If $S_1$ and $S_2$ are regular surfaces and $\phi: U \subset S_1 \to S_2$ is a differentiable mapping of an open set $U \subset S_1$ such that the differential $d\phi_p$ of $\phi$ at $p \in U$ is an isomorphism, then $\phi$ is a local diffeomorphism at $p$.
I think we have to apply Inverse Function Theorem.
Require Hints to proceed with the problem.