Bibliography

Ya. N. Shpilrein. Vector Calculus. A handbook for physicists and engineers. State Publishing House, Moscow, 1925. 324 pp.

The appearance of this book in Russian is of great significance for the dissemination of vector analysis in the Union, notwithstanding the several textbooks on vector calculus that have already appeared.

Any book of an instructional character must be evaluated from two points of view: the freshness and completeness of the material presented in it, and the pedagogical quality of the exposition.

First of all, one must note the extremely consistent and convenient notation introduced by the author; some of it is already becoming established, under the influence of the German book by the same author, in the world literature as well. For the moment the exception is the author’s favored “dot” for denoting the multiplier in a vector product; but here it should be noted that the real meaning of this notation becomes apparent only in the theory of tensors of the second rank.

The content of the book falls into two parts: vector algebra and vector analysis. In the first part, the excellent treatment of rectangular and oblique coordinates attracts attention. In the author’s exposition, the coordinate method becomes a natural part of the coordinate-free method (on which the entire exposition is conducted), and the notions of contra- and covariance acquire a simple and natural geometric meaning.

The second part—vector analysis—consists of chapters on functions of a scalar variable, functions of a point, and the geometry of vector fields. The last chapter is especially interesting and rich in material.

What one must regret, however, is the absence, in the exposition, of the theory of tensors (of the second rank in Euclidean space). In Shpilrein’s German book, more than half the book is devoted to it; one cannot help reproaching the State Publishing House for not having decided to give the author the opportunity, at least in abbreviated form, to present this natural and necessary development of the methods of coordinate-free analysis. And this is all the more so because books such as this determine the level of teaching in higher educational institutions.

It is impossible in a review to speak of the wealth of material covered by the book in the field of vector calculus: throughout the book there are scattered remarks and examples, for the most part found by the author himself, which vividly and subtly illustrate the main material.

But this book is an especially important phenomenon from the standpoint of exposition. Not only are a special simplicity and elegance in the presentation of even difficult questions characteristic of this book, but especially the skillful distribution of material between the text and the problems. It is almost impossible to read this book without at the same time solving the problems it contains. With extraordinary tact, the author draws the reader into independent work; the problems are selected in such a sequence that the reader, in solving them, involuntarily gains confidence both in his own powers and in the merits of vector calculus—and discovers new prospects in this field.

The book has been well produced; the formulas are set in a good typeface, and there are few misprints. The use of two typefaces makes it easier to distinguish the material according to the importance of what is being presented.

The book will be of enormous benefit and should serve as a model of exposition for other theoretical disciplines taught in higher education.

N. Andreev.