Higher Mathematics for Physicsand Engineering

Hiroyuki Shima · Tsuneyoshi Nakayama

Higher Mathematics forPhysics and Engineering

123

Dr. Hiroyuki Shima, Assistant ProfessorDepartment of Applied PhysicsHokkaido UniversitySapporo 060-8628, Japanshima@eng.hokudai.ac.jp

Dr. Tsuneyoshi Nakayama, Professor

Aichi 480-1192, JapanRiken-nakayama@mosk.tytlabs.co.jp

ISBN 978-3-540-87863-6 e-ISBN 978-3-540-87864-3DOI 10.1007/b138494Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2009940406

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Preface

Owing to the rapid advances in the physical sciences and engineering, the de-mand for higher-level mathematics is increasing yearly. This book is designedfor advanced undergraduates and graduate students who are interested in themathematical aspects of their own fields of study. The reader is assumed tohave a knowledge of undergraduate-level calculus and linear algebra.

There are any number of books available on mathematics for physics andengineering but they all fall into one of two categories: the one emphasizesmathematical rigor and the exposition of definitions or theorems, whereas theother is concerned primarily with applying mathematics to practical prob-lems. We believe that neither of these approaches alone is particularly helpfulto physicists and engineers who want to understand the mathematical back-ground of the subjects with which they are concerned. This book is differentin that it provides a short path to higher mathematics via a combination ofthese approaches. A sizable portion of this book is devoted to theorems anddefinitions with their proofs, and we are convinced that the study of theseproofs, which range from trivial to difficult, is useful for a grasp of the generalidea of mathematical logic. Moreover, several problems have been included atthe end of each section, and complete solutions for all of them are presentedin the greatest possible detail. We firmly believe that ours is a better peda-gogical approach than that found in typical textbooks, where there are manywell-polished problems but no solutions.

This book is essentially self-contained and assumes only standard under-graduate preparation such as elementary calculus and linear algebra. Thefirst half of the book covers the following three topics: real analysis, func-tional analysis, and complex analysis, along with the preliminaries and fourappendixes. Part I focuses on sequences and series of real numbers of realfunctions, with detailed explanations of their convergence properties. We alsoemphasize the concepts of Cauchy sequences and the Cauchy criterion thatdetermine the convergence of infinite real sequences. Part II deals with thetheory of the Hilbert space, which is the most important class of infinite vec-tor spaces. The completeness property of Hilbert spaces allows one to develop

VIII Preface

various types of complex orthonormal polynomials, as described in the mid-dle of Part II. An introduction to the Lebesgue integration theory, a subjectof ever-increasing importance in physics, is also presented. Part III describesthe theory of complex-valued functions of one complex variable. All relevantelements including analytic functions, singularity, residue, continuation, andconformal mapping are described in a self-contained manner. A thorough un-derstanding of the fundamentals treated is important in order to proceed tomore advanced branches of mathematical physics.

In the second half of the volume, the following three specific topics arediscussed: Fourier analysis, differential equations, and tensor analysis. Thesethree are the most important subjects in both engineering and the physicalsciences, but their rigorous mathematical structures have hardly been coveredin ordinary textbooks. We know that mathematical rigor is often unnecessaryfor practical use. However, the blind usage of mathematical methods as a toolmay lead to a lack of understanding of the symbiotic relationship betweenmathematics and the physical sciences. We believe that readers who studythe mathematical structures underlying these three subjects in detail will ac-quire a better understanding of the theoretical backgrounds associated withtheir own fields. Part IV describes the theory of Fourier series, the Fouriertransform, and the Laplace transform, with a special emphasis on the proofsof their convergence properties. A more contemporary subject, the wavelettransform, is also described toward the end of Part IV. Part V deals with or-dinary and partial differential equations. The existence theorem and stabilitytheory for solutions, which serve as the underlying basis for differential equa-tions, are described with rigorous proofs. Part VI is devoted to the calculus oftensors in terms of both Cartesian and non-Cartesian coordinates, along withthe essentials of differential geometry. An alternative tensor theory expressedin terms of abstract vector spaces is developed toward the end of Part VI.

