Let $x: U \subset R^2 \to S$ be the parametrization of a surface of revolution S: $$x(u,v)=(f(v)cos(u),f(v)sin(u),g(v)), f(v)>0$$ $$U=\{(u,v) \in R^2; 0<u<2\pi,a<v<b\}$$
a) Show that the map $\phi: U \to R^2$ given by $$\varphi (u,v)=(u,\int \frac{\sqrt{(f'(v))^2+(g'(v))^2}}{f(v)}dv)$$ is a local diffeomorphism.
b) Use part a to prove that a surface of revolution S is locally conformal to a plane in such a way that each local conformal map $\theta:V \subset S → R^2$ takes the parallels and the meridians of the neighborhood V into an orthogonal system of straight lines in $\theta (V) \subset R^2$.
honestly i don't have any idea to proceed. For a) i think that continuity is straightforward, because partial derivatives are continuous, so this is derivable and then continuous. What about inverse? the first component is u=u that is derivable but i don´t know how to work with the second component. For b) any idea is welcome.