Oriented Smooth Atlas on a Smooth Manifold with Boundary

Let $M$ be a smooth manifold with boundary. An oriented smooth atlas on $M$ is a smooth atlas

$$\mathcal{A}=\bigl((U_\alpha,\varphi_\alpha)\bigr)_{\alpha\in A}$$

such that for every $\alpha,\beta\in A$, the charts $(U_\alpha,\varphi_\alpha)$ and $(U_\beta,\varphi_\beta)$ are positively compatible on the interior of $M$ in the sense of Positive Compatibility of Charts on the Interior of a Smooth Manifold with Boundary.