The authors hope and trust that this book will serve as an introductoryguide for the mathematical aspects of the important topics in the physicalsciences and engineering.

Sapporo, Hiroyuki ShimaNovember 2009 Tsuneyoshi Nakayama

Contents

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Basic Notions of a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Set and Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.5 Neighborhood and Contact Point . . . . . . . . . . . . . . . . . . . . 51.1.6 Closed and Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Order of Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Symbols O, o, and ∼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Values of Indeterminate Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.1 l’Hopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.2 Several Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Part I Real Analysis

2 Real Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Sequences of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Convergence of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Bounded Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.3 Monotonic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.4 Limit Superior and Limit Inferior . . . . . . . . . . . . . . . . . . . . 21

2.2 Cauchy Criterion for Real Sequences . . . . . . . . . . . . . . . . . . . . . . . 252.2.1 Cauchy Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Infinite Series of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Limits of Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Cauchy Criterion for Infinite Series . . . . . . . . . . . . . . . . . . 31

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2.3.3 Absolute and Conditional Convergence . . . . . . . . . . . . . . . 322.3.4 Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Convergence Tests for Infinite Real Series . . . . . . . . . . . . . . . . . . . 382.4.1 Limit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.2 Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.3 Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.4 Alternating Series Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1 Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1.2 Continuity of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1.3 Derivative of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.1.4 Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Sequences of Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.1 Pointwise Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.3 Cauchy Criterion for Series of Functions . . . . . . . . . . . . . . 533.2.4 Continuity of the Limit Function . . . . . . . . . . . . . . . . . . . . 543.2.5 Integrability of the Limit Function . . . . . . . . . . . . . . . . . . . 563.2.6 Differentiability of the Limit Function . . . . . . . . . . . . . . . . 57

3.3 Series of Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.1 Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.2 Properties of Uniformly Convergent Series of Functions 623.3.3 Weierstrass M -test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.2 Convergence of an Improper Integral . . . . . . . . . . . . . . . . . 673.4.3 Principal Value Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4.4 Conditions for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 68

Part II Functional Analysis

4 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1.2 Abstract Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1.3 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1.4 Geometry of Inner Product Spaces . . . . . . . . . . . . . . . . . . 764.1.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.6 Completeness of Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 794.1.7 Several Examples of Hilbert Spaces . . . . . . . . . . . . . . . . . . 80

4.2 Hierarchical Structure of Vector Spaces . . . . . . . . . . . . . . . . . . . . . 834.2.1 Precise Definitions of Vector Spaces . . . . . . . . . . . . . . . . . . 83

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4.2.2 Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.3 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.4 Subspaces of a Normed Space . . . . . . . . . . . . . . . . . . . . . . . 864.2.5 Basis of a Vector Space: Revisited . . . . . . . . . . . . . . . . . . . 874.2.6 Orthogonal Bases in Hilbert Spaces . . . . . . . . . . . . . . . . . . 88

4.3 Hilbert Spaces of �2 and L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.1 Completeness of the �2 Spaces . . . . . . . . . . . . . . . . . . . . . . 914.3.2 Completeness of the L2 Spaces . . . . . . . . . . . . . . . . . . . . . . 924.3.3 Mean Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.4 Generalized Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . 954.3.5 Riesz–Fisher Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3.6 Isomorphism between �2 and L2 . . . . . . . . . . . . . . . . . . . . . 98

