Summary: We recast Maxwell's equations into the language of differential forms. This language not only allows for a deeper understanding of Maxwell's equations, and for eventual generalizations to more abstract manifolds, but will also let us recast Maxwell's equations in terms of the calculus of variations via Lagrangians.
The Electromagnetic Two-Form
We start with the definitions:
Definition 11.1.1. Let E = (E1, E2, E3) and B = (B1, B2, B3) be two vector fields. The associated electromagnetic two-form is
F = E1dx ∧ dt + E2dy ∧ dt + E3dz ∧ dt
+ B1dy ∧ dz + B2dz ∧ dx + B3dx ∧ dy.
This two-form is also called the Faraday two-form.
Definition 11.1.2. Let ρ(x,y,z,t) be a function and (J1, J2, J3) be a vector field. The associated current one-form is
J = ρdt − J1dx − J2dy − J3dz.
Maxwell's Equations via Forms
So far we have just repackaged the vector fields and functions that make up Maxwell's equations. That this repackaging is at least reasonable can be seen via
Theorem 11.2.1. Vector fields E, B, and J and function ρ satisfy Maxwell's equations if and only if
dF = 0
⋆ d ⋆ F = J.
Here the star operator is with respect to the Minkowski metric, with basis element dt ∧ dx ∧ dy ∧ dz for Λ(ℝ4). The proof is a long, though enjoyable, calculation, which we leave for the exercises. While the proof is not conceptually hard, it should be noted how naturally the language of differential forms can be used to describe Maxwell's equations. This language can be generalized to different, more complicated, areas of both mathematics and physics.
Potentials
We have rewritten Maxwell's equations in the language of differential forms, via the electromagnetic two-form and the current one-form. The question remains as to how far we can go with this rewriting. The answer is that all of our earlier work can be described via differential forms. In this section we will see how the potential function and the potential vector field can be captured via a single one-form.