Bibliography

COURANT—HILBERT. Methoden der Mathematischen Physik. Bd. 1, XIII+450, in 8°. Berlin, J. Springer, 1924.

Courant—Hilbert. Methods of Mathematical Physics. Vol. I.

The revolution in natural science taking place before our eyes is accompanied also by a revolution in mathematical methods—a revolution perhaps less noticeable than the former, but fraught with no lesser consequences. Not long ago the highest analytical achievement, and at the same time the most convenient expression for natural phenomena, was considered to be differential equations with a small number of unknown functions and independent variables. This simplicity has disappeared since it has proved necessary to study mechanical systems consisting of very many points. Taking the number of points to be infinite, we obtain systems of partial differential equations only in the most favorable cases; in the general case we must deal with integro-differential equations; even in the simplest cases, as, for example, in the theory of figures of equilibrium of rotating fluid bodies, these equations are so complicated that they exceed the possibilities of contemporary mathematical analysis. In their turn, integral and integro-differential equations are connected in a perfectly natural way with the theory of quadratic forms, which likewise has great significance in a number of mechanical and physical theories. On the other hand, their solution can be reduced to the solution of minimal problems, whose prototype is the famous problem of Dirichlet. This is not all: a whole series of paths connects these problems with geometry, in particular with the geometry of non-Euclidean spaces of many dimensions.

This entire aggregate of very complex and heterogeneous problems and theories has still not been linked into any homogeneous whole. Yet a number of paths already present themselves that lead toward this future unity of method. At the present time a new calculus is already beginning to be created—the functional calculus; its chapters are the calculus of variations, the theory of integral equations, the theory of differential equations, and many other branches of analysis. With respect to certain classes of functions, formal methods of calculation already exist which, by algebraic and other analogies, make it possible quickly to solve problems that until recently were practically insoluble. In other problems we seek classes of functions that would satisfy certain predetermined conditions, and we investigate various transformations that would give us functions of the same class. In a third group of problems we investigate operations that make it possible to single out, from the multitude of solutions of a problem, precisely that which is the desired unique solution of a physical or mechanical problem. In a fourth kind of problem we seek the most convenient mathematical garment for the solution of a physical problem. It must be said that at present the mathematical wardrobe is far more extensive than it was a hundred years ago, and that mathematical fashion has changed greatly during these years. Series still remain the chief instrument; only these are not the series with which mathematicians worked earlier:

and physics. Series in orthogonal functions, in the form in which they are presented in the book under review, differ greatly from the old trigonometric series, and those operations on them which are now performed would have seemed impossible and imprecise to the old mathematicians. In particular, they would have been struck by the geometric language used in the modern theory of orthogonal functions.

Courant and Hilbert set themselves the goal of showing the modern physicist what a powerful tool he possesses in the form of modern analysis. The first chapter presents the algebra of linear transformations and quadratic forms, equally important for mathematics, mechanics, and physics. In the second chapter the problem is posed of expansions in series of arbitrary functions, which serves as a natural introduction to the third chapter, where a new exposition of Hilbert’s theory of integral equations is given. In the fourth chapter, the foundations of the calculus of variations are presented from the same point of view. The extensive fifth chapter is devoted to application to the problem of oscillations in mathematical physics; here a number of interesting physical problems are examined. In the sixth chapter the calculus of variations is applied to the same problem. Finally, in the seventh chapter the physicist is given not a method, but, so to speak, a ready-made form for its application in the guise of various systems of orthogonal functions.

In the second volume, which has not yet appeared, the authors propose to give a general survey of the classical differential equations of physics, with a detailed investigation of the question of the existence of solutions and with numerical elaboration; moreover, the methods of the calculus of variations, with applications to questions of contemporary physics, will be brought to the fore.

Such is the content of this important and useful book. Will many physicists become acquainted with it? A few years ago it would have been permissible to doubt this. At present a number of purely mathematical memoirs on the questions touched upon in the book—memoirs written by physicists—show that the situation has changed. Let us hope that this book will prove to be the bridge by which a connection will be established between mathematics and natural science.

V. Kostitsyn.