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What advantage does this have over traditional differential geometry? I scanned through the linked paper briefly, and didn't see much there, other than what se

What advantage does this have over traditional differential geometry? I scanned through the linked paper briefly, and didn't see much there, other than what seemed like a muddying of the concepts of a one-form and a vector, effectively always assuming the standard Euclidean metric tensor to flip from one to the other. Having two directed concepts is very useful, especially once things start to get tricky. My background has a lot of general relativity and plasma in it, though, so my perceptions may be colored by the particulars of those fields.

Dn_Ab 15y ago
I can't be of help because I only started studying this stuff on the side not long ago. But one of the first things I searched for was differential forms vs Geometric algebra. It appeared that the war between the two was old with both sides claiming subsumption. I decided to learn both but started with GA since it seemed easier to compute with. See: http://groups.google.com/group/sci.physics.research/browse_thread/thread/b9296138f521d1d3/1fec3636c1436215 http://groups.google.com/group/sci.physics.research/browse_t... http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node12.html http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node1... http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf
jacobolus 15y ago
The basic advantage is that it’s much more accessible to non-mathematicians, makes the connections between different mathematical abstractions more obvious and explicit, and of a bit more general use because it includes the concept of a “multivector” which combines multiple types of objects. Cartan was directly inspired by Grassmann’s work when creating differential forms (indeed, the word form comes from Grassmann).
- oddthink 15y ago
  I see. It seems more natural for quantum mechanics than for general relativity, though. In GR, the metric is a living, breathing thing, and I don't see how to shoehorn that into GA. Similarly, differential geometry gives you a nice, mechanistic way to think about coordinate transformations, including accelerated/rotating coordinates. Similarly, there's Littlejohn's work on applying "normal" differential geometry to plasma physics. I've not read nearly as much about that as I would like to, but it always seemed like fun stuff. I don't think this kind of notation helps with that.

Dn_Ab 15y ago

I can't be of help because I only started studying this stuff on the side not long ago. But one of the first things I searched for was differential forms vs Geometric algebra. It appeared that the war between the two was old with both sides claiming subsumption. I decided to learn both but started with GA since it seemed easier to compute with. See: http://groups.google.com/group/sci.physics.research/browse_thread/thread/b9296138f521d1d3/1fec3636c1436215 http://groups.google.com/group/sci.physics.research/browse_t... http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node12.html http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node1... http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf

jacobolus 15y ago

The basic advantage is that it’s much more accessible to non-mathematicians, makes the connections between different mathematical abstractions more obvious and explicit, and of a bit more general use because it includes the concept of a “multivector” which combines multiple types of objects. Cartan was directly inspired by Grassmann’s work when creating differential forms (indeed, the word form comes from Grassmann).

oddthink 15y ago
I see. It seems more natural for quantum mechanics than for general relativity, though. In GR, the metric is a living, breathing thing, and I don't see how to shoehorn that into GA. Similarly, differential geometry gives you a nice, mechanistic way to think about coordinate transformations, including accelerated/rotating coordinates. Similarly, there's Littlejohn's work on applying "normal" differential geometry to plasma physics. I've not read nearly as much about that as I would like to, but it always seemed like fun stuff. I don't think this kind of notation helps with that.