Teaching electromagnetic field theory using differential forms

The calculus of differential forms has significant advantages over traditional methods as a tool for teaching electromagnetic (EM) field theory: First, forms clarify the relationship between field intensity and flux density, by providing distinct mathematical and graphical representations for the two types of fields. Second, Ampere’s and Faraday’s laws obtain graphical representations that are as intuitive as the representation of Gauss’s law. Third, the vector Stokes theorem and the divergence theorem become special cases of a single relationship that is easier for the student to remember, apply, and visualize than their vector formulations. Fourth, computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by algebraic rules. In this paper, EM theory and the calculus of differential forms are developed in parallel, from an elementary, conceptually oriented point of view using simple examples and intuitive motivations.We conclude that because of the power of the calculus of differential forms in conveying the fundamental concepts of EM theory, it provides an attractive and viable alternative to the use of vector analysis in teaching electromagnetic field theory.