Non orientable manifold.

Question

I'm looking for a proof of the following statement:

Let $M$ be a smooth manifold covered by two connected charts $(U,\phi) , (V,\psi)$ such that $U\cap V$ has exactly two connected components, with the following propertie: determinant of change of coorinates is positive in one and negative in the other. Then $M$ is not orientable.

Answer 1

See Andrew Hwang's answer to this question: Manifold is not orientable

That's really all you need.

(The question isn't very well phrased, but the answer is still the tool you need!)