Maxwell’s Equations in Differential Forms

To this point, we’ve mostly followed a standard approach to classical electromagnetism, and nothing I’ve said should be all that new to a former physics major, although at some points we’ve infused more mathematical rigor than is typical. But now I want to go in a different direction.

Starting again with Maxwell’s equations, we see all these divergences and curls which, though familiar to many, are really heavy-duty equipment. In particular, they rely on the Riemannian structure on . We want to strip this away to find something that works without this assumption, and as a first step we’ll flip things over into differential forms.

So let’s say that the magnetic field corresponds to a -form , while the electric field corresponds to a -form . To avoid confusion between and the electric constant , let’s also replace some of our constants with the speed of light — . At the same time, we’ll replace with a -form . Now Maxwell’s equations look like:

Now I want to juggle around some of these Hodge stars:

Notice that we’re never just using the -form , but rather the -form . Let’s actually go back and use to represent a -form, so that corresponds to the -form :

In the static case — where time derivatives are zero — we see how symmetric this new formulation is:

For both the -form and the -form , the exterior derivative vanishes, and the operator connects the fields to sources of physical charge and current.