A Course in Mathematical Methods for Physicists, Russell L.Herman, CRC Press/Taylor & Francis, 2014. $89.95 paper (747 pp.). ISBN 978-1-4665-8467-9

In A Course in Mathematical Methods for Physicists, Russell Herman advocates for and successfully practices an example- and application-based introduction to the subject. In that regard, the book is a welcome and refreshing addition to a rich body of literature. For two decades, I have taught a math methods course to advanced undergraduates and incoming graduate students who need a refresher; it uses a more advanced text, Mathematical Methods for Physicists (6th edition, Academic Press, 2005) by George Arfken and Hans Weber.

As Herman, a mathematical physicist, states in the prologue, his book is intended for “undergraduate students who have completed a year-long introductory course in physics,” so a two-semester course based on it should fit into the sophomore or junior year of a typical physics undergraduate curriculum. The text is designed for and will easily accommodate such a course. Engineers and chemistry majors, too, would benefit from taking such an intermediate-level course, perhaps even more so than from a higher-level one. Our own department at Howard University is considering a mid-level math methods course, and I would definitely recommend this textbook as well suited.

A Course in Mathematical Methods for Physicists includes plenty of interesting worked-out examples, many of them quite realistic, and uses them to introduce concepts in a reasonable progression. The book offers a good selection of material, most of which I would expect to be included in a math methods text. I was also pleasantly surprised by the inclusion of nonlinear dynamics in chapter 4 and the very nice discussion of solving inhomogeneous equations in chapter 6.

In chapter 3, Herman introduces and develops linear algebra (matrices, eigenvalues, eigenvectors) and finds nice applications of 3 × 3 and bigger matrices. Then, somewhat unexpectedly given the chapter title, “Linear Algebra,” the presented applications seamlessly turn to systems of linear differential equations and their solutions. The connection between linear algebra and differential equations provided by that fusion is indeed useful, and it highlights the benefits of Herman’s emphasis on examples and applications. It is a pity that the analysis does not go beyond two linear differential equations for two functions, although Herman does find a successful and interesting application of the analysis in the following chapter.

An application-based approach inevitably seems to cause a few closely related concepts (or even one and the same concept) to turn up at distant places in the text. For example, the Bessel equation and its solutions turn up in chapter 6 to introduce the Fourier–Bessel series as an example of a nonsinusoidal harmonic. The reader will have to wait until chapter 11 to be reintroduced to the Bessel equation and its solutions; there Herman provides much more detail, a proper physical motivation, and several concrete and amusing applications to solving the wave equation and the diffusion equation.

I would have expected some of the introduced concepts to be employed further. For example, the gamma function is introduced briefly in chapter 6 and used in chapter 8, but Herman misses the opportunity to show how to use it to compute Gaussian integrals of the form ∫dxxn · exp{–a · x2}, which frequently occur in statistical and quantum physics. Finally, and I have the same qualm with several other textbooks on mathematical physics, Herman introduces the index notation in the vector analysis section of chapter 9, but essentially ignores the difference between covariant and contravariant quantities; the distinction is made only in the last two pages of the chapter.

Although the subject of mathematical methods has inspired many valuable texts, Herman’s approach, motivated by the physics applications, is novel, seldom used by other authors. The myriad well-chosen worked-out examples and other strengths have earned my firm endorsement, and they convincingly outweigh the few shortcomings.

Tristan Hübsch is a professor in the department of physics and astronomy at Howard University in Washington, DC. His research is in mathematical physics and high-energy particle theory.