5 Orthonormal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1 Polynomial Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.1 Weierstrass Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1.2 Existence of Complete Orthonormal sets of Polynomials 1035.1.3 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.1.5 Spherical Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Classification of Orthonormal Functions . . . . . . . . . . . . . . . . . . . . 1145.2.1 General Rodrigues Formula . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2.2 Classification of the Polynomials . . . . . . . . . . . . . . . . . . . . 1165.2.3 The Recurrence Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.4 Coefficients of the Recurrence Formula . . . . . . . . . . . . . . . 1205.2.5 Roots of Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . 1215.2.6 Differential Equations Satisfied by the Polynomials . . . . 1225.2.7 Generating Functions (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.2.8 Generating Functions (II) . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.3.1 Minimax Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.3.2 A Concise Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.3 Discrete Orthogonality Relation . . . . . . . . . . . . . . . . . . . . . 133

5.4 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 1355.4.1 Quantum-Mechanical State in an Harmonic Potential . . 1355.4.2 Electrostatic potential generated by a multipole . . . . . . . 136

6 Lebesgue Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.1 Measure and Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.1.1 Riemann Integral Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 1396.1.2 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.1.3 The Probability Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.1.4 Support and Area of a Step Function . . . . . . . . . . . . . . . . 1446.1.5 α-Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.1.6 Properties of α-summable functions . . . . . . . . . . . . . . . . . . 147

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6.2 Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.2.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.2.2 Definition of the Lebesgue Integral . . . . . . . . . . . . . . . . . . 1516.2.3 Riemann Integrals vs. Lebesgue Integrals . . . . . . . . . . . . . 1526.2.4 Properties of the Lebesgue Integrals . . . . . . . . . . . . . . . . . 1536.2.5 Null-Measure Property of Countable Sets . . . . . . . . . . . . . 1546.2.6 The Concept of Almost Everywhere . . . . . . . . . . . . . . . . . 155

6.3 Important Theorems for Lebesgue Integrals . . . . . . . . . . . . . . . . . 1586.3.1 Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . 1586.3.2 Dominated Convergence Theorem (I) . . . . . . . . . . . . . . . . 1606.3.3 Fatou Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.3.4 Dominated Convergence Theorem (II) . . . . . . . . . . . . . . . 1616.3.5 Fubini Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.4 The Lebesgue Spaces Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.4.1 The Spaces of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.4.2 Holder Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.4.3 Minkowski Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.4.4 Completeness of Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.5 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 1726.5.1 Practical Significance of Lebesgue Integrals . . . . . . . . . . . 1726.5.2 Contraction Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.5.3 Preliminaries for the Central Limit Theorem . . . . . . . . . . 1756.5.4 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.5.5 Proof of the Central Limit Theorem . . . . . . . . . . . . . . . . . 178

Part III Complex Analysis

7 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.1 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.1.1 Continuity and Differentiability . . . . . . . . . . . . . . . . . . . . . 1857.1.2 Definition of an Analytic Function . . . . . . . . . . . . . . . . . . . 1877.1.3 Cauchy–Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . 1897.1.4 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917.1.5 Geometric Interpretation of Analyticity . . . . . . . . . . . . . . 192

7.2 Complex Integrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957.2.1 Integration of Complex Functions . . . . . . . . . . . . . . . . . . . 1957.2.2 Cauchy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1977.2.3 Integrations on a Multiply Connected Region . . . . . . . . . 1997.2.4 Primitive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.3 Cauchy Integral Formula and Related Theorem . . . . . . . . . . . . . 2047.3.1 Cauchy Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.3.2 Goursat Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.3.3 Absence of Extrema in Analytic Regions . . . . . . . . . . . . . 2077.3.4 Liouville Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

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7.3.5 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . 2097.3.6 Morera Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.4 Series Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.4.1 Circle of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.4.2 Singularity on the Radius of Convergence . . . . . . . . . . . . . 2157.4.3 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.4.4 Apparent Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2187.4.5 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.4.6 Regular and Principal Parts . . . . . . . . . . . . . . . . . . . . . . . . 2217.4.7 Uniqueness of Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . 2227.4.8 Techniques for Laurent Expansion . . . . . . . . . . . . . . . . . . . 223

