Mathematics for PhysicsA guided tour for graduate

students

Michael Stoneand

Paul Goldbart

PIMANDER-CASAUBONAlexandria Florence London

ii

Copyright c2002-2008 M. Stone, P. M. GoldbartAll rights reserved. No part of this material can be reproduced, stored ortransmitted without the written permission of the authors. For informationcontact: Michael Stone or Paul Goldbart, Department of Physics, Universityof Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois61801-3080, U.S.A.

Dedication

To the memory of Mikes mother, Aileen Stone: 9 9 = 81.To Pauls mother and father, Carole and Colin Goldbart.

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iv DEDICATION

Acknowledgments

A great many people have encouraged us along the way:Our teachers at the University of Cambridge, the University of California-LosAngeles, and Imperial College London.Our students your questions and enthusiasm have helped shape our under-standing and our exposition.Our colleaguesfaculty and staffat the University of Illinois at Urbana-Champaign how fortunate we are to have a community so rich in bothaccomplishment and collegiality.Our friends and family: Kyre and Steve and Ginna; and Jenny, Ollie andGreta we hope to be more attentive now that this book is done.Our editor Simon Capelin at Cambridge University Press your patience isappreciated.The staff of the U.S. National Science Foundation and the U.S. Departmentof Energy, who have supported our research over the years.Our sincere thanks to you all.

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vi ACKNOWLEDGMENTS

Preface

This book is based on a two-semester sequence of courses taught to incominggraduate students at the University of Illinois at Urbana-Champaign, pri-marily physics students but also some from other branches of the physicalsciences. The courses aim to introduce students to some of the mathematicalmethods and concepts that they will find useful in their research. We havesought to enliven the material by integrating the mathematics with its appli-cations. We therefore provide illustrative examples and problems drawn fromphysics. Some of these illustrations are classical but many are small parts ofcontemporary research papers. In the text and at the end of each chapter weprovide a collection of exercises and problems suitable for homework assign-ments. The former are straightforward applications of material presentedin the text; the latter are intended to be interesting, and take rather morethought and time.

We devote the first, and longest, part (Chapters 1 to 9, and the firstsemester in the classroom) to traditional mathematical methods. We explorethe analogy between linear operators acting on function spaces and matricesacting on finite dimensional spaces, and use the operator language to pro-vide a unified framework for working with ordinary differential equations,partial differential equations, and integral equations. The mathematical pre-requisites are a sound grasp of undergraduate calculus (including the vectorcalculus needed for electricity and magnetism courses), elementary linear al-gebra, and competence at complex arithmetic. Fourier sums and integrals, aswell as basic ordinary differential equation theory, receive a quick review, butit would help if the reader had some prior experience to build on. Contourintegration is not required for this part of the book.

The second part (Chapters 10 to 14) focuses on modern differential ge-ometry and topology, with an eye to its application to physics. The tools ofcalculus on manifolds, especially the exterior calculus, are introduced, and

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viii PREFACE

used to investigate classical mechanics, electromagnetism, and non-abeliangauge fields. The language of homology and cohomology is introduced andis used to investigate the influence of the global topology of a manifold onthe fields that live in it and on the solutions of differential equations thatconstrain these fields.

Chapters 15 and 16 introduce the theory of group representations andtheir applications to quantum mechanics. Both finite groups and Lie groupsare explored.

The last part (Chapters 17 to 19) explores the theory of complex variablesand its applications. Although much of the material is standard, we make useof the exterior calculus, and discuss rather more of the topological aspects ofanalytic functions than is customary.

A cursory reading of the Contents of the book will show that there ismore material here than can be comfortably covered in two semesters. Whenusing the book as the basis for lectures in the classroom, we have found ituseful to tailor the presented material to the interests of our students.

