[sriku]

"simply one equation" that expands to the same set of vector equations which expands to the coordinate-system-specific equations.

∂F=J (barring constants) is a beautiful restatement, but I think it puts the cart before the horse to focus on it because that formulation is possible because of the invariances that hold and that comes from the raw Maxwell's equations .. at least historically.

The abstract formalisation is even harder to convey (at least for me, and so far) since it takes away the familiar "electricity" and "magnetism" and you need to think about the more complex F that combines both. One way perhaps is to start with circuits - which are discrete and circuit laws can be expressed with the same equation and then argue for the continuous case .. but speed of light invariance would still be a long way from that compared to the raw Maxwell's equations route.

Or maybe I misunderstood what you're suggesting.