Maxwell's equation are of two species: the Maxwell-Faraday equations involving
E and B fields and the Maxwell-Ampere equations, involving the excitations, D
It is generally perceived that Maxwell's equations are Lorentz invariant. What
is not so well perceived is that Maxwell's equations - as tensor equations - are
naturally covariant with respect to ALL diffeomorphisms, which include the
Galilean transformation!. (The latter statement fails if a "Lorentzian"
constitutive constraint is imposed upon the Maxwell system relating the D,H
excitations to the E,B fields in an isotropic homogeneous four-fold degenerate
The metric free content of Maxwell's equations was championed by Van Dantzig
(1932), and later by Cartan.
Cartan's methods for the Maxwell Faraday relations start with the idea that there exists a 1-form of potentials, A, on the domain of interest, and that the 2-form F is constructed by exterior differentiation to produce the electro-magnetic intensities from the potentials. The domain of non-zero F=dA defines a manifold that CANNOT be compact without boundary. It is subsumed that the 2-form is differentiable such that ddA=dF=0. When evaluated with respect to a set of variables, (t,x,y,z....), the equations dF=0 that involve the triple of "coordinate" combinations of the first four variables, ALWAYS give the same partial differential system known as the Maxwell-Faraday equations, independent from the dimensionality of the domain (and also independent from the choice of symbols used to "define" the first four coordinate variables)!
The Maxwell-Faraday equations are evidence of C2 differentiability on an
arbitrary domain without metric, and without connection, and without any choice
of gauge - and moreover without choice of dimension! The C2 differentiability
condition disallows any concept of magnetic monopoles.