As a side note there was a historical debate, and the shape of these equations is the end result of the chosen system.
In geometric algebra which did not prevail, these are one equation.
In modern relativisitic formulations using tensors they are one equation also, derivable from the action of the electromagnetic field tensor. This formulation also shows that electric and magnetic fields are the same, simply rotated under a relativisitic transform, which is the modern view.
Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact representation of Maxwell equations:
dF=J
dF=0
Geometric algebra produces two equations too, by the way, not one.
Discovering that the Maxwell equations can be written in this succinct form is kind of mind-blowing, but I'm wondering whether it provides any additional insight? It seems like one needs to do a significant "unpacking" to actually use the equation or gain insight from it. I would love to hear an explanation of electrodynamics starting with "star ddA = J".
Yes, I kinda agree. The underlying physics is, by definition, the same, and what you gain by reducing the number of equations you lose by having more complicated "objects" and needing more advanced maths to handle them (e.g. the electromagnetic field tensor, external calculus and whatnot vs. just vector fields and basic vector calculus).
To get an intuition of the physics, I think the traditional 4 equation form is actually more useful, as you can construct toy examples and study the equations one at a time in isolation.
Where the more advanced formulations are useful, and actually are used, is for stuff like relativistic physics where 4-vectors, curved spacetime etc. are needed and not just a gimmick.
But for more down-to-earth applications of electrodynamics like antennas, field propagation in various forms of matter etc., the classical version is fine.
You get new insights, that F is a curvature 2-form. Written in this way it's explicit that EM is also a geometric theory. This observation opens the door to Yang-Mills theories which are behind all the Standard Model.
Yes, but that's the whole point: math is incredibly information dense. I'm trying to avoid that. Math shouldn't be hard - some concepts are incredibly easy to grasp, but the way that they're taught makes me shudder. It's like seeing code with one letter variable names -written by someone who was scared of losing their job. If you have a way of coming up with an intuitive way of conveying the concepts using geometric algebra, let me know, and I'll be happy to include it in the guide though!
