# Mathematical Techniques and Physical Applications

## 1st Edition

**Authors:**J Killingbeck

**eBook ISBN:**9780323142823

**Imprint:**Academic Press

**Published Date:**1st January 1971

**Page Count:**736

## Description

Mathematical Techniques and Physical Applications provides a wide range of basic mathematical concepts and methods, which are relevant to physical theory. This book is divided into 10 chapters that cover the different branches of traditional mathematics.

This book deals first with the concept of vector, matrix, and tensor analysis. These topics are followed by discussions on several theories of series relevant to physics; the fundamentals of complex variables and analytic functions; variational calculus for presenting the basic laws of many branches of physics; and the applications of group representations. The final chapters explore some partial and integral equations and derivatives of physics, as well as the concept and application of probability theory.
Physics teachers and students will greatly appreciate this book.

## Table of Contents

Preface

Comment on Notation

1. Vector Analysis

1.1. Scalars, Tensors, and Vectors

1.2. Scalar, Vector, and Tensor Fields

1.3. Vector Components, Unit Vectors, Right-Handed Cartesian Axes

1.4. Vector Sums and Products

A. Vector Addition

B. Scalar Product of Vectors

C. Vector Product

D. Matrix Notation for Scalar and Vector Products

E. Triple Scalar and Vector Products

F. Reciprocal Systems of Vectors

1.5. Derivatives of a Vector or Vector Field

A. Gradient Operation and Nabla Operator

B. Potential Vectors, Line Integrals

C. Divergence of a Vector Field

D. Curl of a Vector Field

E. Important Results of Vector Analysis

1.6. Integral Theorems

A. Gauss Divergence Theorem

B. Green's Theorem

C. Stokes' Curl Theorem

D. Field Discontinuities, Surface Curl, and Divergence

1.7. Dyadic Formalism

A. Vector Product and Lie Algebras

1.8. Orthogonal Curvilinear Coordinates

A. Divergence, Gradient, and Curl

B. Common Coordinate Systems

1.9. Uses of Vector Analysis

A. Energy Density of an Electromagnetic Field and the Pointing Vector

B. Electromagnetic Waves

C. Convective Derivative Term in Fluid Theory

D. Electric and Magnetic Dipoles

1.10. Further Examples Involving the Vector Product

A. The Frenet Formulas of Differential Geometry

B. The Vector Potential in Magnetic Theory

C. Additivity of Angular Momentum

D. The Description of Fluid Flow Using a Rotating Coordinate System

2. Matrices

2.1. Simultaneous Linear Equations and Matrix Algebra

2.2. Some Common Types of Matrix

2.3. Inverse of a Matrix, Determinant

A. Singular Matrices as Divisors of Zero

2.4. Theorems Concerning Matrix Products

A. Reversal Rule

B. Determinant of a Product Matrix

C. Triple Products, Families of Equivalent Matrices

2.5. Eigenvectors and Eigenvalues of a Matrix

A. Sum Rules

B. Eigenvalues and Eigencolumns of Hermitian and Unitary Matrices

C. Linear Independence

D. Inhomogeneous Problem: An Orthogonality Criterion for Hermitian Matrices

2.6. Matrices as Representations of Linear Operators

A. Invariant Spaces, Basis Vectors

B. Change of Basis

C. Invariants Associated with a Matrix

2.1. Application of Matrix Theory to Physical Problems

A. Jacobian of a Transformation

B. Homogeneous Strain

C. Normal Modes

D. Lorentz Transformations

E. Matrices in Quantum Mechanics

F. Dirac Relativistic Theory of the Electron

G. Slater Determinantal Wavefunction for Electrons

H. Density Matrix in Quantum Statistical Mechanics

I. Continuous Matrices, Dirac Delta Function

J. Green's Functions

K. Hilbert Spaces

