Is there any textbook or article that has proofs of various vector calculus identities using geometric algebra axioms instead of coordinate expansions ?
Hestenes has published a few but somewhat incomplete articles on that , but as it is a fundamental topic Id hope there would be a more complete systematic overview .
What kind of proofs are you refering to exactly?
Maybe If is something particular we could try to give a proof of it. I’ve actually been studying GC and I would apreciate some practice. To study I’ve been trying to follow directed integration theory from CAGC by hestenes, by trying to derive most of all the identities myself. I thing Is a good practice but in the end it’ll probably not be enough.
We’ll just the basic proofs like what is the gradient of the identity function on a vector space .
I know how to prove this with the integral definition of the vector derivative , or by basis expansion but I don’t think it can be proved otherwise , though hestenes did talk about using the differential to calculate the gradient .
And also just calculating basic vector calculus operations like for example : curl (f A) where f is a scalar function, many more examples can be found on wiki.
PS I saw in another post you mentioned diff geometry on vector manifolds, something that literature on GA also just mentions and then completely avoids any concrete details.
It is mentioned in a paper by hestenes that intrinsic diff hlgeo can be coordinated by extrinsic geometry in basically a vector space if I got that correctly .
However I’d say this misses an opportunity of exploiting multivector manifolds , particularly the potential power of GA to express metrics ( spherical and hyperbolic, even Euclidean ) in a conformal way .
So for example , what would a sphere be as a vector manifold , or what is a spherical metric written in GA .
Grassmann.jl in fact didnt miss this opportunity and it has a manifold implementation which lets you for example manipulate a sphere as a manifold, which I even demonstrated in a not easy to find YouTube video. I havent bothered to expose all this to the general public (despite it all being open source) because the general community is a bunch of backstabbing slime balls and why would I go out of my way to hand everything on a platter to these type of people?
Hi Leo,
Take a look at
J. C. Schindler - Geometric calculus on pseudo-Riemannian manifolds -
He takes a pedagogical path ( a bit different than the one follows by Hestenes) and extend to differential calculus for vector, multivector, and tensor fields.
Thanks for the reply.
Only problem with the text is that it focuses too much on coordinates, which is in stark contrast to what I prefer the most when it comes to hesteneses philosophy, which is the focus in coordinate free basis free methods with coordinates being only a marginal tool at best.