Essentials of Math Methods for Physicists
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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.
Table of Contents
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
Product details
- No. of pages: 960
- Language: English
- Copyright: © Academic Press 1966
- Published: January 1, 1966
- Imprint: Academic Press
- eBook ISBN: 9781483225623
About the Authors
Hans Weber
Affiliations and Expertise
University of Virginia, USA
George B. Arfken
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