Yeah, I have a physics / GR background, so MTW is the lens through which I see all of this. That and Schutz's Geometrical Methods of Mathematical Physics. I ag

Yeah, I have a physics / GR background, so MTW is the lens through which I see all of this. That and Schutz's Geometrical Methods of Mathematical Physics. I agree that we don't need coordinates. Things are things. But what always got me about the geometric algebra stuff was that they used bivectors for areas, which seems like the wrong thing. If you're integrating a vector field over it, you want a 2-form, not a bivector. I suppose the distinction doesn't matter as long as you're just in Euclidean space, but even then, if you need to drop down to coordinates and do the actual integral, you're still going to want to have changes of variables that work. That leaves me with a kernel of doubt that they're doing the right thing.