A. A. Eichenwald. Theoretical Physics. Part One, Field Theory. Gosizdat, 1926. 267 pp. Price 5 rubles in binding.

At the present time, acquaintance with theoretical physics is becoming ever more necessary not only for persons wishing to follow the modern development of the physical sciences, but also for persons working in the field of applied physics and technology. As the author of the book under review quite correctly observes, grain-

of all theoretical physics is field theory, which studies the geometrical relations common to all phenomena occurring in space and time.

Until now, the study of the differential and integral properties of fields was not very accessible to persons without special mathematical knowledge; only the introduction of the vector algorithm has made field theory considerably more accessible and more intuitive than in a coordinate presentation. One need only compare G. N. Suslov’s The Doctrine of the Vector Field (Odessa, 1922), where field theory is set forth in coordinate formulas, with A. A. Eichenwald’s book in order to be convinced what an enormous simplification is afforded by the use of the vector method.

In 267 pages the author has succeeded in presenting not only vector field theory but also tensor theory, carrying the exposition as far as the curvature tensor of a multidimensional non-Euclidean space.

With his characteristic mastery, A. A. Eichenwald sets forth extraordinarily simply rather complex theorems of the differential geometry of the field, and quite imperceptibly leads the reader from the elementary rules of vector addition, through spatial differentiation, to the complicated questions of non-Euclidean geometry. Along the way the book also presents elements of the calculus of variations, the beginnings of the theory of analytic functions, etc. It is understandable that with such an abundance of material the author could not give an exhaustive exposition of the subject; yet precisely therein lies the great merit of the book under review: it enables the unprepared reader, without great effort, to become acquainted with the principal propositions of field theory. Therefore this book should be of great benefit not only to physicists but also to engineers interested in the theoretical questions of physics.

The author writes that, for the sake of simplicity of exposition, he uses direct calculations with coordinates in many cases. In our opinion, what is direct is precisely the invariant calculations with the vectors themselves, and not with coordinates; moreover, in more complicated cases calculations with coordinates are more likely to lead to errors than direct calculations. An example of such an error may be considered the author’s conclusion on p. 188: “the addition and multiplication of symmetric tensors again gives, as a result, a symmetric tensor.” Using the vector algorithm, and not coordinates, one cannot fail to notice the erroneousness of this conclusion: the product of two symmetric tensors depends on the choice of the order of the factors and therefore is not a symmetric tensor.

From the standpoint of invariant calculations, we also find undesirable the author’s use of vector notations: “addition of a matrix, that is, a tensor, with a vector product” on p. 197, the use of the symbol \((A;B)\) to denote a dyad, allowing the operator \(\nabla\) to act through parentheses, etc.

These minor defects cannot, however, diminish the merits of this excellent and beautifully produced book, which undoubtedly deserves wide circulation.

Ya. Shpilrein.