Methods of Mathematical Physics (vol. 2)

Description

Since the first volume of this work came out in Germany in 1937, this book, together with its first volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's final revision of 1961.

Volume I, by the authors, was published in 1953 [MR0065391]. This completely rewritten version of the famous classic on partial differential equations has been greatly influenced by major applied mathematical researches of the 1950's. In fact, modern analytical knowledge of the subject that is valuable in applications is covered to a truly extraordinary extent in this volume. It may be read independently of Volume I, while a shorter Volume III is also promised, concerned with existence proofs (particularly those for elliptic equations) and with the construction of solutions by methods such as finite differences.
In the first chapter, the student is oriented to the subject by means of a wealth of general considerations and special techniques. The next offers a splendidly full account of equations of the first order, beginning with a separate account of the quasi-linear equation; this is of great importance, both in its own right and for introducing the general case whose complexities are here elucidated better than in previous compendia partly because the needs of the user of the methods are kept in mind more constantly. Examples, including equations of eiconal type, of geodetics, of Hamiltonian dynamics and of the calculus of variations, as well as discontinuous (shock-type) solutions of equations of conservational form, further enhance the text's value and the reader's understanding.
The next chapter sets out the classification of higher-order equations into those of elliptic, parabolic, hyperbolic and mixed' type, building on knowledge of characteristic curves already acquired in Chapter II. In particular, linear equations with constant coefficients are used to show the nature of the problems arising with the different types of equation, including problems of waves, Huyghens's principle, scattering, "radiation conditions'', dispersion, the telegraph equation, heat conduction, Fourier-analytic methods and the adjoint differential operator. Chapter IV gives a full account of classical potential theory and of modern extensions, particularly those associated with names such as Bernstein, Bers, Leray and Schauder, which generalise results of the theory to a wide class of elliptic differential equations. Lastly, Chapters V and VI, a remarkable 400-page tour de force, filling half the book, deal with hyperbolic differential equations in two independent variables and in more than two, respectively. Chapter V supplements general considerations (on propagation of discontinuities, domains of dependence and of influence, the application of characteristics as coordinates, the Riemann representation of solutions, iteration methods, and further material on discontinuous solutions of equations of conservational form) with a rich range of illustrative examples, and with a valuable treatment of the Heaviside operational calculus. Next, Chapter VI makes clear all the complicated differential geometry of characteristic surfaces, rays (or bicharacteristics), wave fronts, ray conoids, space-like manifolds and time-like directions, for hyperbolic equations in three or more variables. Modern work on anisotropic wave propagation, with applications to crystal optics and magnetohydrodynamics, is well accounted for. Geometrical optics is developed as an asymptotic theory. The general use of integrals which, in particular physical problems, represent energy is explained. Then a wide range of analytical methods of solution is given, including those dependent on decomposition into plane waves, Hadamard'smethod of descent' and solution of the Cauchy problem for the wave equation, deductions from Darboux's equation for the mean of a function over a sphere, the theory and applications of the delta function and of distributions in general, and Asgeirsson's meanvalue methods for equations with an equal number of space-like and time-like variables. These analytical methods are illustrated by multitudinous applications, which include Lorentz transformations and the theory of elastic waves.
Compared with its first edition, the present volume both contains a much greater volume of information and succeeds in rendering it more easily intelligible, improvements made possible, above all, by the vast volume of research conducted since the appearance of the first edition and based directly or indirectly upon its influence and that of its author. Reviewed by M. J. Lighthill

Ibragimov's Practical Course cites this.

PDF pp. 54ff. are on the Legendre transform.