CALCULUS OF VECTOR FUNCTIONS.

By Richard Crowell and Richard Williamson. Englewood Cliffs, N. J.: PrenticeHall, 1962. 485 pp. $8.25.

Freshman calculus is traditionally concerned with functions of one variable, and sophomore calculus with functions of several variables. But in advanced mathematics it is now customary to regard several (numerical) variables as one (vector) variable. This more sophisticated point of view not only simplifies calculations but also gives new insight into the foundations.

The present book pioneers in bringing undiluted vector methods into the undergraduate calculus curriculum. Every standard topic is re-examined and newly presented in an intrinsic (linear) manner. At the same time, the authors have avoided excessive abstraction by providing a generous store of numerical examples. In particular, old-fashioned coordinate-wise computation is by no means slighted.

The book has already been used at Dartmouth in preliminary form for several years, with unusual success. Moreover, unaware that the text is highbrow, students seem to grasp the many-dimensional concepts of sophomore calculus more readily from Crowell-Williamson than from a standard textbook.

Whether the book will be widely adopted remains to be seen. The very feature that distinguishes it, viz. its coordinate-free point of view, will prejudice many routine calculus teachers against it. On the other hand, there undoubtedly exist research mathematicians who will complain that the coordinate-free philosophy of the authors is still somewhat tentative. Another obvious defect from the standpoint of a traditional curriculum is the absence of a section on line and surface integrals, an especially regrettable lack because the topic can be treated so beautifully by vector methods. But it must be noted that the authors have already started writing a treatment of line and surface integrals and one expects that this material will be included in the second edition.

In conclusion, let us point out a virtue of the book that might otherwise be unnoticed. Anyone who has tried to write a mathematics text knows that in every section it is necessary to make arbitrary choices that affect many other sections. One decides to do topic A in the prettiest possible way but at the expense of having to develop first some topic B that could better be done after A. The authors were forced to make a large number of such decisions in writing Calculusof Vector Functions. But by dint of vigorous debate over a two-year period, a debate which often spilled out of their private offices into the common room of the Dartmouth Mathematics Department, they succeeded in organizing the topics so as to maximize the "naturalness" of each one. This particular virtue contributes strongly to the usefulness of the book.