> I'll just focus on vector calculus. Let me experiment a bit and
> hopefully, I'll be able to decide on which way to go from here.

I's not vector calculus but rather abstract vector algebra
but you might want to look at:

 "Rule-Based Simplification in Vector-Product Spaces"
 by Songxin Liang and David J. Jeffrey.

I have an almost finished implementation/port for Axiom:

 https://github.com/raoulb/fricas_code/tree/master/vecalg

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