Mathematical Methods for Physicists: A Comprehensive Guide

Citation

Arfken, G. B., Weber, H. J., & Harris, F. E. (2012). Mathematical Methods for Physicists: A Comprehensive Guide (7th ed.). Academic Press.

Chapter Summary

The seventh edition has 23 chapters in about 1,200 pages, preceded by a preface with notes "To the Student", "What's New" and "Pathways through the Material". Each chapter ends with a list of additional readings. The publisher describes the edition as retaining the key features of the sixth, with a more careful balance of explanation, theory and examples, improved modular chapters, updated notation and a problem-solving approach. The sequence runs from series and vectors through linear algebra and differential equations to complex analysis, special functions, group theory, transforms, integral equations, the calculus of variations and probability.

Chapters 1–2: Mathematical Preliminaries; Determinants and Matrices

Chapter 1 gathers the prerequisites: infinite series and series of functions, the binomial theorem, mathematical induction, vectors, complex numbers, derivatives and extrema, evaluation of integrals and the Dirac delta function. Chapter 2 covers determinants and matrices.

Chapters 3–4: Vector Analysis; Tensors and Differential Forms

Vector properties, coordinate transformations and rotations, differential vector operators, integral theorems, potential theory and curvilinear coordinates. Chapter 4 moves to tensor analysis, pseudotensors and dual tensors, tensors in general coordinates, Jacobians, and differential forms with their differentiation and integration.

Chapters 5–6: Vector Spaces; Eigenvalue Problems

Function spaces, Gram–Schmidt orthogonalization, and operators (self-adjoint, unitary), with transformations and invariants. Chapter 6 treats eigenvalue equations, Hermitian problems and matrix diagonalization, and normal matrices.

Chapters 7–9: Ordinary Differential Equations; Sturm–Liouville Theory; Partial Differential Equations

First-order equations, constant-coefficient and second-order linear equations, Frobenius series solutions, inhomogeneous and nonlinear equations. Sturm–Liouville theory then covers Hermitian operators, eigenvalue problems and the variation method. The partial differential equation chapter handles first- and second-order equations, separation of variables, and the Laplace, Poisson, wave and heat-flow equations.

Chapter 10: Green's Functions

Green's functions for one-dimensional problems and for problems in two and three dimensions.

Chapters 11–13: Complex Variable Theory; Further Topics in Analysis; Gamma Function

Complex-variable theory from the Cauchy–Riemann conditions through the integral theorem and formula, Laurent expansions, singularities and the calculus of residues, used for definite integrals and sums. Chapter 12 collects orthogonal polynomials, Bernoulli numbers, the Euler–Maclaurin formula, asymptotic series, steepest descents and dispersion relations. Chapter 13 covers the gamma, digamma, beta and zeta functions and Stirling's series.

Chapters 14–16: Bessel Functions; Legendre Functions; Angular Momentum

Bessel functions of several kinds, spherical Bessel functions; Legendre polynomials, associated Legendre functions and spherical harmonics; and angular momentum operators, coupling, spherical tensors and vector spherical harmonics.

Chapters 17–18: Group Theory; More Special Functions

Group representations and symmetry in physics, discrete and continuous groups, the Lorentz group and the Lorentz covariance of Maxwell's equations, and space groups. Chapter 18 adds Hermite, Laguerre and Chebyshev functions, hypergeometric functions, the dilogarithm and elliptic integrals.

Chapters 19–23: Fourier Series; Integral Transforms; Integral Equations; Calculus of Variations; Probability and Statistics

Fourier series and the Gibbs phenomenon; Fourier and Laplace transforms with convolution theorems, signal-processing applications and the discrete Fourier transform; integral equations with Neumann series and Hilbert–Schmidt theory; the Euler equation and constrained variational problems; and probability, the binomial, Poisson and normal distributions, and statistics.

Key Concepts

• Series, vectors and complex numbers: the preliminary toolkit (Chapter 1).
• Tensors and differential forms: the coordinate-independent language of vector analysis (Chapter 4).
• Self-adjoint operators and eigenvalue problems: the linear-algebra structure behind much of physics (Chapters 5–6, 8).
• Frobenius series and Sturm–Liouville theory: solving linear differential equations and relating them to orthogonal functions (Chapters 7–8).
• Green's functions: solving inhomogeneous problems (Chapter 10).
• Calculus of residues: evaluating integrals and sums by complex-variable methods (Chapter 11).
• Special functions: gamma, Bessel, Legendre, Hermite, Laguerre and others (Chapters 13–15, 18).
• Group theory and the Lorentz group: symmetry (Chapter 17).
• Integral transforms: Fourier and Laplace transforms (Chapter 20).

Critical Analysis

Strengths

• Enormous range in one volume. The chapter list covers essentially the whole mathematics of graduate-level physics, including topics that shorter texts omit, such as differential forms, group theory with the Lorentz group, integral equations and dispersion relations. MAA.org's review called it a thorough handbook about mathematics useful in physics.
• Problem-solving orientation. The publisher says the book focuses on problem-solving skills and active learning, with numerous chapter problems, and clearly identified definitions, theorems and proofs. A review in Zentralblatt MATH highlighted the large number of problems and exercises.
• Modular chapters and guidance. The seventh edition advertises improved modular chapters and a "Pathways through the Material" section in the preface, which helps a reader choose which chapters to use.
• Lasting reference value. Each chapter ends with additional readings, and the sections are numerous and specific, so a reader can return to find a particular result.

Limitations

• Size and density. About 1,200 pages and 23 chapters is much more than a single course can use; the book works best as a reference from which chapters are selected.
• Not a first course. Chapter 1 is a compact review of series, vectors, complex numbers and integrals, and the site lists an intermediate mathematical methods course as a prerequisite; students without that background will find the pace steep.
• Little on numerical methods. The contents are devoted to analytic methods; there are no chapters on numerical analysis or computation, so students need separate resources for those.

Where It Fits in the Course

In the mathematical methods course, Arfken, Weber and Harris is the comprehensive graduate text and reference. Riley and Hobson's Foundation Mathematics for the Physical Sciences is a first-year mathematics book that builds calculus, vector, matrix and differential-equation tools from school level, and Schey's Div, Grad, Curl, and All That is a short informal text on vector calculus based on electrostatics; both come well before Arfken. Boas's Mathematical Methods in the Physical Sciences and Hassani's Mathematical Methods: For Students of Physics and Related Fields are undergraduate-level methods books, written so that students can use a method before meeting its full theory (Boas) or with each technique motivated by a physical problem (Hassani). Strogatz's Nonlinear Dynamics and Chaos is a separate subject, an introduction to nonlinear systems and chaos.

The usual route is Boas or Hassani first, then Arfken, Weber and Harris for the topics the course needs in more depth, or for graduate study.

How to Use This Book

The book is for graduate students and advanced undergraduates in physics who have completed calculus, linear algebra, differential equations and an intermediate methods course, and as a reference for researchers.

Start with the preface's pathways section to choose a route. A typical graduate sequence: Chapters 1–3 (preliminaries, matrices, vector analysis), Chapters 5–8 (vector spaces, eigenvalue problems, differential equations and Sturm–Liouville theory), Chapter 11 (complex variables), and then special functions (Chapters 13–15) and Fourier transforms (Chapters 19–20). Group theory (17), Green's functions (10), integral equations (21), calculus of variations (22) and probability (23) can be added as the course requires. Work the end-of-chapter problems, and consult the additional readings for deeper treatments.