I love articles like this, but they all make the same mistake of starting with vector algebra instead of geometric algebra. In 3D space, vector algebra works, but it falls flat on its face in both 2D and 4D scenarios. It's intuitive until it is completely broken.
I would love to see the same style of article, but using bivectors and the like where appropriate, such that the whole thing generalises neatly to 4D space-time, not just 3D space.
I will probably get downvoted for pointing this out, but the reality is that the geometric algebra approach to E&M, while interesting for its own reasons, will not replace the formalism based on Gibbs's vector calculus. One reason is simply that vector calculus is pretty intuitive and easy to learn. The major reason, however, is that the vector calculus approach is totally entrenched in the worlds of engineering and physics. After 100 years, nobody actually practicing those disciplines will make the notation change just so they can replace the 4 Maxwell's equations with one geometric algebra equation.
Also, Gibbs's vector calculus is used in fluid dynamics and other engineering disciplines, and as far as I know, nobody it touting the advantages of geometric algebra to folks working in fluid dynamics. I can be pretty sure that some HN reader will show me I am wrong about this by pointing out one lonely researcher who has found a way to express the Navier-Stokes equations using the geometric product ... but so what? ... My main point is that traditional vector calculus is a language everybody knows how to speak, geometric algebra is just another way to say the same things, so why would anybody change?
The metric system seems like a similar analog to geometric algebra vs vector calculus. You are saying the same thing but the language you are using is much more internally consistent.
Adoption has been bumpy given the US resistance but I think in the long run it (or something even more consistent) will win out. Similarly I think geometric algebra will be adopted. Maybe not in our lifetimes but eventually.
I actually took a look at doing this, but most human minds aren't tuned to 4D space-time, so if you have ideas on presenting this sort of thing to most people let me know and I'll be more than happy to modify my approach!!
tl;dr: GA's geometric product is a mixed-grade differential form, which is quite weird. Why not just think in terms of differential forms? Maxwell's equations are so sweetly summarized as dF=0 and d*F = J.
Just to give a brief answer to those reasonable criticisms:
The mixed-grade already exists in complex numbers (it is very useful there, and even more so in geometric algebra).
Differential forms are included in geometric algebra (the exterior/outer products are isomorphic). Turns out, combining that product with the inner product gives you an invertible product (as Clifford found out). That by itself already is a huge advantage.
Finally, Maxwell's equations are sweetly summarized in differential forms, but even more in geometric algebra: dF = J . Not only it is just one equation instead of two, but in addition the "d" (or "nabla") is directly invertible thanks to the geometric product (which differential forms lack and then have to use more indirect methods, including the Hodge dual).
By the way, I'm very partial to geometric algebra, but wouldn't say it is an "error" not to use it! Maybe just a big missed opportunity :)
You can do that using differential forms as well - using the co-differential δ, we can write a single equation (δ + d)F = J. However, from the perspective of Yang-Mills theory, that's a rather questionable approach as we're stitching together the Bianchi identity and the Yang-Mills equation for no particular reason...
Cool, I didn't know that. Still, the main point of the geometric algebra version is that it's not a "stitching" exercise, but a natural operation in the algebra -- and even better, an invertible one.
I would love to see the same style of article, but using bivectors and the like where appropriate, such that the whole thing generalises neatly to 4D space-time, not just 3D space.