Definition:Compatible Atlases

Definition
Let $M$ be a locally Euclidean space of dimension $d$.

Let $\mathscr F, \mathscr G$ be atlases of class $\mathcal C^k$ on $M$.

We say that $\mathscr F$ and $\mathscr G$ are compatible if for all charts $\left( U, \phi\right) \in \mathscr F$, $\left(V,\psi\right) \in \mathscr G$ the map $\phi \circ \psi^{-1}$ is of class $\mathcal C^k$ from $\psi\left(U \cap V\right)$ to $\phi\left(U \cap V\right)$.