Electromagnetism: a short course in differential forms.

If you know something about electromagnetism, a quick way to develop insight into Cartan's methods is to develop Maxwell's electromagnetic field theory in terms of differential forms.

Start with 1-form, A, over space time, with spatial coefficients as the classic vector potential and a time-like coefficient as the scalar potential. The exterior derivative of the four potentials, F=dA, generates the six components that correspond to the E and B fields of Maxwell. The C2 assumption that dF=ddA = 0 generates the first Maxwell pair of equations, and field intensities (Faraday induction, and div B = 0). Recall that E and B relate to forces.

Next presume the existence of an (dual) N-1 form, which can always be scaled by an appropriate measure to become a closed N-1 current density, J. As dJ = 0, it can be deduced from an N-2 form density, G. That is, J = dG. These equations are equivalent to the second Maxwell pair of field equations, and involve the six components of the field excitations, D and H. The statement dJ = ddG = 0 leads to the classic charge current conservation law. Recall that D and H relate to sources.

Almost everyone is exposed to Gauss' idea that charged points are limit points of the D field lines. Indeed, according to Cartan's Topological Structure, that is to be expected because relative to the Cartan topology, the exterior derivative is a generator of limit points, J = dG. Not so obvious is the conclusion that the E and B field lines are limit sets of the potentials, as F = dA.

It is remarkable that this derivation starts with the existence of a Cartan exterior differential system consisting of two dual forms, A and J, over a four dimensional set. Every such Cartan exterior differential system exhibits the properties that physicists would call electromagnetism. It matters not what the symbols stand for! Note that the derivation makes no use of geometrical metrics or geometric connections.

The concepts of Maxwell field theory, therefore, are of a purely topological nature, valid in any frame of reference [von Dantzig 1934], and not just those that belong to the Lorentz equivalence class. Only when constitutive laws are added, as constraints to the Cartan exterior differential system of A and J, do the geometric perspectives of electromagnetism enter.

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