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?