Physical mathematics

779 Pages·2019·10.809 MB·English

by  Cahill K

PHYSICAL MATHEMATICS
Second edition
Unique in its clarity, examples, and range, Physical Mathematics explains simply
and succinctly the mathematics that graduate students and professional physicists
need to succeed in their courses and research. The book illustrates the mathematics
with numerous physical examples drawn from contemporary research. This second
edition has new chapters on vector calculus, special relativity and artificial intelli-
gence, and many new sections and examples. In addition to basic subjects such as
linear algebra, Fourier analysis, complex variables, differential equations, Bessel
functions, and spherical harmonics, the book explains topics such as the singular
value decomposition, Lie algebras and group theory, tensors and general relativity,
the central limit theorem and Kolmogorov’s theorems, Monte Carlo methods of
experimental and theoretical physics, Feynman’s path integrals, and the standard
model of cosmology.
K E V I N C A H I L Lis Professor of Physics and Astronomy at the University of New
Mexico. He has carried out research at NIST, Saclay, Ecole Polytechnique, Orsay,
Harvard University, NIH, LBL, and SLAC, and has worked in quantum optics,
quantum field theory, lattice gauge theory, and biophysics. Physical Mathematics
is based on courses taught by the author at the University of New Mexico and at
Fudan University in Shanghai.
PHYSICAL MATHEMATICS

Second edition

KEVIN CAHILL
University of New Mexico
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education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781108470032
DOI: 10.1017/9781108555814
First edition ±c K. Cahill 2013
Second edition ±c Cambridge University Press 2019
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2013
Reprinted with corrections 2014
Second edition 2019
Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall, 2019
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Cahill, Kevin, 1941– author.
Title: Physical mathematics / Kevin Cahill (University of New Mexico).
Description: Second edition. | Cambridge ; New York, NY : Cambridge
University Press, 2019. | Includes bibliographical references and index.
Identifiers: LCCN 2019008214 | ISBN 9781108470032 (alk. paper)
Subjects: LCSH: Mathematical physics. | Mathematical physics – Textbooks. |
Mathematics – Study and teaching (Higher)
Classification: LCC QC20 .C24 2019 | DDC 530.15–dc23
LC record available at https://lccn.loc.gov/2019008214
ISBN 978-1-108-47003-2 Hardback
Additional resources for this publication at www.cambridge.org/Cahill2ed
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
To Ginette, Michael, Sean, Peter, Micheon,
Danielle, Rachel, Mia, James, Dylan,
Christopher, and Liam
Brief Contents

Preface page xxi
1 Linear Algebra 1
2 Vector Calculus 84
3 Fourier Series 93
4 Fourier and Laplace Transforms 128
5 Infinite Series 158
6 Complex-Variable Theory 185
7 Differential Equations 248
8 Integral Equations 334
9 Legendre Polynomials and Spherical Harmonics 343
10 Bessel Functions 365
11 Group Theory 390
12 Special Relativity 451
13 General Relativity 466
14 Forms 536
15 Probability and Statistics 564
16 Monte Carlo Methods 632
vii
viii Brief Contents
17 Artificial Intelligence 643
18 Order, Chaos, and Fractals 647
19 Functional Derivatives 661
20 Path Integrals 669
21 Renormalization Group 718
22 Strings 727
References 737
Index 744
Contents
Preface page xxi
1 Linear Algebra 1
1.1 Numbers 1
1.2 Arrays 2
1.3 Matrices 4
1.4 Vectors 7
1.5 Linear Operators 10
1.6 Inner Products 12
1.7 Cauchy–Schwarz Inequalities 15
1.8 Linear Independence and Completeness 16
1.9 Dimension of a Vector Space 17
1.10 Orthonormal Vectors 18
1.11 Outer Products 19
1.12 Dirac Notation 20
1.13 Adjoints of Operators 25
1.14 Self-Adjoint or Hermitian Linear Operators 26
1.15 Real, Symmetric Linear Operators 27
1.16 Unitary Operators 27
1.17 Hilbert Spaces 29
1.18 Antiunitary, Antilinear Operators 30
1.19 Symmetry in Quantum Mechanics 30
1.20 Determinants 30
1.21 Jacobians 38
1.22 Systems of Linear Equations 40
1.23 Linear Least Squares 41
1.24 Lagrange Multipliers 41
1.25 Eigenvectors and Eigenvalues 43
ix
x Contents
1.26 Eigenvectors of a Square Matrix 44
1.27 A Matrix Obeys Its Characteristic Equation 48
1.28 Functions of Matrices 49
1.29 Hermitian Matrices 54
1.30 Normal Matrices 59
1.31 Compatible Normal Matrices 61
1.32 Singular-Value Decompositions 65
1.33 Moore–Penrose Pseudoinverses 70
1.34 Tensor Products and Entanglement 72
1.35 Density Operators 76
1.36 Schmidt Decomposition 77
1.37 Correlation Functions 78
1.38 Rank of a Matrix 80
1.39 Software 80
Exercises 81
2 Vector Calculus 84
2.1 Derivatives and Partial Derivatives 84
2.2 Gradient 85
2.3 Divergence 86
2.4 Laplacian 88
2.5 Curl 89
Exercises 92
3 Fourier Series 93
3.1 Fourier Series 93
3.2 The Interval 96
3.3 Where to Put the 2pi’s 97
3.4 Real Fourier Series for Real Functions 98
3.5 Stretched Intervals 102
3.6 Fourier Series of Functions of Several Variables 103
3.7 Integration and Differentiation of Fourier Series 104
3.8 How Fourier Series Converge 104
3.9 Measure and Lebesgue Integration 108
3.10 Quantum-Mechanical Examples 110
3.11 Dirac’s Delta Function 117
3.12 Harmonic Oscillators 120
3.13 Nonrelativistic Strings 122
3.14 Periodic Boundary Conditions 123
Exercises 125
Contents xi
4 Fourier and Laplace Transforms 128
4.1 Fourier Transforms 128
4.2 Fourier Transforms of Real Functions 131
4.3 Dirac, Parseval, and Poisson 132
4.4 Derivatives and Integrals of Fourier Transforms 136
4.5 Fourier Transforms of Functions of Several Variables 141
4.6 Convolutions 142
4.7 Fourier Transform of a Convolution 144
4.8 Fourier Transforms and Green’s Functions 145
4.9 Laplace Transforms 146
4.10 Derivatives and Integrals of Laplace Transforms 148
4.11 Laplace Transforms and Differential Equations 149
4.12 Inversion of Laplace Transforms 150
4.13 Application to Differential Equations 150
Exercises 156
5 Infinite Series 158
5.1 Convergence 158
5.2 Tests of Convergence 159
5.3 Convergent Series of Functions 161
5.4 Power Series 162
5.5 Factorials and the Gamma Function 163
5.6 Euler’s Beta Function 168
5.7 Taylor Series 168
5.8 Fourier Series as Power Series 169
5.9 Binomial Series 170
5.10 Logarithmic Series 172
5.11 Dirichlet Series and the Zeta Function 172
5.12 Bernoulli Numbers and Polynomials 174
5.13 Asymptotic Series 175
5.14 Fractional and Complex Derivatives 177
5.15 Some Electrostatic Problems 178
5.16 Infinite Products 181
Exercises 182
6 Complex-Variable Theory 185
6.1 Analytic Functions 185
6.2 Cauchy–Riemann Conditions 186
6.3 Cauchy’s Integral Theorem 187
6.4 Cauchy’s Integral Formula 193