Is there something in sage that does the same thing that
TransformedField in Mathematica 9 ?
The idea is that
CoordinateTransform is given some coordinates, e.g. (r,th) and asked to transform them from "polar" to "cartesian", thus gives the expression of the cartesian coordinates in terms of the polar coordinates, e.g.
(x(r,th), y(r,th)) = (r*cos(th), r*sin(th))
Obvioulsly, it also works with other coordinates systems.
TransformedField makes the transformation between a scalar, vector, or tensor field in, say, cartesian coordinates, to spherical coordinates.
These actions are not very complicated, nor difficult to implement when needed, but they are also very common.