I am looking for some sensible convention for naming functions that represent scalar and vector fields in different coordinates systems.
Currently I am saying something like this:
Let $f$ be the scalar field in the plane given by the function $g(x, y) = x^2 + y^2$ in the Cartesian coordinates. Then in the polar coordinates the field $f$ is given by the function $h(r,\theta) = r^2$.
I would have much appreciated not having to invent arbitrary names for representations in different coordinate systems of a field named "$f$".
Have differential geometers or vector analysts invented anything?