Field Theory Handbook

Abstract

Let us first state exactly what this book is and what it is not. It is a compendium of equations for the physicist and the engineer working with electrostatics, magne- tostatics, electric currents, electromagnetic fields, heat flow, gravitation, diffusion, optics, or acoustics. It tabulates the properties of 40 coordinate systems, states the Laplace and Helmholtz equations in each coordinate system, and gives the separation equations and their solutions. But it is not a textbook and it does not cover relativistic and quantum phenomena. The history of classical physics may be regarded as an interplay between two ideas, the concept of action-at-a-distance and the concept of a field. Newton's equation of universal gravitation, for instance, implies action-at-a-distance. The same form of equation was employed by COULOMB to express the force between charged particles. AMPERE and GAUSS extended this idea to the phenomenological action between currents. In 1867, LUDVIG LORENZ formulated electrodynamics as retarded action-at-a-distance. At almost the same time, MAXWELL presented the alternative formulation in terms of fields. In most cases, the field approach has shown itself to be the more powerful. A partial differential equation is solved, and boundary conditions are fitted to give a unique solution of the problem. The partial differential equations of classical physics, considered in this book, are the Laplace equation, Poisson equation, diffusion equation, scalar wave equation, and vector wave equation. Several methods of handling these equations are possible, but separation of variables is generally the most valuable. The procedure is as follows: (1) Transform the partial differential equation into the coordinate system that fits the geometry of the problem. (2) Separate this equation into three ordinary differential equations. (3) Obtain solutions of these ordinary differential equations. (4) Build up the unique solution that fits the boundary conditions, using as building blocks the particular solutions obtained in (3). The amount of labor involved in solving a practical problem by this method, particularly when the required coordinates are unfamiliar, is rather formidable. One may well hesitate about embarking on such a program; and this hesitancy is probably responsible for th~ dearth of engineering solutions of field problems. Most of the labor, however, occurs in the first three steps; and these parts of the solution can be completed, once for all, and the results tabulated. This is the purpose of the Handbook-to remove the routine drudgery from field solutions so that the scientist can concentrate on (4), the unique and important part of the work. No such tables have been available previously. The range of problems that can be handled by separation of variables depends to a marked extent on the number of available coordinate systems. Accordingly, we have not limited ourselves to the eleven systems of EISENHART but have chosen a number of others, bringing the total to 40. This by no means constitutes the totality of possible coordinates that allow simple separation or R-separation, though it includes most systems of reasonable simplicity and usefulness. Methods of obtaining further coordinates are explained in our book Field Theory for Engineers (D. Van Nostrand Co., Princeton, N. j., U.S.A., 1961). For each of the 40 coordinate systems are given the relations to rectangular coordinates, the Stackel matrix, metric coefficients, gradient, divergence, and curl, the Laplace and Helmholtz equations, the separation equations, and the solutions of these equations. Also listed are a tabulation of all the ordinary differential equations of field theory and their solutions, also a bibliography of works dealing with the mathematical functions involved. Every equation has been checked independently by the two authors. But in a work that includes so many mathematical expressions, many of them given here for the first time, complete freedom from error would approach the mira- culous. Any suggestions regarding errors or improvements will be greatly ap- preciated. To a pure mathematician, our tabulations will seem ludicrously redundant. We have listed all special cases, even when they are obtainable from the basic form by a simple functional transformation. In this respect, the Handbook is similar to a table of integrals, whose practical value resides precisely in its re- dundancy. We hope that the book will help the research worker in two ways: (a) By providing new coordinate systems, thus extending the range of engineer- ing problems that can be handled by separation of variables, and (b) by freeing him from much of the annoyance and wasted effort usually associated with the routine part of the solution of partial differential equations