This volume in the IAS/Park City Mathematical Subseries of the *Student Mathematical Library* shares with many other volumes of that series an approach that is freshly considered, accelerated and challenging. The authors take their cue from Richard Feynman: “Imagine that you are explaining your ideas to your former smart, though ignorant, self, at the beginning of your studies!”

The book is about differential equations, mostly those describing a system evolving in time, and mainly focused on initial value problems. Prerequisites include solid introductions to multivariable calculus and linear algebra. The approach is highly conceptual and aims to develop both a mature intuition and good analytical skills. The organization of the book is somewhat unusual: there is “text” — five chapters — and “appendices” — seven mini-chapters. The five chapters constitute about two-thirds of the book and are intended to be relatively easy reading. More technical material is relegated to the appendices. For example, existence and uniqueness theorems are stated and discussed in the first chapter, and the proofs are provided in Appendix B. That’s not to say that the main text reads like a newspaper; instead, it makes serious demands on the reader. This is not a text to use casually with the average undergraduate.

The topics all sound very conventional. The five chapters treat the basic elements of ordinary differential equations and their solutions, linear differential equations, second-order differential equations and the calculus of variations, Newtonian mechanics, and numerical methods. Yet the treatment is quite sophisticated. In the first chapter, just a few pages into the text, we already see continuity of solutions with respect to initial conditions and smoothness of solutions with respect to parameters. This is followed by an introduction to chaotic dynamics and a bit about analytic solutions of ordinary differential equations, all in the first chapter.

It is also surprising to see the calculus of variations introduced so early in the third chapter in the context of second order equations. That third chapter starts out with a discussion of tangent vectors and the tangent bundle. The authors are clearly intent on building a deeper conceptual understanding and offering correspondingly sophisticated tools. The fourth chapter, on classical mechanics, serves to pull together the earlier material and provide an extended series of applications. Here too the treatment is subtle and aimed at developing a mature appreciation of important applications.

The last chapter takes up numerical methods. The treatment is extensive, complete and competent, but it would be dramatically enhanced with more illustrations, supporting software, and some actual computation. The authors’ introduction discusses their joint interests in visualization and the intended “Web Companion” designed to support this book. They promise graphics, animation, Mathematica, Matlab and Maple notebooks. Their introduction says, “We have used traditional diagrams in the text where we felt that they would be useful, and in addition we have placed a much richer assortment of visual material online to accompany the text … Here, organized by chapter and section, you will find visualizations that go far beyond anything we could hope to put in the pages of a book…”

The book’s web site is www.ams.org/bookpages/stml-51/, and it contains a link to http://vmm.math.uci.edu/ODEandCM/, which is the companion web site. This site has the appearance of being assembled from a variety of independent pieces. It is not particularly coherent or well-organized. Nor are the individual pieces very well crafted. The part of the book that most needs graphical support — the chapter on numerical methods — has very little associated content on the website. Perhaps the site remains a work in progress.

This book offers a sophisticated introduction to differential equations that strong student would likely find very attractive. It would also function nicely for independent or guided self-study. However, the book would benefit considerably from better graphics — either in the text itself or on the Companion site.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.