That books seems very interesting, but is it really about Geometric Algebra? I'm not talking about geometry in general, or differential geometry, which I'm a bit familiar with. And not Algebraic Geometry either, for that matter, which is also a fascinating subject.
I mean specifically Geometric Algebra.
It sort of seems like a notation, but it has almost a cult like following and perhaps it's more than a notation, is it a theory, a branch of mathematics?
Re:What is the inverse of a vector?
by musgravepeter on Wednesday November 03, 2021 @05:22AM
(#29092510)
There is Geometric Algebra for Physicists by Doran and Lasenby (2003). It recasts mechanics, E&M up to gauge theories and GR into geometric algebra. I stalled out at mechanics but I've now taken it off the Tsundoku pile and may give it another chance.
Thanks for the tip!
Re:What is the inverse of a vector?
by yuppiemephisto on Wednesday November 03, 2021 @05:11AM
(#29092446)
It really is useful even at the advanced level. Following it far enough leads to the Atiyah-Singer Index Theorem and Hodge Theory. The advantage over the exterior algebra is that you have that and an interior algebra, which leads to many formulas in differential geometry becoming very natural (like Cartan's magic formula).
Re:What is the inverse of a vector?
by contravariant on Wednesday November 03, 2021 @05:35AM
(#29092578)
Is there some canonical metric on Lie algebras that makes this possible? Otherwise geometric algebra won't get you far.
Every Lie algebra is a bivector algebra (see 'Lie groups as Spin groups' http://geocalc.clas.asu.edu/pdf/LGasSG.pdf).
Additionally the GA formalism enables closed form solutions for the exponential map for all bivector algebras. (see 'Graded symmetry Groups' https://www.researchgate.net/publication/353116859_Graded_Sy...).
It is an absolute necessity for general relativity.