Contents

Dedication iii

Acknowledgments v

Preface vii

1 Calculus of Variations 1

1.1 What is it good for? . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Variable endpoints . . . . . . . . . . . . . . . . . . . . . . . . 29

1.5 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . 36

1.6 Maximum or minimum? . . . . . . . . . . . . . . . . . . . . . 40

1.7 Further exercises and problems . . . . . . . . . . . . . . . . . 42

2 Function Spaces 55

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.2 Norms and inner products . . . . . . . . . . . . . . . . . . . . 57

2.3 Linear operators and distributions . . . . . . . . . . . . . . . . 74

2.4 Further exercises and problems . . . . . . . . . . . . . . . . . 85

3 Linear Ordinary Differential Equations 95

3.1 Existence and uniqueness of solutions . . . . . . . . . . . . . . 95

3.2 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.3 Inhomogeneous equations . . . . . . . . . . . . . . . . . . . . . 103

3.4 Singular points . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5 Further exercises and problems . . . . . . . . . . . . . . . . . 108

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4 Linear Differential Operators 111

4.1 Formal vs. concrete operators . . . . . . . . . . . . . . . . . . 111

4.2 The adjoint operator . . . . . . . . . . . . . . . . . . . . . . . 114

4.3 Completeness of eigenfunctions . . . . . . . . . . . . . . . . . 128

4.4 Further exercises and problems . . . . . . . . . . . . . . . . . 145

5 Green Functions 155

5.1 Inhomogeneous linear equations . . . . . . . . . . . . . . . . . 155

5.2 Constructing Green functions . . . . . . . . . . . . . . . . . . 156

5.3 Applications of Lagranges identity . . . . . . . . . . . . . . . 167

5.4 Eigenfunction expansions . . . . . . . . . . . . . . . . . . . . . 170

5.5 Analytic properties of Green functions . . . . . . . . . . . . . 171

5.6 Locality and the Gelfand-Dikii equation . . . . . . . . . . . . 184

5.7 Further exercises and problems . . . . . . . . . . . . . . . . . 185

6 Partial Differential Equations 193

6.1 Classification of PDEs . . . . . . . . . . . . . . . . . . . . . . 193

6.2 Cauchy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.3 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.4 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.5 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.6 Further exercises and problems . . . . . . . . . . . . . . . . . 249

7 The Mathematics of Real Waves 257

7.1 Dispersive waves . . . . . . . . . . . . . . . . . . . . . . . . . 257

7.2 Making waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

7.3 Non-linear waves . . . . . . . . . . . . . . . . . . . . . . . . . 274

7.4 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

7.5 Further exercises and problems . . . . . . . . . . . . . . . . . 289

8 Special Functions 295

8.1 Curvilinear co-ordinates . . . . . . . . . . . . . . . . . . . . . 295

8.2 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . 302

8.3 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.4 Singular endpoints . . . . . . . . . . . . . . . . . . . . . . . . 332

8.5 Further exercises and problems . . . . . . . . . . . . . . . . . 340

CONTENTS xi

9 Integral Equations 3479.1 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3479.2 Classification of integral equations . . . . . . . . . . . . . . . . 3489.3 Integral transforms . . . . . . . . . . . . . . . . . . . . . . . . 3509.4 Separable kernels . . . . . . . . . . . . . . . . . . . . . . . . . 3589.5 Singular integral equations . . . . . . . . . . . . . . . . . . . . 3619.6 Wiener-Hopf equations I . . . . . . . . . . . . . . . . . . . . . 3659.7 Some functional analysis . . . . . . . . . . . . . . . . . . . . . 3709.8 Series solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 3789.9 Further exercises and problems . . . . . . . . . . . . . . . . . 382

10 Vectors and Tensors 38710.1 Covariant and contravariant vectors . . . . . . . . . . . . . . . 38710.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39010.3 Cartesian tensors . . . . . . . . . . . . . . . . . . . . . . . . . 40410.4 Further exercises and problems . . . . . . . . . . . . . . . . . 415

11 Differential Calculus on Manifolds 41911.1 Vector and covector fields . . . . . . . . . . . . . . . . . . . . 41911.2 Differentiating tensors . . . . . . . . . . . . . . . . . . . . . . 42511.3 Exterior calculus . . . . . . . . . . . . . . . . . . . . . . . . . 43411.4 Physical