Mathematical Methods for Physicists by Arfken and Weber

[Greg Bernhardt]

• Author: George B. Arfken (Author), Hans J. Weber (Author), Frank E. Harris (Author)
• Title: Mathematical Methods for Physicists
• Amazon Link: https://www.amazon.com/dp/0123846544/?tag=pfamazon01-20
• Prerequisities: Calculus 1,2,3

Table of Contents:

[LIST]
[*] Preface
[*] Vector Analysis
[LIST]
[*] Definitions, Elementary Approach
[*] Rotation of the Coordinate Axes
[*] Scalar or Dot Product
[*] Vector or Cross Product
[*] Triple Scalar Product, Triple Vector Product
[*] Gradient, ∇
[*] Divergence, ∇
[*] Curl, ∇x
[*] Successive Applications of ∇
[*] Vector Integration
[*] Gauss' Theorem
[*] Stokes' Theorem
[*] Potential Theory
[*] Gauss' Law, Poisson's Equation
[*] Dirac Delta Function
[*] Helmholtz's Theorem
[*] Additional Readings
[/LIST]
[*] Vector Analysis in Curved Coordinates and Tensors
[LIST]
[*] Orthogonal Coordinates in R^3
[*] Differential Vector Operators
[*] Special Coordinate Systems: Introduction
[*] Circular Cylinder Coordinates
[*] Spherical Polar Coordinates
[*] Tensor Analysis
[*] Contraction, Direct Product
[*] Quotient Rule
[*] Pseudotensors, Dual Tensors
[*] General Tensors
[*] Tensor Derivative Operators
[*] Additional Readings
[/LIST]
[*] Determinants and Matrices
[LIST]
[*] Determinants
[*] Matrices
[*] Orthogonal Matrices
[*] Hermitian Matrices, Unitary Matrices
[*] Diagonalization of Matrices
[*] Normal Matrices
[*] Additional Readings
[/LIST]
[*] Group Theory
[LIST]
[*] Introduction to Group Theory
[*] Generators of Continuous Groups
[*] Orbital Angular Momentum
[*] Angular Momentum Coupling
[*] Homogeneous Lorentz Group
[*] Lorentz Covariance of Maxwell's Equations
[*] Discrete Groups
[*] Differential Forms
[*] Additional Readings
[/LIST]
[*] Infinite Series
[LIST]
[*] Fundamental Concepts
[*] Convergence Tests
[*] Alternating Series
[*] Algebra of Series
[*] Series of Functions
[*] Taylor's Expansion
[*] Power Series
[*] Elliptic Integrals
[*] Bernoulli Numbers, Euler-Maclaurin Formula
[*] Asymptotic Series
[*] Infinite Products
[*] Additional Readings
[/LIST]
[*] Functions of a Complex Variable I Analytic Properties, Mapping
[LIST]
[*] Complex Algebra
[*] Cauchy-Riemann Conditions
[*] Cauchy's Integral Theorem
[*] Cauchy's Integral Formula
[*] Laurent Expansion
[*] Singularities
[*] Mapping
[*] Conformal Mapping
[*] Additional Readings
[/LIST]
[*] Functions of a Complex Variable II
[LIST]
[*] Calculus of Residues
[*] Dispersion Relations
[*] Method of Steepest Descents
[*] Additional Readings
[/LIST]
[*] The Gamma Function (Factorial Function)
[LIST]
[*] Definitions, Simple Properties
[*] Digamma and Polygamma Functions
[*] Stirling's Series
[*] The Beta Function
[*] Incomplete Gamma Function
[*] Additional Readings
[/LIST]
[*] Differential Equations
[LIST]
[*] Partial Differential Equations
[*] First-Order Differential Equations
[*] Separation of Variables
[*] Singular Points
[*] Series Solutions—Frobeniusy Method
[*] A Second Solution
[*] Nonhomogeneous Equation—Green's Function
[*] Heat Flow, or Diffusion, PDF
[*] Additional Readings
[/LIST]
[*] Sturm-Liouville Theory—Orthogonal Functions
