[Certhas]

For a physicist it eventually becomes necessary to understand exterior algebra.

This is often done in the context of differential forms, but of course can be brought back to vectors easily. With those well established tools GA doesn't offer much. This blog post seems to point out exactly this fact.

https://en.m.wikipedia.org/wiki/Exterior_algebra

[oddthink]

I always come back to Schutz's Geometrical Methods of Mathematical Physics as my reference for notation, but I agree. I came to this by way of General Relativity, so that colors my perceptions. The few treatments of GA that I've looked at (briefly) weren't very clear about the distinction between 1-forms and 1-vectors and seemed to assume Euclidean metric everywhere, so I left thinking that it seemed a little weird and not quite trusting it.

In any case, my experience is that the coordinate-free manipulations only go so far, but that you pretty quickly need to drop to some coordinates to actually get work done. d*F=J is nice and all, but it won't calculate your fields for you.