Many authors of advanced texts in electromagnetic field theory opt for a treatment which postulates Maxwell’s equations, ab initio, and proceeds therefrom. However, this approach is not entirely satisfactory. The field vectors are not defined uniquely by Maxwell’s equations and a rigorous development along conventional lines demands a disturbing proliferation of postulates of one form or another.
The author seeks an alternative treatment which admits “unequivocal definition of the electromagnetic field quantities, which bypasses the unjustifiable extrapolations of the inductive method, and which demands fewer postulates.” Such a development was presented as early as 1897 by Levi-Civita, who wrote: “We can find the essentials of Maxwell’s theory even while starting from the classical laws. It is sufficient to complete them by the hypothesis that the actions at a distance are propagated with a finite volocity.” In other words, if \(\phi\) and \(\vec A\) represent appropriately - retarded forms of the scalar and vector potentials conventionally associated with time-invariant distributions of charge and current, and if \(\vec E\) and \(\vec B\) are defined by \(\vec E=-\text{grad} \phi -1/c\partial \vec A/\partial t\); \(\vec B=curl \vec A\), we can deduce the dynamical form of Maxwell’s equations for a vacuum. The ancillary relationship div \(\vec J=-\partial \rho /\partial t\) follows immediately from the postulate of the conservation of source strength or from the discrete physical model.
In this approach Maxwell’s equations are derived by mathematical manipulation of the space and time derivatives of potential functions defined in terms of scalar and vector source densities. Since such analysis, qua analysis, does not require that the source densities be based upon an electrical model, but only that they satisfy the equation of continuity, the proposed procedure transfers Maxwell’s equations from the realm of physics to that of pure mathematics - to what may be called a branch of retarded potential theory. In these circumstances the physics of electromagnetic theory is introduced through the constitutive relationships and the Lorentz force formula. The fact that these describe interactions of charge complexes by expressions which involve the symbols of potential theory in no way requires that such symbols be necessarily given a physical interpretation. This division of electromagnetics into pure and applied aspects brings with it a comprehensive simplification of fundamental principles which are delineated in this text.
This monograph is concerned with a systematic treatment of retarded potential theory insofar as it is relevant to the study of classical electromagnetics. To highlight its purely analytical nature overt mention of electricity and magnetism has been eschewed, although standard symbolism has been retained. The notes consist of seven chapters entitled the differential and integral calculus of vectors, curvilinear coordinate systems, Green’s theorem and allied topics, unretarded potential theory, retarded potential theory, Helmholtz’s formula and allied topics, exponential potential theory.