Differential Forms - Teaching
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
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
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.