[LIST]
[*] Self-Adjoint ODEs
[*] Hermitian Operators
[*] Gram-Schmidt Orthogonalization
[*] Completeness of Eigenfunctions
[*] Green's Function—Eigenfunction Expansion
[*] Additional Readings
[/LIST]
[*] Bessel Functions
[LIST]
[*] Bessel Functions of the First Kind, J_v(x)
[*] Orthogonality
[*] Neumann Functions
[*] Hankel Functions
[*] Modified Bessel Functions, I_v(x) and K_v(x)
[*] Asymptotic Expansions
[*] Spherical Bessel Functions
[*] Additional Readings
[/LIST]
[*] Legendre Functions
[LIST]
[*] Generating Function
[*] Recurrence Relations
[*] Orthogonality
[*] Alternate Definitions
[*] Associated Legendre Functions
[*] Spherical Harmonics
[*] Orbital Angular Momentum Operators
[*] Addition Theorem for Spherical Harmonics
[*] Integrals of Three Y's
[*] Legendre Functions of the Second Kind
[*] Vector Spherical Harmonics
[*] Additional Readings
[/LIST]
[*] More Special Functions
[LIST]
[*] Hermite Functions
[*] Laguerre Functions
[*] Chebyshev Polynomials
[*] Hypergeometric Functions
[*] Confluent Hypergeometric Functions
[*] Mathieu Functions
[*] Additional Readings
[/LIST]
[*] Fourier Series
[LIST]
[*] General Properties
[*] Advantages, Uses of Fourier Series
[*] Applications of Fourier Series
[*] Properties of Fourier Series
[*] Gibbs Phenomenon
[*] Discrete Fourier Transform
[*] Fourier Expansions of Mathieu Functions
[*] Additional Readings
[/LIST]
[*] Integral Transforms
[LIST]
[*] Integral Transforms
[*] Development of the Fourier Integral
[*] Fourier Transforms—Inversion Theorem
[*] Fourier Transform of Derivatives
[*] Convolution Theorem
[*] Momentum Representation
[*] Transfer Functions
[*] Laplace Transforms
[*] Laplace Transform of Derivatives
[*] Other Properties
[*] Convolution (Faltungs) Theorem
[*] Inverse Laplace Transform
[*] Additional Readings
[/LIST]
[*] Integral Equations
[LIST]
[*] Introduction
[*] Integral Transforms, Generating Functions
[*] Neumann Series, Separable (Degenerate) Kernels
[*] Hilbert-Schmidt Theory
[*] Additional Readings
[/LIST]
[*] Calculus of Variations
[LIST]
[*] A Dependent and an Independent Variable
[*] Applications of the Euler Equation
[*] Several Dependent Variables
[*] Several Independent Variables
[*] Several Dependent and Independent Variables
[*] Lagrangian Multipliers
[*] Variation with Constraints
[*] Rayleigh-Ritz Variational Technique
[*] Additional Readings
[/LIST]
[*] Nonlinear Methods and Chaos
[LIST]
[*] Introduction
[*] The Logistic Map
[*] Sensitivity to Initial Conditions and Parameters
[*] Nonlinear Differential Equations
[*] Additional Readings
[/LIST]
[*] Probability
[LIST]
[*] Definitions, Simple Properties
[*] Random Variables
[*] Binomial Distribution
[*] Poisson Distribution
[*] Gauss'Normal Distribution
[*] Statistics
[*] Additional Readings
[/LIST]
[*] General References
[*] Index
[/LIST]

 

[Jorriss]

I actually think Arfken & Weber is a very good book. It's not a 'mathematical physics' book aiming to teach the structure of physical theories - it's just a methods text. If you need to do an integral and forgot a method, look it up in here, and to that end, I think it serves its purpose well.

