> @Raoul: I have not yet taken a course in abstract algebra though I am 
> currently enrolled in a course of linear algebra. I can read it up of 
> course. Do you foresee any problems with not knowing abstract algebra
> just yet?

Oh, this one is not so much about "algebra" in the sense of either
polynomial or commutative algebra but the main point is that the
vectors are component free, this is what "algebra" refers to here.

The math behind this is fairly easy, mostly how to make the clever 
use of about 10 axioms or "rules".

This is of course no full GSOC project. But thinking about it
might help you to make a good proposal for general vector calculus.

Even if this is beyond your scope and project but in the long term,
component-wise vectors calculus and component-free vector algebra
should go hand in hand. (The situation is similar to the one of
matrix calculus with explicit matrices/entries and matrix expressions
ignoring the entries at all.)

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