This question regards making sympy's geometric algebra module use both covariant and contravariant vector forms to make the output much more compact. So far I am able to use one or the other, but not both together. It may be that I don't know the maths well enough, and the answer is in the documentation after all.
Some background:
I have a system of equations that I want to solve in a complicated non-orthogonal coordinate system. The metric tensor elements of this coordinate system are known, but their expressions are unwieldy so I'd like to to keep them hidden and simply use gij, the square root of its determinant J, and gij. Also it's useful to describe vectors, V, in either their contravariant or their covariant forms,
V = ∑Viei = ∑Viei,
and transform between them where necessary.
Here ei = ∇u(i) and u(i) is the ith coordinate, and ei = ∂R/∂u(i). This notation is the same as that used in this invaluable text, which I cannot recommend more. Specifically, chapter 2 will be useful for this question.
There are many curls and divergence operations in the system of equations I'm trying to solve. The former is most simply expressed with the contravariant form of the a vector, and the latter with the covariant:
∇.V = 1/J ∑∂u(i)JVi,
∇ x V = εijk/J (∂u(i)Vi)ei,
where εijk is the Levi-Cevitta symbol. I would consider this question answered if I could print the above two equations using sympy's geometric algebra module.
How does one configure sympy's geometric algebra module to express calculations in this manner i.e. using covariant and contravariant vector expressions in order to hide away the complicated nature of the coordinate system?
Maybe there is an alternative toolbox that does exactly this?
