Differential Forms - Teaching

History

In 1992, a paper by Deschamps (IEEE Proc., 69, pp. 676-696, 1981) sparked our interest in the use of differential forms in electromagnetics. As we made our way through the literature, Burke's treatment of mathematical physics using forms (Applied Differential Geometry, Cambridge Univ. Press, 1985) and the general relativity text by Misner, Thorne, and Wheeler (Gravitation, Freeman, 1973) along with work by others showed that forms are a powerful tool for making the fundamentals of EM theory intuitive. We developed a treatment of forms and EM theory aimed at the undergraduate level, and began to insert segments using differential forms into various electromagnetics courses offerred by the BYU Department of Electrical and Computer Engineering.

Favorable responses by students convinced us to expand the use of differential forms. In the Fall semester of 1995, we completely reworked our beginning electromagnetics course to use differential forms, and developed a set of course notes for use in the class. At the end of the semester, students completed a written evaluation of the course. The results were overwhelmingly positive in favor of the use of differential forms. Since that time, we have continued to use differential forms in our EM courses, refining and expanding our course materials in the process.

Advantages of Differential Forms

In our experience, the key advantages of differential forms in teaching electromagnetics are the following, ranked in approximate order of importance (in our opinion):

  • Pictures of differential forms provide simple, intuitive, and useful graphical representations for Maxwell's laws, the field and source quantities, operations on differential forms, identities and theorems, path, surface and volume integration and boundary conditions. While vectors are ideal for representing displacement and velocity, differential forms are ideal for representing field intensity and flux density, leading to the the single most important advantage of differential forms: an intuitive representation of Ampere's and Faraday's laws and the curl operation.

  • Differential forms are closely related to vectors, and one can move easily between the vector and differential form representations. This allows students to benefit from the advantages of forms while retaining the ability to follow existing literature. In practice, familiarity with differential forms actually helps students understand what the vector operations and quantities really mean.

  • Field intensity and flux density obtain different graphical and mathematical representations. These representations clarify the distinct physical properties of the two types of quantities, so that the reasons for requiring two quantities to represent one field become obvious.

  • Stokes' theorem and the divergence theorem are replaced by a single, simple law, the generalized Stokes theorem.

  • Path, surface, and volume integrals are easier to evaluate using differential forms than with vectors, due to the simpler behavior of forms under coordinate transformation.

  • Derivative formulas in curvilinear coordinates are just as simple as those for rectangular coordinates--no need to look up the curl operator for spherical coordinates in a table.

  • Vector identities are replaced by simple, general properties of the exterior product, exterior derivative, and other operators, eliminating much of the need for tables of identities.

    Using the Curriculum

    We have made available our course materials (stemming from the paper Teaching Electromagnetic Field Theory Using Differential Forms, IEEE Trans. Ed., Vol. 40, No. 1, pp. 53-68, 1997), and would like to offer support to those interested in using the material. We hope to develop problem sets and expand the course notes incorporating feedback from other instructors.