Bruce R. Kusse and ErikA. Westwig Mathematical Physics Related Titles Vaughn, M. T. Introduction to Mathematical Physics 2006. Approx. 650 pages with 50 figures. Softcover ISBN 3-527-40627-1 Lambourne, R., Tinker, M. Basic Mathematics for the Physical Sciences 2000.688 pages. Softcover ISBN 0-47 1-85207-4 Tinker, M., Lambourne, R. Further Mathematics for the Physical Sciences 2000.744 pages. Softcover ISBN 0-471-86723-3 Courant, R., Hilbert, D. Methods of Mathematical Physics Volume 1 1989. 575 pages with 27 figures. Softcover ISBN 0-47 1-50447-5 Volume 2 1989. 852 pages with 61 figures. Softcover ISBN 0-471-50439-4 Trigg, G. L. (ed.) Mathematical Tools for Physicists 2005.686 pages with 98 figures and 29 tables. Hardcover ISBN 3-527-40548-8 Bruce R. Kusse and Erik A. Westwig Mathematical Physics Applied Mathematics for Scientists and Engineers 2nd Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA 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 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 prerequisite for 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, curvilinearc oordinate systems, 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 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. E.A.W. 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. BRUCER . KUSSE ERIKA . WESTWIG Ithaca, New York CONTENTS 1 A Review of Vector and Matrix Algebra Using SubscriptlSummation Conventions 1 1.1 Notation, I 1.2 Vector Operations, 5 2 Differential and Integral Operations on Vector and Scalar Fields 18 2.1 Plotting Scalar and Vector Fields, 18 2.2 Integral Operators, 20 2.3 Differential Operations, 23 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 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 Transformations Between Coordinate Systems, 7 1 4.4 Tensor Diagonalization, 78 4.5 Tensor Transformations in 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 6.5 Contour Deformation, 146 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 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 8.2 Orthogonality, 253 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 10.1 Terminology, 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 X CONTENTS 11 Solutions to Laplace’s Equation 424 11.1 Cartesian Solutions, 424 11 .2 Expansions With Eigenfunctions, 433 11.3 Cylindrical Solutions, 441 1 1.4 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 13.1 Multivalued Functions, 509 13.2 The Method of Steepest Descent, 542 14 Tensors in Non-Orthogonal Coordinate 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 Pseudovectors and the Mirror Test 653