[sbrorson]

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?

[ptarjan]

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.

[BoiledCabbage]

Interesting paper to skim that seems to look into it.

https://vixra.org/pdf/1206.0021v1.pdf

[nathan_compton]

Field theorists pretty much already have abandoned the vector calculus version of the equations, though.