So, I have $ f(x,y) = (x^2-y^2, 2xy) $, which is a local $\mathcal C^1$ isomorphism in $\mathbb R^2 \setminus \{(0,0)\}$.
I have to write this function in polar coordinates:
$f(x,y) = f(r\cos\phi, r\sin\phi).$
My beginnings: I know that
$df(r, \phi) = \cos\phi -r\sin\phi \sin\phi r\cos\phi.$
But I really have no clue how to work this one out. It is a single exercise of this kind in my book; so, I probably don't need it for the test, but I would like to know.