Mathematical Methods in the Physical Sciences

Citation

Boas, M. L. (2005). Mathematical Methods in the Physical Sciences (3rd ed.). Wiley.

Chapter Summary

The book has 15 chapters and 864 pages, followed by references, answers to selected problems and an index. The publisher describes its purpose as helping students develop, in a short time, a basic competence in each of the many areas of mathematics needed in advanced physics, chemistry and engineering courses, with enough depth for a solid foundation but without detailed proofs that suit mathematics students better. Chapters end with sets of "Miscellaneous Problems".

Chapter 1: Infinite Series, Power Series

Begins with the geometric series and moves through convergence tests for series of positive terms, alternating and conditionally convergent series, power series and their interval of convergence, and techniques for obtaining expansions. It also covers accuracy of series approximations, some uses of series and a short section on using a computer.

Chapter 2: Complex Numbers

Real and imaginary parts, the complex plane, complex algebra, complex infinite series, Euler's formula, powers and roots, the exponential, trigonometric and hyperbolic functions, logarithms and inverse functions, with a section of applications.

Chapter 3: Linear Algebra

The longest chapter (106 pages). It runs from matrices, row reduction and determinants through vectors, lines and planes, matrix operations and linear operators, linear dependence, special matrices and linear vector spaces, to eigenvalues, eigenvectors and diagonalization with its applications. It ends with a brief introduction to groups and a section on general vector spaces. The publisher notes that matrix diagonalization was moved here from Chapter 10 in the third edition, with more detail on linear vector spaces.

Chapters 4–6: Partial Differentiation; Multiple Integrals; Vector Analysis

Multivariable calculus: total differentials, the chain rule, maxima and minima including Lagrange multipliers, change of variables and Leibniz' rule; double and triple integrals with Jacobians and surface integrals; and vector analysis, from triple products and the gradient to line integrals, Green's theorem, and the divergence and Stokes' theorems.

Chapter 7: Fourier Series and Transforms

Periodic functions, Fourier coefficients, the Dirichlet conditions, complex form, even and odd functions, an application to sound, Parseval's theorem and Fourier transforms. Per the publisher, Fourier integrals were moved here in the third edition.

Chapter 8: Ordinary Differential Equations

Separable and first-order linear equations, second-order linear equations with constant coefficients, the Laplace transform and its use in solving equations, convolution, the Dirac delta function and a brief introduction to Green functions. The Laplace transform and the expanded delta-function treatment are new placements in this edition.

Chapters 9–10: Calculus of Variations; Tensor Analysis

The Euler equation, the brachistochrone, Lagrange's equations and isoperimetric problems; then Cartesian tensors, the inertia tensor, the Kronecker delta and Levi-Civita symbol, pseudovectors, curvilinear coordinates and non-Cartesian tensors. The publisher says the tensor treatment was expanded in this edition.

Chapters 11–12: Special Functions; Series Solutions of Differential Equations; Legendre, Bessel, Hermite, and Laguerre Functions

The gamma and beta functions, the error function, asymptotic series, Stirling's formula and elliptic integrals; then Legendre's equation and polynomials, the method of Frobenius, Bessel functions and Hermite and Laguerre functions with ladder operators.

Chapters 13–15: Partial Differential Equations; Functions of a Complex Variable; Probability and Statistics

Laplace's, diffusion, wave and Poisson's equations in rectangular, cylindrical and spherical settings; analytic functions, contour integrals, Laurent series, the residue theorem and conformal mapping; and probability, random variables, binomial, normal and Poisson distributions, and statistics for experimental measurements.

Key Concepts

Chapters 1–3: Series, Complex Numbers, Linear Algebra

• Convergence of series: the tests that decide whether an infinite series is usable.
• Euler's formula: the link between complex exponentials and trigonometric functions.
• Diagonalization: finding eigenvalues and eigenvectors to simplify matrix problems.

