Perhaps the slickest utilization of Cartan's exterior differential forms is the
derivation of Maxwell's first pair of equations on a variety without metric and
without connection. Maxwell's equations are statements of topology, not
geometry. It is also remarkable that the Maxwell equations are the first four
elements of a nested set in any number of dimensions!
The second pair of Maxwell equations take a little more doing. The fashionable
way is to claim that the seond Maxell pair is the "dual" of the first Maxwell
pair. Such self dual constraints and limitations are the basis of Yang Mill
type theories. Such theories do not explain optical activity or the Faraday
effect. See "Parity and Time-reversal Symmetry Breaking..."
The first Maxwell pair has to do with forces, the second Maxwell pair has to do
with sources. The first Maxwell pair is on a non-compact symplectic manifold.
The second Maxwell pair can be on a compact domain.