L. Schmidt Orthogonalization Procedure

M. Indefinite Metric Formalism

N. Metric Spaces

O. Schwartz Inequality

3. Tensor Analysis

3.1. Cartesian Tensors

A. Scalars and Vectors

B. Second-Rank Cartesian Tensors as Operators

C. Higher Rank Cartesian Tensors, Summation Convention

D. Symmetric and Antisymmetric Tensors

E. Contraction of Tensors, Inner Products

F. Comment on Translational Transformations

G. Spherical Tensors

H. Transformations in Four Dimensions

I. Principal Axes

3.2. Tensors in Nonorthogonal Frames

A. Covariant and Contravariant Vector Components

B. Second-Rank Tensors

C. Line Element, Fundamental Tensor

D. Raising and Lowering of Indices, The Tensor gij

3.3. General Tensors

A. Covariant and Contravariant Vectors

B. Fundamental Tensor gij and Its Symmetry

C. Summary

D. Eddington's Theorem

E. The "Curling" Operation

3.4. The Christoffel Symbols

A. Length of a Vector and the Angle between Two Vectors

3.5. Length of a Curve, Geodesies

3.6. Covariant Derivatives

A. Alternative Approach to the Covariant Derivative

3.7. The Determinant |g|, Tensor Densities

3.8. Tensor Form of Gradient, Divergence, and Curl

3.9. Curvature Tensor

3.10. Theory of Elasticity

3.11. Lorentz Covariance of Maxwell's Equations

3.12. A Summary of Tensor Theory

A. Groups of Transformations

B. Riemann Metric Formulation of Geometry

C. Affine Connections and Parallel Displacement

4. Sequences and Series

4.1. Sequences, Cauchy Sequences, Convergence

4.2. Series, Absolute and Conditional Convergence

4.3. Convergence Tests for Series

A. Alternating Series Test

B. Comparison Test

C. Ratio Test

D. Root Test

4.4. Multiplication and Addition of Series

4.5. Sequences and Series of Functions, Uniform Convergence

4.6. Radius of Convergence of a Series, Term-by-Term Differentiation and Integration

4.7. Dirichlet Conditions for Fourier Series

A. Comment on Euclidian Spaces

4.8. Exponential Function

4.9. Results Involving Integrals

4.10. Series in Physical Theory

A. Pade Approximants

4.11. Convergence of Iterative Processes

4.12. Perturbation Theory

4.13. Partial Summation Procedures

A. Example Involving a Repeated Fraction

5. Complex Variables and Analytic Functions

5.1. Complex Numbers and Polynomial Equations

5.2. Argand Diagram

5.3. de Moivre's Theorem

A. Anharmonic Oscillator Example

5.4. Complex Numbers in Physical Problems

5.5. Differentiation, Analytic Functions

5.6. Taylor Series for the Complex Variable

A. Cauchy Theorem

B. Cauchy Formula

C. Derivatives of f(z)

D. Uniqueness

5.7. Analytic Continuation

A. The Gamma Function

5.8. Singularities, Poles, and Residues

A. Branch Points, Many-Valued Functions

B. Branch Points and Analytic Continuation

5.9. Quaternions

A. Entire and Meromorphic Functions

5.10. Principal Part of an Integral, Kramers-Kronig Relations

5.11. Fourier Transforms

A. Jordan's Lemma

5.12. Truncated Fourier Series for Real Variables

A. Gibbs' Phenomenon

5.13. Laplace Transform

A. Yukawian Potentials

5.14. Laplace's Equation

A. General Use of Analytic Functions in Treating Laplace's Equation

5.15. Use of Closed Contour Integrals in Physics

A. Example of Nonanalytic Function in Physics

B. Partition Function in Statistical Mechanics, Saddle-Point Integration

C. Spin Eigenfunctions in Quantum Mechanics

D. Example of a Many-Valued Function in Solid-State Physics

E. Two-Dimensional Periodicity, Elliptic Functions

F. Complex Numbers in Alternating Current Theory

G. Electrical Ladder Network

H. Complex Energy Values in Quantum Mechanics for Nonstationary States

I. Delta Function and Energy Conservation in Quantum Mechanics

