Diffeomorphism Between Surfaces Preserves Orientability

Question

From Do Carmo (Exercise 2.6.2).

Let $S_2$ be an orientable regular surface and $\varphi:S_1\rightarrow S_2$ be a local diffeomorphism at every $p\in S_1$. Prove $S_1$ is orientable.

Up until this point in the book, there is no mention of atlases or charts, so the other stack posts similar to this aren't really of much help. I'm not sure how to begin. What do I need to show? Any guidance would be appreciated.

Answer 1

This exercise reads a bit different in Montiel and Ros's book Curves and Surfaces, 2nd edition:

Let $f: S_1 \longrightarrow S_2$ be a local diffeomorphism between two surfaces and let $S_2$ be orientable. Let $N_2$ be a unit normal field on $S_2$. Define a map $N_1: S_1 \longrightarrow \Bbb{R}^3$ as follows: if $p \in S_1$ put $$ N_1(p) = \frac{a \wedge b}{|a \wedge b|} $$ where $\{a, b\}$ is a basis for $T_pS_1$ satisfying $$ \det ((df)_p(a), (df)_p(b), N_2(f(p))) > 0. $$ Show that $N_1$ is a unit normal field on $S_1$ and hence $S_1$ is orientable.

Can you continue from there? Let me know if you need further clarifications.