7.5 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 2287.5.1 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.5.2 Kutta–Joukowski Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2297.5.3 Blasius Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8 Singularity and Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.1 Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.1.1 Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.1.2 Nonisolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 2358.1.3 Weierstrass Theorem for Essential Singularities . . . . . . . . 2368.1.4 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8.2 Multivaluedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2408.2.1 Multivalued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2408.2.2 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.2.3 Branch Point and Branch Cut . . . . . . . . . . . . . . . . . . . . . . 243

8.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458.3.1 Continuation by Taylor Series . . . . . . . . . . . . . . . . . . . . . . . 2458.3.2 Function Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2468.3.3 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.3.4 Conservation of Functional Equations . . . . . . . . . . . . . . . . 2508.3.5 Continuation Around a Branch Point . . . . . . . . . . . . . . . . 2528.3.6 Natural Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528.3.7 Technique of Analytic Continuations . . . . . . . . . . . . . . . . . 2548.3.8 The Method of Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

9 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2599.1 Calculus of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

9.1.1 Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2599.1.2 Remarks on Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2619.1.3 Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2629.1.4 Ratio Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.1.5 Evaluating the Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

9.2 Applications to Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2679.2.1 Classification of Evaluable Real Integrals . . . . . . . . . . . . . 267

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9.2.2 Type 1: Integrals of f(cos θ, sin θ) . . . . . . . . . . . . . . . . . . . . 2689.2.3 Type 2: Integrals of Rational Function . . . . . . . . . . . . . . . 2689.2.4 Type 3: Integrals of f(x)eix . . . . . . . . . . . . . . . . . . . . . . . . . 2709.2.5 Type 4: Integrals of f(x)/xα . . . . . . . . . . . . . . . . . . . . . . . . 2719.2.6 Type 5: Integrals of f(x) log x . . . . . . . . . . . . . . . . . . . . . . . 273

9.3 More Applications of Residue Calculus . . . . . . . . . . . . . . . . . . . . . 2779.3.1 Integrals on Rectangular Contours . . . . . . . . . . . . . . . . . . . 2779.3.2 Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2799.3.3 Summation of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2819.3.4 Langevin and Riemann zeta Functions . . . . . . . . . . . . . . . 283

9.4 Argument Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2859.4.1 The Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2859.4.2 Variation of the Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 2889.4.3 Extentson of the Argument Principle . . . . . . . . . . . . . . . . 2899.4.4 Rouche Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

9.5 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939.5.1 Principal Value Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939.5.2 Several Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2959.5.3 Dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2979.5.4 Kramers–Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 2989.5.5 Subtracted Dispersion Relation . . . . . . . . . . . . . . . . . . . . . 2999.5.6 Derivation of Dispersion Relations . . . . . . . . . . . . . . . . . . . 300

10 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30510.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

10.1.1 Conformal Property of Analytic Functions . . . . . . . . . . . . 30510.1.2 Scale Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30710.1.3 Mapping of a Differential Area . . . . . . . . . . . . . . . . . . . . . . 30810.1.4 Mapping of a Tangent Line . . . . . . . . . . . . . . . . . . . . . . . . . 30910.1.5 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31110.1.6 Singular Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

10.2 Elementary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31510.2.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31510.2.2 Bilinear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31610.2.3 Miscellaneous Transformations . . . . . . . . . . . . . . . . . . . . . . 31710.2.4 Mapping of Finite-Radius Circle . . . . . . . . . . . . . . . . . . . . . 32110.2.5 Invariance of the Cross ratio . . . . . . . . . . . . . . . . . . . . . . . . 322

10.3 Applications to Boundary-Value Problems . . . . . . . . . . . . . . . . . . 32510.3.1 Schwarz–Christoffel Transformation . . . . . . . . . . . . . . . . . . 32510.3.2 Derivation of the Schwartz–Christoffel Transformation . 32610.3.3 The Method of Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

10.4 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 33210.4.1 Electric Potential Field in a Complicated Geometry . . . . 33210.4.2 Joukowsky Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

