Marcus Pivato's Linear Partial Differential Equations and Fourier Theory should be on the shelf of every one who teaches mathematics or physics. Every third-semester calculus student should study the first chapter, where the the author has included excellent intuitive discussions of divergence and the gradient function with accessible examples from physics. The fifteen pages of examples from electromagnetics and thermodynamics in the first chapter of this book constitute a discussion which is conspicuously absent from the traditional calculus textbook. The remaining 43 pages contain discussion and examples from quantum theory as well. These pages should be included in any course on partial differential equations.

The text is an excellent blend of mathematical theory and physical examples. The development of the theory is justified and supported by physical examples. Pivato maintains mathematical rigor in all his discussions, yet he keeps every discussion grounded upon readily accessible physical applications from thermodynamics, quantum theory, engineering, and chemistry. The physics student can appreciate the mathematical rigor, and the mathematics student can understand the applications from physics.

Each chapter begins with a listing of sections of the book which contain prerequisite material for the current chapter. This makes it an excellent source of supplementary readings for any calculus or differential equations course. Furthermore, exercises which are used to advance the discussion or to develop essential notions are inserted into the reading rather than left at the ends of the chapters. These exercises are clearly indicated by symbols in the margin. The book also contains a reasonable number of problems at the end of each chapter.

Pivato's book, like any book, has its peculiarities. For one, it bills itself in the preface as a book which is "written for third-year undergraduate students in mathematics, physics, engineering, and other mathematical sciences." In the preface, the book purports that it is suitable for a 15-week course and that the only prerequisites are the standard three-semester calculus sequence and the customary undergraduate linear algebra course. I believe that Pivato is omitting that a good deal of mathematical maturity (usually fostered through a senior-level real analysis class—or two!) is also required. For example, one section of Chapter 10 is devoted to proving that the the set \(C^k_{per}[-\pi, \pi]\) of \(C^k\) functions on \([-\pi, \pi]\) with periodic boundary values is dense in \(L^2[-\pi , \pi]\)—a proof which requires the Cauchy-Schwarz Inequality and a discussion of convolutions. A number of the proofs in this section use "classic strateg[ies] in real analysis." The fact that the author attempts to define \(L^2[-\pi , \pi]\) through the Riemann integral instead of the Lebesgue integral does not mitigate the difficulty and laconism of this particular discussion. The author's claim that the book is based only upon the first two years of undergraduate mathematics reminds me of a jesting comment made by Professor Helmut Hofer when one of my fellow graduate students continued to prod him for information on a particular homework problem. Hofer replied to him that everything is "just a little calculus."

Chapter 10 is arguably the densest 29 pages of the book. It contains material which a student with two semesters of senior-level real analysis would find approachable but challenging. Yet, the appendices of the book contain material which is peculiarly elementary when compared to the examples, proofs, and theorems given in the chapters. For example, Appendix A deals with the elementary notions of set and function at a level presumed to be understood well before the student begins the first semester of calculus. Appendix C deals with elementary notions from complex numbers and functions \(f : \mathbb{C} \to \mathbb{C}\); this appendix even contains a discussion of how to add, multiply, and conjugate complex numbers. Appendix D reminds the reader of the cylindrical and spherical coordinate systems.

Indeed, many readers would appreciate these elementary reminders found in the appendices of the book; however, in my opinion, the very elementary nature of the material in the appendices stands in contradiction to the level of mathematical maturity required for the last sixteen chapters of the book.

I strongly recommend that faculty buy this book as a source of readings for numerous courses throughout the curriculum, and I would recommend that physicists and engineers wanting to enhance their understanding of the mathematical foundations of their fields read this book. I would also recommend this book for a one- or two-semester course on partial differential equations for students who have completed at least one semester of senior-level real analysis.