# Book:Frederick W. Byron, Jr./Mathematics of Classical and Quantum Physics

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## Frederick W. Byron, Jr. and Robert W. Fuller: *Mathematics of Classical and Quantum Physics*

Published $1992$, **Dover Publications, Inc.**

- ISBN 0-486-67164-X.

A republication of *Mathematics of Classical and Quantum Physics: Volume One* and *Mathematics of Classical and Quantum Physics: Volume Two* in one volume.

### Subject Matter

### Contents

- Preface

- VOLUME ONE

**1 Vectors in Classical Physics**- Introduction

- 1.1 Geometric and Algebraic Definitions of a Vector
- 1.2 The Resolution of a Vector into Components
- 1.3 The Scalar Product
- 1.4 Rotation of the Coordinate System: Orthogonal Transformations
- 1.5 The Vector Product
- 1.6 A Vector Treatment of Classical Orbit Theory
- 1.7 Differential Operations on Scalar and Vector Fields
- *1.8 Cartesian-Tensors

**2 Calculus of Variations**- Introduction

- 2.1 Some Famous Problems
- 2.2 The Euler-Lagrange Equation
- 2.3 Some Famous Solutions
- 2.4 Isoperimetric Problems - Constraints
- 2.5 Application to Classical Mechanics
- 2.6 Extremization of Multiple Integrals
- *2.7 Invariance Principles and Noether's Theorem

**3 Vectors and Matrices**- Introduction

- 3.1 Groups, Fields, and Vector Spaces
- 3.2 Linear Independence
- 3.3 Bases and Dimensionality
- 3.4 Isomorphisms
- 3.5 Linear Transformations
- 3.6 The Inverse of a Linear Transformation
- 3.7 Matrices
- 3.8 Determinants
- 3.9 Similarity Transformations
- 3.10 Eigenvalues and Eigenvectors
- *3.11 The Kronecker Product

**4 Vector Spaces in Physics**- Introduction

- 4.1 The Inner Product
- 4.2 Orthogonality and Completeness
- 4.3 Complete Orthonormal Sets
- 4.4 Self-Adjoint (Hermitian and Symmetric) Transformations
- 4.5 Isometries - Unitary and Orthogonal Transformations
- 4.6 The Eigenvalues and Eigenvectors of Self-Adjoint and Isometric Transformations
- 4.7 Diagonalization
- 4.8 On the Solvability of Linear Equations
- 4.9 Minimum Principles
- 4.10 Normal Modes
- 4.11 Perturbation Theory - Nondegenerate Case
- *4.12 Perturbation Theory - Degenerate Case

**5 Hilbert Space-complete Orthonormal Sets of Functions**- Introduction

- 5.1 Function Space and Hilbert Space
- 5.2 Complete Orthonormal Sets of Functions
- 5.3 The Dirac $\delta$-Function
- 5.4 Weierstrass's Theorem: Approximation by Polynomials
- 5.5 Legendre Polynomials
- 5.6 Fourier Series
- 5.7 Fourier Integrals
- 5.8 Spherical Harmonics and Associated Legendre Functions
- 5.9 Hermite Polynomials
- 5.10 Sturm-Liouville Systems - Orthogonal Polynomials
- 5.11 A Mathematical Formulation of Quantum Mechanics

- VOLUME TWO

**6 Elements and Applications of the Theory of Analytic Functions**- Introduction

- 6.1 Analytic Functions - The Cauchy-Riemann Conditions
- 6.2 Some Basic Analytic Functions
- 6.3 Complex Integration - The Cauchy-Goursat Theorem
- 6.4 Consequences of Cauchy's Theorem
- 6.5 Hilbert Transforms and the Cauchy Principal Value
- 6.6 An Introduction to Dispersion Relations
- 6.7 The Expansion of an Analytic Function in a Power Series
- 6.8 Residue Theory - Evaluation of Real Definite Integrals and Summation of Series
- 6.9 Applications to Special Functions and Integral Representations

**7 Green's Functions**- Introduction

- 7.1 A New Way to Solve Differential Equations
- 7.2 Green's Functions and Delta Functions
- 7.3 Green's Functions in One Dimension
- 7.4 Green's Functions in Three Dimensions
- 7.5 Radial Green's Functions
- 7.6 An Application to the Theory of Diffraction
- 7.7 Time-dependent Green's Functions: First Order
- 7.8 The Wave Equation

**8 Introduction to Integral Equations**- Introduction

- 8.1 Iterative Techniques - Linear Integral Operators
- 8.2 Norms of Operators
- 8.3 Iterative Techniques in a Banach Space
- 8.4 Iterative Techniques for Nonlinear Equations
- 8.5 Separable Kernels
- 8.6 General Kernels of Finite Rank
- 8.7 Completely Continuous Operators

**9 Integral Equations in Hilbert Space**- Introduction

- 9.1 Completely Continuous Hermitian Operators
- 9.2 Linear Equations and Perturbation Theory
- 9.3 Finite-Rank Techniques for Eigenvalue Problems
- 9.4 The Fredholm Alternative For Completely Continuous Operators
- 9.5 The Numerical Solution of Linear Equations
- 9.6 Unitary Transformations

**10 Introduction to Group Theory**- Introduction

- 10.1 An Inductive Approach
- 10.2 The Symmetric Groups
- 10.3 Cosets, Classes, and Invariant Subgroups
- 10.4 Symmetry and Group Representations
- 10.5 Irreducible Representations
- 10.6 Unitary Representations, Schur's Lemmas, and Orthogonality Relations
- 10.7 The Determination of Group Representations
- 10.8 Group Theory in Physical Problems

- General Bibliography

- Index to Volume One

- Index to Volume Two