The Original Maxwell's Equations
Maxwell, actually, was not using vectors so much as differential forms and he did (in fact) make the underlying Grassmann algebra explict at one point in his treatise. He wrote much of chapter 1 on the same topics covered by more modern treatments of differential forms and integration thereof (even touching into basic homology theory with the discussion about cyclomatic number, cyclosis, or whatever he called it).
The fields (in more modern notation) were
D = (D^x dy dz + D^y dz dx + D^z dx dy)
H = (H_x dx + H_y dy + H_z dz)
E = (E_x dx + E_y dy + E_z dz)
B = (B^x dy dz + B^y dz dx + B^z dx dy),
the potentials
A = (A_x dx + A_y dy + A_z dz)
phi
the velocity, G, relative to the distinguished frame of reference, and
the sources
J = (J^x dy dz + J^y dz dx + J^z dx dy)
rho dx dy dz
and the "total" current C = J + dD/dt. There was no decomposition D = epsilon_0 E + P, with a separate polarization vector because a central thesis of Maxwell was that the vacuum, itself, was a dielectric medium with the ability to engender charge screening (thus effecting a distinction between bare vs. dressed charges -- a point he touched on), vacuum polarization (which he touched on in his thought experiment explaining the impossibility of point-like and line-like sources), and dielectric breakdown producing electromagnetic energy (the section on the "glow" phenomenon) which -- in the other context -- would have meant also the ability of the vacuum, itself, to break down and produce electromagnetic radiation (i.e. matter/anti-matter annihilation).
The distinguished frame of reference is that where electromagnetic radiation proceeds in a sphere centered on a fixed point. There was no explicit discussion of which frame might be so distinguished and what the Earth's motion relative to it was, because the section on the measurement and values of light speed only had the speed resolved to 100,000 km/sec, which is not precise enough for the Earth's motion to become an issue.
The equations posed by are NOT equivalent to the modern equations. In particular, for the electric field, he had
E = -dA/dt - grad(phi) + G x B.
If you take the *modern* definition (E = -dA/dt - grad(phi)), what this amounts to is that Maxwell's constitutive law (D = K E) becomes in modern notation
D = epsilon (E + G x B).
This is the constitutive law that is uniquely identified by the requirement that the remaining part of Maxwell's equations be invariant under the Galilean transformations that define Newtonian physics. The other constitutive law (not explicitly mentioned) would then have to be B = mu (H - G x D). There was a brief mention implying the -GxD term in a footnote to a later edition of the treatise, however.
The 4 macroscopic equations were present (div D = rho; div B = 0; curl H = C = J + dD/dt; curl E - dB/dt = 0). Maxwell wasn't clear on how the last equation meshed with the relation (E = ... + G x B). He never did a comprehensive analysis, in fact, to determine how his entire system transformed under a change in reference frame. That was the deficit Einstein picked up on and resolved.
Ohm's law (J = sigma E) was included, as well as the continuity equation (d(rho)/dt + div J = 0), and the derivation of B from the potential (B = curl A). The continuity law is deriveable and Maxwell didn't realize that early on, so the earliest form he posed for a system was actually UNDER-determined; not perfectly matched with 20 variables and 20 independent equations. The other constitutive law (B = mu H) was listed with several variants, and no clear, definitive, statement was made on what it ought to be. That's partly linked to the point made above about Maxwell not having done a clear analysis on his system to check their Galilean invariance. Had he done so, he would have clearly seen that the only relation possible is B = mu (H - G x D) and that, right there, might have raised a few eyebrows, implying as it does, a dependence of the B field on the electric displacement in frames that are in motion relative to the distinguished frame.
The Lorentz relations (D = epsilon_0 E, B = mu_0 H) therefore represent a completely INEQUIVALENT and fundamental departure from Maxwell's Galilean invariant (D = epsilon_0 (E + G x B), B = mu_0 (H - G x D)). As soon as you write down the former -- whether you realize it or not -- you're in Relativity, for they are only invariant under the LORENTZ transformations, not the Galilean transformations.
So, a major historical myth is also debunked here: Einstein did not get rid of the Aether from an "otherwise-equivalent" Aether theory; he (and Lorentz, already) posed a system of constitutive relations that were fundamentally different from Maxwell's system and never had an Aether frame in it. It's Maxwell's consitutive laws that had the G in it, not Lorentz's. The Aether version (with Maxwell's G) is NOT equivalent to the Aether-less version (with Lorentz's relations with the G). In the latter, G would be 0 in every frame of reference. In the former, G would change. They are completely different, empirically, and have different empirical consequences.
