How can we change variables from $(x,y)$ to $(r,\theta)$ for the metric on the open disc $r<\delta$ defined by $(dx^2+dy^2)\over g(\sqrt{x^2+y^2})^2$ where $g(\sqrt{x^2+y^2})>0$ $\forall r<\delta$?
I am tempted to say the transformed metric is $dr^2\over g(r)^2$, but There might be some monkey business with Jacobians or such?