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Are there any rigorous books on analysis bases on geometric/Clifford algebra? I am searching for one which contains a detailed proof of theorems like stokes' theorem and the helmholtz decomposition.

Thank you!

Something fun in the process: I thought the subject could be called geometric analysis or Clifford analysis, but both turned out to be different subjects already.

Ben Grossmann
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Libertas
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  • check the controversies at the comments of https://math.stackexchange.com/questions/2210804/is-hesteness-geometric-algebra-widely-accepted – janmarqz Apr 13 '20 at 19:13

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See Fundamental Theorem of Calculus (Sobczyk), and Geometric Algebra for Physicists. In his book, Vector and Geometric Calculus, Alan MacDonald does a really nice job building up the basics required to be able to understand the statement of the theorem. I have a derivation of the Helmholtz decomposition in chapter II of my book (which I'll claim has "an Engineer's rigor" and no more.)

Some work may be required to beef up the rigor in all these (and other sources), as noted in MacDonald's paper. Spivak's "Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus" (pg 129-130) also hints at problems with triangularization independent of geometric algebra (unbounded area for some triangularizations), and provides some references for "this tricky topic" (that I haven't read yet.)

Peeter Joot
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  • Very interesting answer. The first paper you linked is amazing and is exactly what I was searching for. Thank you, – Libertas May 29 '20 at 22:19