Let $M$ be a smooth manifold. If $M$ can be covered by two charts $(U,\varphi)$ and $(V,\phi)$ in such a way that the intersection $U\cap V$ is a connected subset of $M$, then $M$ is orientable.

This is an example in Do Carmo's *Riemannian Geometry*, but how to prove it? I have no ideas. Can anyone help me?