$X=U\cup V$ with two connected coordinates neighborhood and $U\cap V $ is connected, then $X$ is orientable.

Question

After reading this post Show that $S$ is non-orientable came up with this question

Let $X$ be a connected surface with two connected coordinate neighborhoods $U$ and $V$ with $X=U\cup V$. If $U\cap V$ is connected, I want to show $X$ is orientable.

Answer 1

Choose some orientation on $U$ and choose orientation on $V$ so that they coincide in some point $x_0 \in U \cap V.$ Then if $s(x)$ is the sign of the Jacobi matrix, $s(x_0) = 1$ and $s$ is locally constant. So $s \equiv 1$ on $U \cap V.$