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Part IV Fourier Analysis

11 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33911.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

11.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33911.1.2 Dirichlet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34011.1.3 Fourier Series of Periodic Functions . . . . . . . . . . . . . . . . . . 34211.1.4 Half-range Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 34311.1.5 Fourier Series of Nonperiodic Functions . . . . . . . . . . . . . . 34411.1.6 The Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 34611.1.7 Fourier Series in Higher Dimensions . . . . . . . . . . . . . . . . . 347

11.2 Mean Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 35111.2.1 Mean Convergence Property . . . . . . . . . . . . . . . . . . . . . . . . 35111.2.2 Dirichlet and Fejer Integrals . . . . . . . . . . . . . . . . . . . . . . . . 35311.2.3 Proof of the Mean Convergence of Fourier Series . . . . . . 35511.2.4 Parseval Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35611.2.5 Riemann–Lebesgue Theorem . . . . . . . . . . . . . . . . . . . . . . . . 357

11.3 Uniform Convergence of Fourier series . . . . . . . . . . . . . . . . . . . . . . 36011.3.1 Criterion for Uniform and Pointwise Convergence . . . . . . 36011.3.2 Fejer theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36011.3.3 Proof of Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . 36111.3.4 Pointwise Convergence at Discontinuous Points . . . . . . . 36311.3.5 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36511.3.6 Overshoot at a Discontinuous Point . . . . . . . . . . . . . . . . . . 366

11.4 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 37111.4.1 Temperature Variation of the Ground . . . . . . . . . . . . . . . . 37111.4.2 String Vibration Under Impact . . . . . . . . . . . . . . . . . . . . . . 373

12 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37712.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

12.1.1 Derivation of Fourier Transform . . . . . . . . . . . . . . . . . . . . . 37712.1.2 Fourier Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 37912.1.3 Proof of the Fourier Integral Theorem. . . . . . . . . . . . . . . . 38012.1.4 Inverse Relations of the Half-width . . . . . . . . . . . . . . . . . . 38112.1.5 Parseval Identity for Fourier Transforms . . . . . . . . . . . . . . 38212.1.6 Fourier Transforms in Higher Dimensions . . . . . . . . . . . . . 384

12.2 Convolution and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38712.2.1 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38712.2.2 Cross-Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . 38812.2.3 Autocorrelation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 390

12.3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39112.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39112.3.2 Inverse Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39212.3.3 Nyquest Frequency and Aliasing . . . . . . . . . . . . . . . . . . . . . 393

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12.3.4 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39412.3.5 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39612.3.6 Matrix Representation of FFT Algorithm. . . . . . . . . . . . . 39812.3.7 Decomposition Method for FFT . . . . . . . . . . . . . . . . . . . . . 400

12.4 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 40112.4.1 Fraunhofer Diffraction I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40112.4.2 Fraunhofer Diffraction II . . . . . . . . . . . . . . . . . . . . . . . . . . . 40312.4.3 Amplitude Modulation Technique . . . . . . . . . . . . . . . . . . . 404

13 Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40713.1 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

13.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40713.1.2 Several Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40813.1.3 Significance of Analytic Continuation . . . . . . . . . . . . . . . . 40913.1.4 Convergence of Laplace Integrals . . . . . . . . . . . . . . . . . . . . 41013.1.5 Abscissa of Absolute Convergence . . . . . . . . . . . . . . . . . . . 41113.1.6 Laplace Transforms of Elementary Functions . . . . . . . . . . 412

13.2 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 41513.2.1 First Shifting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41513.2.2 Second Shifting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 41613.2.3 Laplace Transform of Periodic Functions . . . . . . . . . . . . . 41713.2.4 Laplace Transform of Derivatives and Integrals . . . . . . . . 41813.2.5 Laplace Transforms Leading to Multivalued Functions . 420

