Two point tensors?

[Chris Bradley]

Hi all,

I'm new to Sage so forgive me if this is a dumb question but does Sage deal with two-point tensors? By this I mean a second order tensor with one contravariant index in the tangent space of one manifold and one contravariant index in the tangent space of another manifold. The particular application is in solid mechanics and is the deformation gradient tensor which is given by the derivative of the map between the two manifolds i.e., \chi: M -> S where M has chart coordinates X and S has chart coordinates x and the deformation gradient tensor is given by F = \del \chi/ \del X. All I've managed to find, documentation wise, involves creating tensors from the tangent/cotangent space of a single manifold rather than a tensor from the tangent space of M and the tangent space of S. Thanks in advance.

Best wishes

Chris

[Chris Bradley]

Apologies, must read email when my brain is working. The deformation gradient has a covariant index on M and a contravariant index on S. As such it maps a vector on M to a (deformed m) vector on S i.e., x^i = F^i_J X^J where F^i_J = \del chi(X^i) / X^J where x  = \chi(X).

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[Eric Gourgoulhon]

Hi,

Could it be that you are looking for the pushforward operator associated to a differentiable map between two manifolds?

cf. https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/diff_map.html#sage.manifolds.differentiable.diff_map.DiffMap.pushforward

Eric.