The book *A Course on Partial Differential Equations* by Walter Craig is a textbook for a course on partial differential equations (PDEs). While this sentence apparently states the obvious, there are two important points for discussion contained therein. First, it is a textbook, as in meant for students. Second, it is a textbook for a course on PDEs. What requires elaboration are the two questions, 1) what type of students and with what background is it a textbook for?, and 2) exactly what type of course on PDEs is it a textbook for, that is, what would or should the goals of a course on PDEs that might use Craig's book as a course text be?

The first question is relatively easy to address, *A Course on Partial Differential Equations* is appropriate for any student that has a solid background in multivariable calculus, linear algebra, and elementary real analysis (a typical one-semester course at the undergraduate level). What I mean by this is that any such student should be able to actually read the book and learn from it without having to refer to many (if any at all) outside references. A first-year graduate student should certainly be able to learn from Craig's book.

Partial differential equations as a field of study and research forms a vast field within mathematics. Both *The Princeton Companion to Mathematics* and *The Princeton Companion to Applied Mathematics* contain entries on partial differential equations broadly (written by Sergiu Klainerman and Lawrence C. Evans respectively) as well as many entries on various specific PDEs or specific aspects of PDEs. Furthermore, the American Mathematical Society (AMS) Graduate Studies in Mathematics (GSM) book series, the series to which *A Course on Partial Differential Equations* belongs, contains around twenty books in which PDEs are a major, if not the only, focus. At the time of writing, the GSM series contained around 200 books total, so the subject of PDEs represents around ten percent of the GSM catalog. Despite the large expanse of the area of PDEs, there are some prominent themes in the field: the discovery of formulas for solutions to specific PDEs, understanding solutions to PDEs in the physical context that the equation(s) model (what Craig calls applications), the analysis of general properties of solutions to PDEs (what Craig calls phenomenology), and the development of rigorous frameworks in which the PDEs can be studied (what Craig calls theory). A course with the goal of introducing students to these aspects of PDEs is the type of course that *A Course on Partial Differential Equations* would make an appropriate textbook for.

Essentially, *A Course on Partial Differential Equations* provides a modern treatment of classical partial differential equations by way of a survey of equations, techniques, results, and applications. In *A Course on Partial Differential Equations*, a student will learn how to develop solutions to the heat, wave, and Laplace's equations. The role of the Fourier transform in the study of solutions of PDEs is treated in Craig's book as are several other major topics like dispersion, conservation laws, energy methods, and Green's functions. There are three stand-out features of *A Course on Partial Differential Equations* that I would like to mention: 1) while the treatment is modern, the methods are elementary in that the author largely avoids using more sophisticated machinery to derive results, 2) the author frequently makes connections between PDEs and other areas of mathematics, e.g. complex function theory, 3) in addition to exercises, the chapters of the book (with the exception of the first) also contain projects. The projects are more involved than the exercises (which are excellent) and give the student a taste of what it might be like to enter into research in the field of partial differential equations.

I really enjoyed reading *A Course on Partial Differential Equations*. The writing is clear and engaging, the derivations are very well-developed, and the author does a nice job connecting the mathematics with the physical motivation that underlies many of the PDEs discussed in the text. The author also uses clear well-chosen notation and I found very few typos or other errors in the text. I think that a student would find *A Course on Partial Differential Equations* very appealing as a course text, I wish that it would have been available when I was a student.

As I hinted at before, *A Course on Partial Differential Equations* is not without its competitors. First of all, there is *Partial Differential Equations 4th Ed.* by John and *Partial Differential Equations: An Introduction* by Strauss. Craig's book is more modern than these books, and also contains some interesting topics that John and Strauss do not, for example, the coverage of Schrödinger's equation and Hamiltonian equations. Already in the GSM series there is *A Basic Course in Partial Differential Equations* by Han and *Partial Differential Equations: An Accessible Route through Theory and Applications* by Vasy which are quite similar to Craig's book. These two books are also modern in the same way as Craig's book. In my opinion, all three books, the ones by Han, Vasy, and Craig are excellent. While there is substantial overlap between the three, they are also all sufficiently different to be enjoyed individually or in parallel. There are three things about *A Course on Partial Differential Equations* that might make it more appealing to some that the books by Han or Vasy; 1) it is shorter in length, 2) it maintains a closer connection to physics so that it is more of a book on mathematical physics than a book on pure mathematics, 3) the projects. Of course, there are a number of textbooks on PDEs that are written at a lower level of sophistication than Craig's as well as many that are written at a higher level of sophistication or are more specialized in their scope. Craig provides references to several sources to which one may turn for further treatment of PDEs once one has mastered the material from *A Course on Partial Differential Equations*.

My final point is that *A Course on Partial Differential Equations* is an excellent book that provides an modern introduction to a fascinating field of mathematics and I highly recommend reading it.

Jason Graham is an Associate Professor in the Department of Mathematics at the University of Scranton. He received his PhD from the program in Applied Mathematical and Computational Sciences at the University of Iowa. His professional interests are in applied mathematics and mathematical biology.