I need help with the the questions in the attachment. For the first I think I will have to use the regular value theorem for part of it. For the second, I need to use the theorem that f is a local isometry iff there is a basis e,e' of the tangent space such that df(e).df(e')=e.e', df(e).df(e)=e.e and df(e').df(e')=e'.e'. I want to find a basis by choosing a suitable co-ordinate chart for the domain. I'm a bit unsure how to deal with the fact that f is a function of rcos(theta) and rsin(theta) rather than r and theta.