13.3 Convergence Theorems for Laplace Integrals . . . . . . . . . . . . . . . . 42213.3.1 Functions of Exponential Order . . . . . . . . . . . . . . . . . . . . . 42213.3.2 Convergence for Exponential-Order Cases . . . . . . . . . . . . 42413.3.3 Uniform Convergence for Exponential-Order Cases . . . . 42513.3.4 Convergence for General Cases . . . . . . . . . . . . . . . . . . . . . . 42713.3.5 Uniform Convergence for General Cases . . . . . . . . . . . . . . 42913.3.6 Distinction Between Exponential-Order Cases

and General Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43113.3.7 Analytic Property of Laplace Transforms . . . . . . . . . . . . . 432

13.4 Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43213.4.1 The Two-Sided Laplace Transform . . . . . . . . . . . . . . . . . . 43213.4.2 Inverse of the Two-Sided Laplace Transform . . . . . . . . . . 43413.4.3 Inverse of the One-Sided Laplace Transform . . . . . . . . . . 43613.4.4 Useful Formula for Inverse Laplace Transformation . . . . 43613.4.5 Evaluating Inverse Transformations . . . . . . . . . . . . . . . . . . 43913.4.6 Inverse Transform of Multivalued Functions . . . . . . . . . . . 441

13.5 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 44513.5.1 Electric Circuits I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44513.5.2 Electric Circuits II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

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14 Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44914.1 Continuous Wavelet Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

14.1.1 Definition of Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44914.1.2 The Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45114.1.3 Correlation Between Wavelet and Signal . . . . . . . . . . . . . . 45214.1.4 Actual Application of the Wavelet Transform . . . . . . . . . 45514.1.5 Inverse Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . 45614.1.6 Noise Reduction Technique . . . . . . . . . . . . . . . . . . . . . . . . . 457

14.2 Discrete Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46014.2.1 Discrete Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . 46014.2.2 Complete Orthonormal Wavelets . . . . . . . . . . . . . . . . . . . . 46214.2.3 Multiresolution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 46314.2.4 Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 46414.2.5 Constructing an Orthonormal Basis . . . . . . . . . . . . . . . . . . 46614.2.6 Two-Scale Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46714.2.7 Constructing the Mother Wavelet . . . . . . . . . . . . . . . . . . . 46914.2.8 Multiresolution Representation . . . . . . . . . . . . . . . . . . . . . . 471

14.3 Fast Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47614.3.1 Generalized Two-Scale Relations . . . . . . . . . . . . . . . . . . . . 47614.3.2 Decomposition Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 47814.3.3 Reconstruction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Part V Differential Equations

15 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48315.1 Concepts of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

15.1.1 Definition of Ordinary Differential Equations . . . . . . . . . . 48315.1.2 Explicit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48415.1.3 Implicit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48515.1.4 General and Particular Solutions . . . . . . . . . . . . . . . . . . . . 48615.1.5 Singular Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48815.1.6 Integral Curve and Direction Field . . . . . . . . . . . . . . . . . . 489

15.2 Existence Theorem for the First-Order ODE . . . . . . . . . . . . . . . . 49115.2.1 Picard Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49115.2.2 Properties of Successive Approximations . . . . . . . . . . . . . 49315.2.3 Existence Theorem and Lipschitz Condition . . . . . . . . . . 49515.2.4 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49715.2.5 Remarks on the Two Theorems . . . . . . . . . . . . . . . . . . . . . 498

15.3 Sturm–Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50015.3.1 Sturm–Liouville Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 50015.3.2 Conversion into a Sturm–Liouville Equation . . . . . . . . . . 50115.3.3 Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50215.3.4 Required Boundary Condition . . . . . . . . . . . . . . . . . . . . . . 50315.3.5 Reality of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

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16 System of Ordinary Differential Equations . . . . . . . . . . . . . . . . . 50916.1 Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

