Electromagnetics and differential forms

Differential forms of various degrees go hand in hand with multiple integrals. They obviously constitute an essential tool in expressing the laws of physics. Some of their structures, however (exterior algebra, exterior differential operators, and others), are not widely known or used. This article concentrates on the relevance of the "exterior calculus" to electromagnetics. It is shown that the association of differential forms with electromagnetic quantities is quite natural. The basic relations between these quantities, displayed in flow diagrams, make use of a single operator "d" (exterior differential) in place of the familiar curl, grad and div operators of vector calculus. Their covariance properties (behavior under change of variables) are discussed. These formulas in space-time have a strikingly concise and elegant expression. Furthermore, they are also invariant under any change of coordinates involving both space and time. Physical dimensions of the electromagnetic forms are such that only two units (coulomb and weber, or e and g) are needed. A few sample applications of the exterior calculus are discussed, mostly to familiarize the reader with the aspect of equations written in this manner. The transition from differential to integral formulas is uniformly performed by means of Stokes' theorem (concisely expressed in terms of forms). When integrations over moving domains are involved, the concepts of flow and Lie derivative come into play. The relation of the topology of a region to the existence of potentials valid in that region is illustrated by two examples: the magnetic field due to a steady electric current and the vector potential of the B-field due to a Dirac magnetic monopole. An extensive appendix reviews most results needed in the main text.