Chapter 4.1 - Maxwell’s equations and numerical electromagnetic modeling in the context of the theory of differential forms

Abstract

In the framework of classical theory, the electromagnetic (EM) field is described by the electric and magnetic vector fields, and Maxwell’s equations represent a system of differential equations with respect to these vector fields. An alternative approach to the formulation of Maxwell’s equations is based on the algebraic theory of differential forms, and results in a very compact and symmetric system of differential form equations. I demonstrate in this chapter that Maxwell’s equations appear naturally from the basic equations for the differential forms. The basic laws of electromagnetism are actually imprinted in the fundamental differential relationships between the vector fields and differential forms. The new equations contain the differentials of the flux and work of the electric and magnetic fields. This fact indicates that the electric and/or magnetic flux and work should be treated as the major characteristics of the EM field, instead of using the conventional vectorial representations. The differential form approach corresponds well to geophysical experiments, which involve, as a rule, the measurement of the flux and the work (or voltage) of electric and magnetic fields. It is shown that a similar approach can be used also for effective numerical modeling of EM fields.