Suppose we have two coordinate systems: (representing the usual Cartesian coordinates, for example) and (representing cylindrical polars, say). I think it's true that:
but I can't rigorously prove this. (It feels right based on the chain rule but I can't write a formal proof, and haven't found anything online.) Can anyone help please?
On a similar note, for functions is it true that
I'm less sure that this second formula holds in general.