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.