Orientable and Oriented Smooth Manifold with Boundary

Let $M$ be a smooth manifold with boundary. We say that $M$ is orientable if it admits an oriented smooth atlas in the sense of Oriented Smooth Atlas on a Smooth Manifold with Boundary.

An oriented smooth manifold with boundary is a smooth manifold with boundary together with a chosen oriented smooth atlas. Such a chosen oriented smooth atlas is called an orientation of $M$.