Definition:Compatible Atlases/Definition 2

Definition
Let $M$ be a Hausdorff space.

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

Then $\mathscr F$ and $\mathscr G$ are compatible for every pair of charts $\left({U, \phi}\right) \in \mathscr F$ and $\left({V, \psi}\right) \in \mathscr G$, the transition mapping


 * $\phi \circ \psi^{-1}: \psi \left({U \cap V}\right) \to \phi \left({U \cap V}\right)$

is of class $C^k$.