Introduction to Mathematical Physics Methods and Concepts 2nd Edition by Chun Wa Wong 0199641390 9780199641390

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ISBN 10: 0199641390 

ISBN 13: 9780199641390

Author: Chun Wa Wong

Mathematical physics provides physical theories with their logical basis and the tools for drawing conclusions from hypotheses. Introduction to Mathematical Physics explains to the reader why and how mathematics is needed in the description of physical events in space. For undergraduates in physics, it is a classroom-tested textbook on vector analysis, linear operators, Fourier series and integrals, differential equations, special functions and functions of a complex variable. Strongly correlated with core undergraduate courses on classical and quantum mechanics and electromagnetism, it helps the student master these necessary mathematical skills. It contains advanced topics of interest to graduate students on relativistic square-root spaces and nonlinear systems. It contains many tables of mathematical formulas and references to useful materials on the Internet. It includes short tutorials on basic mathematical topics to help readers refresh their mathematical knowledge. An appendix on Mathematica encourages the reader to use computer-aided algebra to solve problems in mathematical physics. A free Instructor’s Solutions Manual is available to instructors who order the book for course adoption.

Table of contents:

1 Vectors and fields in space

1.1 Concepts of space

1.2 Vectors in space

1.3 Permutation symbols

1.4 Vector differentiation of a scalar field

1.5 Vector differentiation of a vector field

1.6 Path-dependent scalar and vector integrations

1.7 Flux, divergence and Gauss’s theorem

1.8 Circulation, curl and Stokes’s theorem

1.9 Helmholtz’s theorem

1.10 Orthogonal curvilinear coordinate systems

1.11 Vector differential operators in orthogonal curvilinear coordinate systems

Appendix 1 Tables of mathematical formulas

2 Transformations, matrices and operators

2.1 Transformations and the laws of physics

2.2 Rotations in space: Matrices

2.3 Determinant and matrix inversion

2.4 Homogeneous equations

2.5 The matrix eigenvalue problem

2.6 Generalized matrix eigenvalue problems

2.7 Eigenvalues and eigenvectors of Hermitian matrices

2.8 The wave equation

2.9 Displacement in time and translation in space: Infinitesimal generators

2.10 Rotation operators

2.11 Matrix groups

Appendix 2 Tables of mathematical formulas

3 Relativistic square-root spaces[sup(*)]

3.1 Introduction

3.2 Special relativity and Lorentz transformations

3.3 Relativistic kinematics and the mass–energy equivalence

3.4 Quaternions

3.5 Dirac equation, spinors and matrices

3.6 Symmetries of the Dirac equation[sup(*)]

3.7 Weyl and Majorana spinors, symmetry violations[sup(*)]

3.8 Lorentz group

3.9 Cartan spinors and spin transformations in square-root space

3.10 Dyadics

3.11 Cartesian tensors

3.12 Tensor analysis

Appendix 3 Tables of mathematical formulas

4 Fourier series and Fourier transforms

4.1 Wave–particle duality: Quantum mechanics

4.2 Fourier series

4.3 Fourier coeffcients and Fourier-series representation

4.4 Complex Fourier series and the Dirac Δ function

4.5 Fourier transform

4.6 Green function and convolution

4.7 Heisenberg’s uncertainty principle

4.8 Conjugate variables and operators in wave mechanics

4.9 Generalized Fourier series and Legendre polynomials

4.10 Orthogonal functions and orthogonal polynomials

4.11 Mean-square error and mean-square convergence

4.12 Convergence of Fourier series

4.13 Maxwell equations in Fourier spaces

4.14 3D Fourier transforms: Helmholtz decomposition theorem

Appendix 4A: Short table of Fourier cosine series

Appendix 4B: Short table of Fourier sine series

Appendix 4C: Short table of Fourier transforms

Appendix 4D: Short table of 3D and 4D Fourier transforms

Appendix 4E: Tables of mathematical formulas

5 Differential equations in physics

5.1 Introduction

5.2 Linear differential equations

5.3 First-order differential equations

5.4 Second-order linear differential equations

5.5 The second homogeneous solution and an inhomogeneous solution

5.6 Green functions

5.7 Series solution of the homogeneous second-order linear differential equation

5.8 Differential eigenvalue equations and orthogonal functions

5.9 Partial differential equations of physics

5.10 Separation of variables and eigenfunction expansions

5.11 Boundary and initial conditions

5.12 Separation of variables for the Laplacian

5.13 Green functions for partial differential equations

Appendix 5 Tables of mathematical formulas

6 Nonlinear systems[sup(*)]

6.1 Introduction

6.2 Nonlinear instabilities

6.3 Logistic map and chaos

6.4 Strange attractor

6.5 Driven dissipative linear pendula

6.6 Chaos in parametrically driven dissipative nonlinear pendula

6.7 Solitons

6.8 Traveling kinks

6.9 Nonlinear superposition of solitons

6.10 More general methods for multi-solitons[sup(*)]

Appendix 6 Tables of mathematical formulas

7 Special functions

7.1 Introduction

7.2 Generating function for Legendre polynomials

7.3 Hermite polynomials and the quantum oscillator

7.4 Orthogonal polynomials

7.5 Classical orthogonal polynomials[sup(*)]

7.6 Associated Legendre polynomials and spherical harmonics

7.7 Bessel functions

7.8 Sturm-Liouville equation and eigenfunction expansions

Appendix 7 Tables of mathematical formulas

8 Functions of a complex variable

8.1 Introduction

8.2 Functions of a complex variable

8.3 Multivalued functions and Riemann surfaces

8.4 Complex differentiation: Analytic functions and singularities

8.5 Complex integration: Cauchy integral theorem and integral formula

8.6 Harmonic functions in the plane

8.7 Taylor series and analytic continuation

8.8 Laurent series

8.9 Residues

8.10 Complex integration: Calculus of residues

8.11 Poles on the contour and Green functions

8.12 Laplace transform

8.13 Inverse Laplace transform

8.14 Construction of functions and dispersion relations

8.15 Asymptotic expansions[sup(*)]

Appendix 8 Tables of mathematical formulas

Appendix A: Tutorials

A.1 Complex algebra

A.2 Vectors

A.3 Simple and partial differentiations

A.4 Simple and multiple integrals

A.5 Matrices and determinants

A.6 Infinite series

A.7 Exponential functions

Appendix B: Mathematica and other computer algebra systems

Appendix C: Computer algebra (CA) with Mathematica

C.1 Introduction to CA

C.2 Equation solvers

C.3 Drawing figures and graphs

C.4 Number-intensive calculations

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Tags: Chun Wa Wong, Introduction, Mathematical, Physics