This is a practice exercise from a geometry textbook by P. Wilson. Suppose we have a Riemannian metric of the form $|dz|^2/h(r)^2$ on an open disc of radius $\delta>0$ centered on the origin in $\mathbb C$, where $h(r)>0 $ for all $r<\delta$. I want to show that the gaussian curvature is K=hh'/r+hh"-(h')^2. Could someone please kindly help me out? I am thinking of using $K=-\sqrt{G}_{uu}/\sqrt{G}$ for a first fundamental form being $du^2+G(u,v)dv^2$. I can change variables to get the first fundamental form into the form $da^2+db^2$ but it isn't helpful at all here... Please help!
Perhaps a more fundamental question: What is $|dz|^2$? What is it in terms of $r$, say?