Chapters 4–8: Calculus, Vectors, Fourier, Differential Equations

• Lagrange multipliers: maximizing or minimizing a quantity under a constraint.
• Divergence theorem and Stokes' theorem: integral relations of vector analysis.
• Fourier series and transforms: expansion of periodic functions and its extension to transforms.
• Laplace transform and Green functions: tools for solving differential equations.

Chapters 9–15: Variations, Tensors, Special Functions, PDEs, Complex Variables, Probability

• Euler equation: the condition for an integral to be stationary.
• Cartesian tensors: quantities such as the inertia tensor that transform in definite ways.
• Legendre and Bessel functions: solutions of the equations that arise in separated PDEs.
• Residue theorem: evaluating definite integrals through complex contours.

Critical Analysis

Strengths

• Wide coverage in one volume. The contents range from series and linear algebra to partial differential equations, complex analysis and probability. Chapter-end sections of miscellaneous problems give practice across topics.
• Methods first, by design. The publisher states that the book emphasizes methods over applications, aims for depth without heavy proofs, and is "not a recipe book". Section titles such as "Applications of Series", "Applications of Diagonalization" and "Some Applications of Conformal Mapping" show where applications are placed.
• Revisions with a reason. The third edition reorders material: diagonalization moves to the linear algebra chapter, the treatment of basis functions is carried through the Fourier, differential-equation, series-solution and PDE chapters, and the Laplace transform and delta function sit in the differential-equations chapter. Computer algebra systems are mentioned, with their usefulness and pitfalls, throughout.
• Useful for later courses. Sections on the inertia tensor, Lagrange's equations, ladder operators and the Schrödinger equation connect the mathematics to mechanics and quantum mechanics.

Limitations

• Light on proof. By the publisher's own description, detailed proofs are left out, so readers who want rigorous justification of convergence, transforms or distributions need a mathematics text.
• Breadth limits depth. Several topics get brief treatment, for example groups in a single seven-page section and the Laplace transform in three short sections. A graduate course will need a fuller reference.
• Little computational content. No chapter deals with numerical methods; computing appears as remarks and as a short "Using a Computer" section in Chapter 1.
• Topics outside the book. Nonlinear dynamics and chaos do not appear in the contents.

Where It Fits in the Course

Boas is the broad, one-volume methods text for students in the middle of an undergraduate physics sequence. Albano's Physics Today review (2001) of the first edition of Hassani's Mathematical Methods judged it most nearly comparable to Boas, then in her second edition, in target audience, pace and field coverage; Hassani delays complex numbers until Chapter 18 and has chapters on tensors, integral transforms, chaos and probability, whereas Boas puts complex numbers in Chapter 2.

Before Boas, Riley and Hobson's Foundation Mathematics for the Physical Sciences builds calculus, vectors and matrices from school-level mathematics, and Schey's Div, Grad, Curl, and All That gives an informal route into vector calculus that helps with Chapter 6. After it, Arfken, Weber and Harris's Mathematical Methods for Physicists is the graduate-level reference, and Strogatz's Nonlinear Dynamics and Chaos covers a subject that Boas leaves out.

How to Use This Book

The publisher addresses students who have had a two- or three-semester introductory calculus course. The book then builds the other tools itself, starting differential equations and linear algebra from introductions.

Start with Chapters 1 and 2: series and complex numbers recur in later chapters, such as the complex form of Fourier series and the chapter on functions of a complex variable. Chapters 3–6 supply the linear algebra and multivariable calculus used in physics courses and make a reasonable first term. Chapters 7, 8, 12 and 13 (Fourier methods, differential equations, special functions and PDEs) are the core for electromagnetism and quantum mechanics courses. Chapters 9, 10, 11, 14 and 15 can be taken as needed alongside mechanics, relativity, quantum mechanics or laboratory work. Solve problems as you read; answers to selected problems are at the back, and treat the book's remarks on the pitfalls of computer algebra as prompts to check machine output by hand.