\begin{definition}[Definition:Compatible Charts]
Let $M$ be a topological space.
Let $d$ be a natural number.
Let $\struct {U, \phi}$ and $\struct {V, \psi}$ be $d$-dimensional charts of $M$.
Then $\struct {U, \phi}$ and $\struct {V, \psi}$ are '''$C^k$-compatible''' {{iff}} their transition mapping:
:$\psi \circ \phi^{-1}: \map \phi {U \cap V} \to \map \psi {U \cap V}$
is of class $C^k$.
\end{definition}