J. Patterson Function of X-Ray Crystallography

K. Examples Involving the Kramers-Kronig Relations

6. Variational Calculus

6.1. Stationary and Extreme Values of Ordinary Functions

6.2. Functionals and Functional Derivatives

A. Generalization to a Greater Number of Variables

B. Additive Functionals

6.3. Variation with Auxiliary Conditions, Lagrange Multipliers

6.4. Variational Principles in Mechanics

A. Generalized Coordinates, Motion with Constraints

B. Least-Action Principle

C. Hamilton Canonical Formalism

D. Conservation Rules

E. Poisson Brackets and Operator Commutators

F. Four-Dimensional Variational Principle

G. Canonical Transformations, Hamilton-Jacobi Equation

6.5. Schrödinger Equation and Related Variational Principles

A. Rayleigh-Ritz Procedure

6.6. Continuous Fields, Wave Equation

A. Variational Principle from Electrostatic Theory

6.7. Bending of Beams

A. An Orthogonality Criterion

6.8. Example from Electrodynamics

6.9. Invariance Properties of Functionals

6.10. Equation of a Geodesic

A. Einstein's Equations of the Gravitational Field

B. Stationary Phase Method, Feynman Path Integral Approach to Quantum Mechanics

6.11. Functionals of a Plane Curve, the Curl Operation, and Contour Integration

7. Group Representations

7.1. Group Postulates and Multiplication Table

7.2. Comments on the Generalized Product Concept

7.3. Isomorphism, Representations

7.4. Subgroups, Lagrange Theorem, Cyclic Groups

7.5. Regular Representation, Permutation Groups

7.6. Classes, Character of a Matrix Representation

7.7. Change of Basis

7.8. Reduction of a Representation, Irreducible Representations

A. Some Important Results Concerning Irreducible Representations

B. Reducibility and Complete Reducibility

7.9. Projection Operators

7.10. Direct Product Representations and Groups

7.11. Factor Groups

7.12. Rotation Group R3

7.13. Infinitesimal Rotation Operators

A. Half-Integral Representations, Pauli Matrices

B. Gn2, Double Groups

C. Four-Dimensional Rotation Group

7.14. Symmetry Considerations in the Theory of Equations

7.15. Basic Theorem Concerning Eigenvalue Equations

A. Galilean and Lorentz Groups

B. Role of Boundary Conditions

7.16. Determinant and Permanent of a Matrix as Projections, Bosons and Fermions

7.17. Crystal-Field Theory, Point Groups

A. Dh as a Permutation Group

B. Rotation Group Notation

7.18. Translational Symmetry, Bloch Functions

7.19. Gn3, Group, Elementary Particles, Atomic Spectroscopy

A. Quarks

B. I-Spin

7.20. Selection Rules and the Wigner-Eckart Theorem

A. Recoupling Transformations, 6j Symbols

B. Configuration Interaction and Selection Rules for the Be Atom

7.21. Young Tableaux and Their Properties

7.22. Dimensional Analysis and Scaling Transformations

A. A Comment on Schur's Lemma

B. Other Uses of Group Theory.

8. Some Differential Equations of Physics

8.1. Some Topics Involving Partial Derivatives

A. Pfaffians

B. Euler's Theorem on Homogeneous Functions

C. Homogeneous Functions and Irreducible Representations of the Rotation Group

8.2. Systems of First-Order Total Differential Equations

A. First Integrals, Liouville Equation

B. The Simple Pendulum as an Example

8.3. Second-Order and Higher Order Equations

A. Homogeneous Equations

B. Initial Conditions for First-Order and Second-Order Differential Equations

C. Inhomogeneous Equations

D. Method of Variation of Parameters

E. van der Pol Equation

8.4. Sturm-Liouville Equation

A. A Link with the Schmidt Orthogonalization Process

8.5. Hypergeometric Series and Hypergeometric Equation

A. A Comment on Power Series Solutions

