Van Dyke's book is mainly about singular perturbation theory, although it begins with a discussion of some regular perturbation problems.
It is the first systematic book on the subject. Principal emphasis is placed on the "method of matched asymptotic expansions," the systematic extension of Prandtl's boundary layer theory, which has now been brought to a relatively complete state, at lesst formally, as a result of the work of Kaplun, Lagerstrom, Proudman and Pearson, and others. A second technique, Lighthill's "method of strained coordinates," is also discussed at length, and a brief discussion of the method of multiple scales, which has recently found increasing application, is presented. The presentation is almost entirely done by examples, largely chosen from aeronautical applications of fluid mechanics.
On the whole, these examples seem very well chosen to illustrate significant aspects of the techniques, and their presentation is lively and clear, with many interesting sidelights. This method of presentation is certainly the best, since at present a satisfactory general mathematical theory of singular perturbation problems does not exist. Even if one did, Van Dyke's presentation might still be found most useful to those primarily interested in applications.
This book should be welcomed as a very useful reference for research workers-its main limitation in this regard is the somewhat specialized character of the examples, which means that even though the techniques described have much wider applications, the reader is expected to be fairly familiar with fluid mechanics. This is of course consistent with the title, and one cannot really criticize the book for this. Aside from this one limiting factor, the book should be very useful for teaching, and we may hope that with its availability will come a greater appreciation by students of the real significance of nonuniformities in limiting processes.