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.
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 :)