Maxwell's equations, in differential form, are
$$\vec\nabla\times\vec B~=~\mu_0\vec J+\epsilon_0\mu_0\partial\vec E/\partial t,$$
$$\vec\nabla\times\vec E~=~-\partial\vec B/\partial t,$$
which are, respectively, scalar, vector, pseudovector and pseudoscalar equations. Is this purely a coincidence, or is there a deeper reason for having one of each type of equation?
If I'm not mistaken, these objects correspond to (or, at least, a correspondence can be made with) the ranks of differential forms on a 3-dimensional manifold, so I guess there might be some connection with the formulation of Maxwell's equations in terms of differential forms. If this is this case, is there an underlying physical reason that our expression of the equations turns out with one of each rank of equation, or is it a purely mathematical thing?
As a final note, I might also be totally wrong about the rank of each equation. I'm going on the contents of the right-hand side. (e.g. a "magnetic charge density" would be pseudoscalar.)