 

[marcusl]

I also like Arfken. It's a good reference book, and because examples are taken from all areas of physics, I always learn something new when I use it.

 

[rhombusjr]

It probably works well if you already know/knew the material. From experience, I don't recommend trying to learn the techniques for the first time with this book. The explanations are too brief.

 

[jesse73]

A reference book, not great to learn from but good to recall

 

[jedishrfu]

I've found some mistakes in the book, but one of which stumped me for a awhile:

chapter 1 vector diagram showing the three axes of XYZ space and a vector with a projection onto the XY plane and with three arcs to indicate the vector cosines. The mistake is one arc goes from the x-axis to the dashlined vector projection and not to the vector itself.

Earlier editions had two diagrams here so some copyeditor/artist combined them and introduced the mistake (I think from 4th edition to the present, I haven't seen the 7th edition yet)

 

[Astronuc]

I've found some mistakes in the book, but one of which stumped me for a awhile:

chapter 1 vector diagram showing the three axes of XYZ space and a vector with a projection onto the XY plane and with three arcs to indicate the vector cosines. The mistake is one arc goes from the x-axis to the dashlined vector projection and not to the vector itself.

Earlier editions had two diagrams here so some copyeditor/artist combined them and introduced the mistake (I think from 4th edition to the present, I haven't seen the 7th edition yet)

The figure (1.5 in 6th edition) is fixed in the 7th edition (Figure 1.9), at least according to the book review on Google books.

http://books.google.com/books?id=JOpHkJF-qcwC&printsec=frontcover#v=onepage&q&f=false

The 7th edition has been restructured as follows:

1 Mathematical Preliminaries
1.1 Infinite Series
1.2 Series of Functions
1.3 Binomial Theorm
1.4 Mathematical Induction
1.5 Operations on Series Expansions of Functions
1.6 Some Important Series
1.7 Vectors
1.8 Complex Numbers and Functions
1.9 Derivatives and Extrema
1.10 Evaluation of Integrals
1.11 Dirac Delta Function

2 Matrices and Determinants
2.1 Determinants
2.2 Matrices

3 Vector Analysis

4 Tensors and Differential Forms

5 Vector Spaces

6 Eigenvalue Problems

7 Ordinary Differential Equations

8 Sturm-Liouville Theory

9 Partial Differential Equations

10 Green's Functions

11 Complex Variable Theory

12 Further Topics in Analysis

13 Gamma Function

14 Bessel Functions

15 Legendre Functions

16 Angular Momentum

17 Group Theory

18 More Special Functions

19 Fourier Series

20 Integral Transforms

21 Integral Equations

22 Calculus of Variations

23 Probability and Statistics

Index

 

[Orodruin]

There are still some mistakes in the 7th edition though. I just discovered one which is obviously not a typo but an author mistake. All because I already knew the answer due to computing it when someone asked a question here on PF.

 

[vanhees71]

Great! Could you point out the mistake? Perhaps it's good to correct it in our personal copies. I also learned the hard way that you shouldn't trust formulae without checking them yourself .

 

[Orodruin]

Great! Could you point out the mistake? Perhaps it's good to correct it in our personal copies. I also learned the hard way that you shouldn't trust formulae without checking them yourself .

What? And deprive you of the joy of finding out for yourself?

Spoiler

Page 1085: "If the path is not required to be a great circle, any deviation from Path 2 will increase the length." This is also restated page 1103: "because even the longer path is of minimum length relative to small deformations."

The truth is that the longer great circle between the two points is a saddle point. Aug 22, 2014

 

[Jesca]

The description of the integral Fourier transform in chapter 20 (7th edition) is with an inverted signal, could you confirm?

 

[MidgetDwarf]

I found it very clear coming from a pure math background, and wanting to read about how some of these maths are used in an applied setting.

 

[jbergman]

I used this book as an undergrad. It's section on vector calculus is ok, but I've found it hopeless for learning anything else.