Essentials of Math Methods for Physicists

Essentials of Math Methods for Physicists aims to guide the student in learning the mathematical language used by physicists by leading them through worked examples and then practicing problems. The pedagogy is that of introducing concepts, designing and refining methods and practice them repeatedly in physics examples and problems. Geometric and algebraic approaches and methods are included and are more or less emphasized in a variety of settings to accommodate different learning styles of students.

Comprised of 19 chapters, this book begins with an introduction to the basic concepts of vector algebra and vector analysis and their application to classical mechanics and electrodynamics. The next chapter deals with the extension of vector algebra and analysis to curved orthogonal coordinates, again with applications from classical mechanics and electrodynamics. These chapters lay the foundations for differential equations, variational calculus, and nonlinear analysisin later discussions. High school algebra of one or two linear equations is also extended to determinants and matrix solutions of general systems of linear equations, eigenvalues and eigenvectors, and linear transformations in real and complex vector spaces. The book also considers probability and statistics as well as special functions and Fourier series. Historical remarks are included that describe some physicists and mathematicians who introduced the ideas and methods that were perfected by later generations to the tools routinely used today.

This monograph is intended to help undergraduate students prepare for the level of mathematics expected in more advanced undergraduate physics and engineering courses.

Preface

1 Vector Analysis

1.1 Elementary Approach

1.2 Scalar or Dot Product

1.3 Vector or Cross Product

1.4 Triple Scalar Product, Triple Vector Product

1.5 Gradient, ∇

1.6 Divergence, ∇

1.7 Curl, ∇x

1.8 Successive Applications of ∇

1.9 Vector Integration

1.10 Gauss' Theorem

1.11 Stokes' Theorem

1.12 Potential Theory

1.13 Gauss' Law, Poisson's Equation

1.14 Dirac Delta Function

2 Vector Analysis in Curved Coordinates and Tensors

2.1 Special Coordinate Systems: Introduction

2.2 Circular Cylinder Coordinates

2.3 Orthogonal Coordinates

2.4 Differential Vector Operators

2.5 Spherical Polar Coordinates

2.6 Tensor Analysis

2.7 Contraction, Direct Product

2.8 Quotient Rule

2.9 Dual Tensors

3 Determinants and Matrices

3.1 Determinants

3.2 Matrices

3.3 Orthogonal Matrices

3.4 Hermitian Matrices, Unitary Matrices

3.5 Diagonalization of Matrices

4 Group Theory

4.1 Introduction to Group Theory

4.2 Generators of Continuous Groups

4.3 Orbital Angular Momentum

4.4 Homogeneous Lorentz Group

5 Infinite Series

5.1 Fundamental Concepts

5.2 Convergence Tests

5.3 Alternating Series

5.4 Algebra of Series

5.5 Series of Functions

5.6 Taylor's Expansion

5.7 Power Series

5.8 Elliptic Integrals

5.9 Bernoulli Numbers, Euler-Maclaurin Formula

5.10 Asymptotic Series

6 Functions of a Complex Variable I 389

6.1 Complex Algebra

6.2 Cauchy-Riemann Conditions

6.3 Cauchy's Integral Theorem

6.4 Cauchy's Integral Formula

6.5 Laurent Expansion

6.6 Mapping

6.7 Conformal Mapping

7 Functions of a Complex Variable II

7.1 Singularities

7.2 Calculus of Residues

7.3 Method of Steepest Descents

8 Differential Equations

8.1 Introduction

8.2 First Order Differential Equations

8.3 Second Order Differential Equations

8.4 Singular Points

8.5 Series Solutions-Frobenius's Method

8.6 A Second Solution

8.7 Numerical Solutions

8.8 Introduction to Partial Differential Equations

8.9 Separation of Variables

9 Sturm-Liouville Theory—Orthogonal Functions

9.1 Self-Adjoint ODEs

9.2 Hermitian Operators

9.3 Gram-Schmidt Orthogonalization

9.4 Completeness of Eigenfunctions

10 The Gamma Function (Factorial Function)

10.1 Definitions, Simple Properties

10.2 Digamma and Polygamma Functions

10.3 Stirling's Series

10.4 The Incomplete Gamma Function and Related Functions

11 Legendre Polynomials and Spherical Harmonics

11.1 Introduction

11.2 Recurrence Relations and Special Properties

11.3 Orthogonality

11.4 Alternate Definitions of Legendre Polynomials

11.5 Associated Legendre Functions

12 Bessel Functions

12.1 Bessel Functions of the First Kind Jv(x)

12.2 Neumann Functions, Bessel Functions of the Second Kind

12.3 Asymptotic Expansions

12.4 Spherical Bessel Functions

13 Hermite and Laguerre Polynomials

13.1 Hermite Polynomials

13.2 Laguerre Functions

14 Fourier Series

14.1 General Properties

14.2 Advantages, Uses of Fourier Series

14.3 Complex Fourier Series

14.4 Properties of Fourier Series

15 Integral Transforms

15.1 Introduction, Definitions

15.2 Fourier Transform

15.3 Development of the Inverse Fourier Transforms

15.4 Fourier Transforms-Inversion Theorem

15.5 Fourier Transform of Derivatives

15.6 Convolution Theorem

15.7 Momentum Representation

15.8 Laplace Transforms

15.9 Laplace Transform of Derivatives

15.10 Other Properties

15.11 Convolution or Faltungs Theorem

15.12 Inverse Laplace Transform

16 Partial Differential Equations

16.1 Examples of PDEs and Boundary Conditions

16.2 Heat Flow or Diffusion PDE

16.3 Inhomogeneous PDE-Green's Function

17 Probability

17.1 Definitions, Simple Properties

17.2 Random Variables

17.3 Binomial Distribution

17.4 Poisson Distribution

17.5 Gauss' Normal Distribution

17.6 Statistics

18 Calculus of Variations

18.1 A Dependent and an Independent Variable

18.2 Several Dependent Variables

18.3 Several Independent Variables

18.4 Several Dependent and Independent Variables

18.5 Lagrangian Multipliers: Variation with Constraints

18.6 Rayleigh-Ritz Variational Technique

19 Nonlinear Methods and Chaos

19.1 Introduction

19.2 The Logistic Map

19.3 Sensitivity to Initial Conditions and Parameters

19.4 Nonlinear Differential Equations

Appendix: Real Zeros of a Function

Index