Does anyone have a good summary of how this relates to the differential geometry world with its n-forms and n-vectors? For example, I'm used to thinking of the wedge product as operating over n-forms and requiring a metric (or volume element) transformation to work over vectors. Similarly, I don't see any discussion of behaviors under coordinate transformations.
> I don't see any discussion of behaviors under coordinate transformations.
That's because, as the article emphasizes, you don't need coordinates (or basis vectors, which is the term used in the article) to work with geometric objects. You can use them, but you don't need to.
Similarly, vectors, bivectors, tensors, etc. can all be defined without making use of their behavior under coordinate transformations. Textbooks that emphasize coordinates might make it appear otherwise, but that's not the case.
So, like many things, this is both true and false. The object isn't the coordinates, but you do need two kinds of things to make physics work. (Maybe you don't for geometry, though, not sure.)
If you have a displacement vector, and a potential gradient, they can't be the same "kind" of thing (i.e. both vectors), because their dot-product should be preserved. If the slope gets half as long, it's twice as steep.
> you do need two kinds of things to make physics work
I'm not sure what two kinds of things you are referring to, but it doesn't seem like they are "geometric objects" and "coordinates". All I'm saying is that you don't need coordinates; they are a convenience, not a necessity. I am not saying you need only one kind of geometric object.
> If you have a displacement vector, and a potential gradient, they can't be the same "kind" of thing
Agreed. You need both vectors and covectors, or more generally "things with upper indexes" and "things with lower indexes". But you don't need coordinates to work with those things. The indexes do not have to represent components. They can represent "slots" (at least that's what Misner, Thorne, and Wheeler call them in their classic GR textbook), in which you can insert vectors (for lower index slots) or covectors (for upper index slots) in order to obtain other geometric objects (and ultimately numbers, which are what you compare with actual measurements).
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.
When I looked at differential geometry, I'd already seen Hestene's Geometric Algebra (a Clifford Algebra) and I was confused that you couldn't combine n-forms with m-forms in DG. The link is (I believe!) that Clifford Algebras are _graded_ algebras, whereas DG's n-forms are not graded, so in DG you can't combine forms of different degree arbitrarily.
For a better explanation I can recommend the accepted answer here:
The wedge product is inherent in all linear algebra, and is a much simpler and more basic tool than differential forms.
More generally the basics of geometric algebra (“real Clifford algebra”) should be taught to all undergraduate students taking technical courses, and could be profitably pushed back into high school.
The exterior algebra can be defined over any vector space. It just happens that the exterior algebra over the cotangent space is useful for differential geometry.
Thanks, this is basically what I had concluded. The exterior algebra exists on both the tangent and cotangent space, and that I was mostly used to seeing it in the cotangent world, due to its applications there.