16.1.1 Systems of the First-Order ODEs . . . . . . . . . . . . . . . . . . . . 50916.1.2 Column-Vector Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51016.1.3 Reducing the Order of ODEs . . . . . . . . . . . . . . . . . . . . . . . 51016.1.4 Lipschitz Condition in Vector Spaces . . . . . . . . . . . . . . . . . 512

16.2 Linear System of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51316.2.1 Basic Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51316.2.2 Vector Space of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51416.2.3 Fundamental Systems of Solutions . . . . . . . . . . . . . . . . . . . 51616.2.4 Wronskian for a System of ODEs . . . . . . . . . . . . . . . . . . . . 51716.2.5 Liouville Formula for a Wronskian . . . . . . . . . . . . . . . . . . . 51816.2.6 Wronskian for an nth-Order Linear ODE . . . . . . . . . . . . . 51916.2.7 Particular Solution of an Inhomogeneous System . . . . . . 522

16.3 Autonomous Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52516.3.1 Autonomous System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52516.3.2 Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52616.3.3 Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52716.3.4 Stability of a Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . 52716.3.5 Linear Autonomous System . . . . . . . . . . . . . . . . . . . . . . . . . 528

16.4 Classification of Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53016.4.1 Improper Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53016.4.2 Saddle Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53116.4.3 Proper Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53216.4.4 Spiral Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53316.4.5 Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53316.4.6 Limit Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

16.5 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 53616.5.1 Van der Pol Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

17 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53917.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

17.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53917.1.2 Subsidiary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54017.1.3 Linear and Homogeneous PDEs . . . . . . . . . . . . . . . . . . . . . 54017.1.4 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54117.1.5 Second-Order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54317.1.6 Classification of Second-Order PDEs . . . . . . . . . . . . . . . . . 544

17.2 The Laplacian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54617.2.1 Maximum and Minimum Theorem . . . . . . . . . . . . . . . . . . . 54617.2.2 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54817.2.3 Symmetric Properties of the Laplacian . . . . . . . . . . . . . . . 548

17.3 The Diffusion Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55017.3.1 The Diffusion Equations in Bounded Domains . . . . . . . . 55017.3.2 Maximum and Minimum Theorem . . . . . . . . . . . . . . . . . . . 55117.3.3 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

Contents XIX

17.4 The Wave Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55217.4.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55217.4.2 Homogeneous Wave Equations . . . . . . . . . . . . . . . . . . . . . . 55317.4.3 Inhomogeneous Wave Equations . . . . . . . . . . . . . . . . . . . . . 55517.4.4 Wave Equations in Finite Domains . . . . . . . . . . . . . . . . . . 556

17.5 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 55917.5.1 Wave Equations for Vibrating Strings . . . . . . . . . . . . . . . . 55917.5.2 Diffusion Equations for Heat Conduction . . . . . . . . . . . . . 561

Part VI Tensor Analyses

18 Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56518.1 Rotation of Coordinate Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

18.1.1 Tensors and Coordinate Transformations . . . . . . . . . . . . . 56518.1.2 Summation Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56618.1.3 Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . 56718.1.4 Rotation of Coordinate Axes . . . . . . . . . . . . . . . . . . . . . . . . 56818.1.5 Orthogonal Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56918.1.6 Matrix Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57018.1.7 Determinant of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

18.2 Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57618.2.1 Cartesian Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57618.2.2 A Vector and a Geometric Arrow . . . . . . . . . . . . . . . . . . . . 57718.2.3 Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57818.2.4 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

18.3 Pseudotensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58018.3.1 Improper Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58018.3.2 Pseudovectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58218.3.3 Pseudotensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58418.3.4 Levi–Civita Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

18.4 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58618.4.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . 58618.4.2 Contraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58718.4.3 Outer and Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . 58718.4.4 Symmetric and Antisymmetric Tensors . . . . . . . . . . . . . . . 58918.4.5 Equivalence of an Antisymmetric Second-Order

