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While trying to understand Maxwell's equations (having learned a bit about manifolds) I encounter the following issue:

Gauss' Law seems to integrate the electric field $E$ over a $2$-manifold, seemingly implying that $E$ is a $2$-form.

Ampere's Law of Induction, however, integrates the electric field over a $1$-manifold, seemingly implying that $E$ is a $1$-form.

Which one is it? Is $E$ a $1$-form or a $2$-form? And if $E$ is an $m$-form, how is integration carried out on a $n$-manifold for $m\neq n$?

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Neither. When you are coming down to computing all these things in detail, then you have to treat both the E and B fields together as one. The beginner's introduction with E and B fields separately, might be easier to teach and simpler looking for students, but ultimately causes confusion when we start considering relativity. In fact, we tended to introduce E and B fields as vectors, i.e. the dual entities to forms; However, with a natural metric to convert between forms and vectors, this difference is not too drastic.

The correct unification of E and B fields into one EM field, is in terms of the Faraday tensor $F_{ab}$, and this is a 2-form. Its Hodge dual is also a 2-form, and it is these that appear in the differential form version of Maxwell's equations. Needless to say, the integration of this tensor to recreate the standard set of 4 Maxwell's equations is not exactly trivial, although it is also not considered that difficult.

Those integrals that you are familiar with, that seem to be about loops and surfaces, closed surfaces and volumes, are actually 2-form and 3-form integrals. The reason why one dimension seems to disappear, is because we have to insert the "4-velocity vector of the observer's frame of reference" as one of the possible vectors to integrate over. This is trivial in the Gauß's law, where we do not insert this time vector, and hence it is a divergence's volume integral, whereas in the Ampère's law with Maxwell's addition, the time vector is used to occupy one of the vector's slots, leaving just two others for the surface's vectors.

All these things would be easier if you can just find a treatment of EM using differential forms in the first place. There are plenty of good treatments.

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    $\begingroup$ 'The beginner's introduction with E and B fields separately, is a hack-job that won't work. ' I would phrase it differently. To treat E and B separately only works for stationary charges and currents (even though currents are of course originating in moving charges). Hence, historically, E and B were separate. Physicists maintain this tradition against better judgement. $\endgroup$
    – my2cts
    Commented Dec 4, 2023 at 12:27
  • $\begingroup$ Oh, I really like your version. I'm going to revise and not copy yours, though. @my2cts $\endgroup$
    – naturallyInconsistent
    Commented Dec 4, 2023 at 13:05

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