6.2 Functions of a Complex Variable, 138 6.3 Derivatives of Complex Functions, 140 6.4 The Cauchy Integral Theorem, 144 6.6 The Cauchy Integrd Formula, 147 6.7 Taylor and Laurent Series, 150 6.8 The Complex Taylor Series, 153 6.9 The Complex Laurent Series, 159 6.10 The Residue Theorem, 171 6.1 1 Definite Integrals and Closure, 175 6.12 Conformal Mapping, 189 www.com CONTENTS ix 7 Fourier Series 219 7.1 The Sine-Cosine Series, 219 7.2 The Exponential Form of Fourier Series, 227 7.3 Convergence of Fourier Series, 231 7.4 The Discrete Fourier Series, 234 8 Fourier Transforms 250 8.1 Fourier Series as To -+ m, 250 8.3 Existence of the Fourier Transform, 254 8.4 The Fourier Transform Circuit, 256 8.5 Properties of the Fourier Transform, 258 8.6 Fourier Transforms-Examples, 267 8.7 The Sampling Theorem, 290 9 Laplace Transforms 303 9.1 Limits of the Fourier Transform, 303 9.2 The Modified Fourier Transform, 306 9.3 The Laplace Transform, 313 9.4 Laplace Transform Examples, 314 9.5 Properties of the Laplace Transform, 318 9.6 The Laplace Transform Circuit, 327 9.7 Double-Sided or Bilateral Laplace Transforms, 331 10 Differential Equations 339 10.2 Solutions for First-Order Equations, 342 10.3 Techniques for Second-Order Equations, 347 10.4 The Method of Frobenius, 354 10.5 The Method of Quadrature, 358 10.6 Fourier and Laplace Transform Solutions, 366 10.7 Green’s Function Solutions, 376 www.com X CONTENTS 11 Solutions to Laplace’s Equation 424 11.2 Expansions With Eigenfunctions, 433 11.4 Spherical Solutions, 458 12 Integral Equations 491 12.1 Classification of Linear Integral Equations, 492 12.2 The Connection Between Differential and Integral Equations, 493 12.3 Methods of Solution, 498 13 Advanced Topics in Complex Analysis 509 13.2 The Method of Steepest Descent, 542 14 Tensors in Non-OrthogonalCoordinate Systems 562 14.1 A Brief Review of Tensor Transformations, 562 14.2 Non-Orthononnal Coordinate Systems, 564 15 Introduction to Group Theory 597 15.1 The Definition of a Group, 597 15.2 Finite Groups and Their Representations, 598 15.3 Subgroups, Cosets, Class, and Character, 607 15.4 Irreducible Matrix Representations, 612 15.5 Continuous Groups, 630 Appendix A The Led-Cidta Identity 639 Appendix B The Curvilinear Curl 641 Appendiv C The Double Integral Identity 645 Appendix D Green’s Function Solutions 647 Appendix E Pseudovectorsand the Mirror Test 653 www.com CONTENTS xi Appendix F Christoffel Symbols and Covariant Derivatives 655 Appendix G Calculus of Variations 661 Errata List 665 Bibliography 671 Index 673 www.com This Page Intentionally Left Blank www.com 1 A REVIEW OF VECTOR AND MATRIX ALGEBRA USING SUBSCRIPTISUMMATION CONVENTIONS This chapter presents a quick review of vector and matrix algebra. The intent is not to cover these topics completely, but rather use them to introduce subscript notation and the Einstein summation convention. These tools simplify the often complicated manipulations of linear algebra.1 NOTATION Standard, consistent notation is a very important habit to form in mathematics.
Good notation not only facilitatescalculationsbut, like dimensionalanalysis, helps to catch and correct errors. Thus, we begin by summarizing the notational conventions that will be used throughout this book, as listed in Table 1. Notational Conventions Symbol Quantity a A real number A complex number A vector component A matrix or tensor element An entire matrix A vector @, A basis vector - T A tensor L An operator 1 www.com 2 A R E W W OF VECTOR AND MATRIX ALGEBRA A three-dimensionalvector can be expressed as v = VX& + VY&,+ VZ&, (1.1) where the components (Vx, V,, V,) are called the Cartesian components of and (ex.e,, $) are the basis vectors of the coordinate system. This notation can be made more efficient by using subscript notation, which replaces the letters (x, y, z ) with the numbers (1,2,3).
That is, we define: Equation 1.1 becomes or more succinctly, i= 1,2,3 Figure 1.1 shows this notational modification on a typical Cartesian coordinate sys- tem. Although subscript notation can be used in many different types of coordinate systems, in this chapter we limit our discussion to Cartesian systems. Cartesian basis vectors are orthonormal and position independent. Orthonoml means the magnitude of each basis vector is unity, and they are all perpendicular to one another.
Position independent means the basis vectors do not change their orientations as we move around in space. Non-Cartesian coordinate systems are covered in detail in Chapter 3.4 can be compactedeven further by introducingthe Einstein summation convention, which assumes a summation any time a subscript is repeated in the same term.1 The Standard Cartesian System www.com NOTATION 3 We refer to this combination of the subscript notation and the summation convention as subscripthummation notation. Now imagine we want to write the simple vector relationship This equation is written in what we call vector notation. Notice how it does not depend on a choice of coordinate system.
In a particular coordinate system, we can write the relationship between these vectors in terms of their components: C1 = A1 + B1 C2 = A2 + B2 (1. With subscript notation, these three equations can be written in a single line, where the subscript i stands for any of the three values (1,2,3).