The scope of this book is narrower than its title might imply, as its author makes clear from the start. The book is about some mathematics inspired by physics rather than about all the math a physicist needs to know. It covers topics important in traditional applied mathematics, especially transform methods, traveling waves and special functions.
Overall, it is a pleasant book to read, providing insight and substantial exercises. The level of exposition is consistently good for a graduate-level text: the right amount of detail is given, and there is a narrative line organizing the first half of the book. The book gives its prerequisites as “familiarity with linear algebra and functions of a complex variable.” This is accurate, noting that “linear algebra” here includes the notions of dual and quotient spaces, as well as eigenvectors and eigenvalues.
The first half of the book provides a detailed description of the theory of distributions, culminating with the Fourier and Laplace transforms. In order to carry this off, Schwartz introduces Lebesgue integration theory, although largely without proof. Despite the fact that some proofs are skipped, theorems are stated and applied rigorously.
Schwartz’s treatment of integration theory and distribution theory is graceful and sheds light on the physical motivation for the theory of distributions. His discussion of potential theory is also notable in this regard. One might expect this, since Schwartz won his Fields Medal for the theory of distributions, but mathematical and expository skill are not always paired as they are here.
In the second half of the book, there is a long chapter on the wave and heat equations. This chapter is organized by the physics involved instead of the mathematics. Wave-like phenomena, including transverse and longitudinal vibrations in different media, vibrating membranes, and waves in space, get the most attention. The relevant partial differential equations are derived, with a careful consideration of the physical hypotheses involved. The standard solution techniques are developed in more detail than in most texts, but not with full mathematical rigor, which would swamp the flow of ideas.
It is surprising, given the first part of the book, how little the theory of distributions is used in this chapter. In part, this is because the application of distribution theory to the heat equation occurs in an earlier chapter.
The final two chapters feel more like extended appendices than part of the text proper. Schwartz introduces numerous results about the gamma function and Bessel functions, many of which are used in the earlier chapters. He maintains the same level of exposition as in the earlier parts of the book, but physical applications are not discussed in these chapters.
John Curran is Assistant Professor of Mathematics at Eastern Michigan University, where he also coordinates the actuarial science program. He worked for a Wall Street firm for several years before obtaining his Ph.D. in applied mathematics from Brown University.