Tensor to a Pseudovector . . . . . . . . . . . . . . . . . . . . . . . . . . . 59018.4.6 Quotient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59218.4.7 Quotient Theorem for Two-Subscripted Quantities . . . . . 593

18.5 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 59618.5.1 Inertia Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59618.5.2 Tensors in Electromagnetism in Solids . . . . . . . . . . . . . . . 59818.5.3 Electromagnetic Field Tensor . . . . . . . . . . . . . . . . . . . . . . . 59818.5.4 Elastic Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

XX Contents

19 Non-Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60119.1 Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

19.1.1 Local Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60119.1.2 Reciprocity Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60319.1.3 Transformation Law of Covariant Basis Vectors . . . . . . . 60419.1.4 Transformation Law of Contravariant Basis Vectors . . . . 60619.1.5 Components of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60619.1.6 Components of a Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60819.1.7 Mixed Components of a Tensor . . . . . . . . . . . . . . . . . . . . . 60919.1.8 Kronecker Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

19.2 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61119.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61119.2.2 Geometric Role of Metric Tensors . . . . . . . . . . . . . . . . . . . 61219.2.3 Riemann Space and Metric Tensor . . . . . . . . . . . . . . . . . . . 61319.2.4 Elements of Arc, Area, and Volume . . . . . . . . . . . . . . . . . . 61419.2.5 Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61619.2.6 Representation of Basis Vectors in Derivatives . . . . . . . . 61719.2.7 Index Lowering and Raising . . . . . . . . . . . . . . . . . . . . . . . . 617

19.3 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62119.3.1 Derivatives of Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 62119.3.2 Nontensor Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62219.3.3 Properties of Christoffel Symbols . . . . . . . . . . . . . . . . . . . . 62319.3.4 Alternative Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

19.4 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62719.4.1 Covariant Derivatives of Vectors . . . . . . . . . . . . . . . . . . . . . 62719.4.2 Remarks on Covariant Derivatives . . . . . . . . . . . . . . . . . . . 62819.4.3 Covariant Derivatives of Tensors . . . . . . . . . . . . . . . . . . . . 62919.4.4 Vector Operators in Tensor Form . . . . . . . . . . . . . . . . . . . . 630

19.5 Applications in Physics and Engineering . . . . . . . . . . . . . . . . . . . . 63419.5.1 General Relativity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 63419.5.2 Riemann Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63519.5.3 Energy–Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 63619.5.4 Einstein Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

20 Tensor as Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63920.1 Vector as a Linear Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

20.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63920.1.2 Vector Spaces Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64020.1.3 Vector Spaces of Linear Functions . . . . . . . . . . . . . . . . . . . 64020.1.4 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64120.1.5 Equivalence Between Vectors and Linear Functions . . . . 642

20.2 Tensor as Multilinear Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64320.2.1 Direct Product of Vector Spaces . . . . . . . . . . . . . . . . . . . . . 64320.2.2 Multilinear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64420.2.3 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

Contents XXI

20.2.4 General Definition of Tensors . . . . . . . . . . . . . . . . . . . . . . . 64520.3 Components of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

20.3.1 Basis of a Tensor Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64620.3.2 Transformation Laws of Tensors . . . . . . . . . . . . . . . . . . . . . 64820.3.3 Natural Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64820.3.4 Inner Product in Tensor Language . . . . . . . . . . . . . . . . . . . 65120.3.5 Index Lowering and Raising in Tensor Language . . . . . . . 652

Part VII Appendixes

A Proof of the Bolzano–Weierstrass Theorem . . . . . . . . . . . . . . . . 657A.1 Limit Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657A.2 Cantor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658A.3 Bolzano–Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

B Dirac δ Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661B.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661B.2 Representation as a Limit of Function . . . . . . . . . . . . . . . . . . . . . . 662B.3 Remarks on Representation 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

C Proof of Weierstrass Approximation Theorem . . . . . . . . . . . . . 667

D Tabulated List of Orthonormal Polynomial Functions . . . . . . 671

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677