8.6. Confluent Hypergeometric Series

8.7. Classification of Second-Order Partial Differential Equations

8.8. Some Important Forms of the Schrödinger Equation

A. One-Dimensional Harmonic Oscillator

B. Electron Moving in Spherically Symmetric Field

8.9. Floquet's Theorem, Mathieu Functions

8.10. Potential Theory

A. Generation of Solutions of the Laplace Equation

B. Generating Function for the Legendre Polynomials

C. Legendre Differential Equation

D. Associated Legendre Equation: Spherical Harmonics

E. Dirichlet Problem, Uniqueness Theorems

F. An Application to Geomagnetism

8.11. Wave Equation and Its Solutions

A. The (JWKB) Approximation

8.12. Dispersion Relations, Group Velocity

8.13. Coupled Wave Equations

A. General Discussion

8.14. Scattering and Diffraction of Waves

A. Reflection of Waves

8.15. Surface Waves

A. Physical Considerations

8.16. Retarded Potentials, Hertz Oscillator

8.17. Equation of Continuity, Diffusion Phenomena

A. Classical Fluid Flow

B. Quantum Mechanical Probability Current

C. Thermal Conduction in a Solid

D. Diffusion

E. Euler Equation and Navier-Stokes Equation

F. Vorticity

8.18. Liouville Equation of Statistical Mechanics

A. Schrödinger Equation and Hamilton-Jacobi Equation

8.19. Theory of Orbits in General Relativity

8.20. Matrix Methods and Systems of Linear Equations

A. Rate Equations for a Two-Level System

B. Radioactive Decay Chains

C. Stability Theory of Liapunov

8.21. Lie Algebra Approach to the Special Functions

9. Integral Equations

9.1. Some Important Types of Integral Equations

A. A Metric for Operators

9.2. Neumann Series Solution

9.3. Fredholm Solution

9.4. Schmidt-Hilbert Approach and the Hilbert Bilinear Expansion

A. Representations of the Delta Function

9.5. Boundary Conditions for Integral Equations

A. A One-Dimensional Example

9.6. Further Comments on Green's Functions

A. Green's Functions as Kernels of Integral Equations; Boundary Conditions.

9.7. Response Functions, the Causality Condition, and the Kramers-Kronig Relations

9.8. Hartree-Fock Equations as Integro-Differential Equations

9.9. Born-Green Equation of the Classical Statistical Theory of Fluids

9.10. Neumann Series for the Schrödinger Equation

9.11. Second Quantization, The Propagator Green's Functions

9.12. Equation of Motion Method

9.13. Green's Function Approach to Solid State Band Theory

9.14. Feynman Path Integral Approach to the Propagator Green's Function

10. Probability Theory

10.1. Frequential and a Priori Probabilities

10.2. Discrete and Continuous Distributions

10.3. Averages

10.4. Some Standard Integrals of Statistical Mechanics

10.5. Random Walk in One Dimension

10.6. Binomial and Poisson Distributions

10.7. Gaussian Distribution, Least-Squares Fitting Procedures

10.8. Moments of a Distribution

10.9. Correlation: Conditional Probability

10.10. Types of Counting in Quantum Statistical Mechanics

10.11. Uncertainty Principle and Operator Formalism in Quantum Mechanics

10.12. Stochastic Processes, the Wiener-Khinchin Theorem

10.13. Cosine Transforms

10.14. Markov Chains and Stochastic Matrices

10.15. Correlation Functions and Linear Response Coefficients for Physical Systems

10.16. Entropy Concept in Probability Theory

10.17. Boltzmann Transport Equation

10.18. Monte-Carlo Method for the Ising Model

10.19. Continuous Markov Processes and the Generalized Liouville Equation

Literature Survey

Index

## Details

- No. of pages:
- 736

- Language:
- English

- Copyright:
- © Academic Press 1971

- Published:
- 1st January 1971

- Imprint:
- Academic Press

- eBook ISBN:
- 9780323142823