Definition:Differentiable Mapping between Manifolds/Point/Definition 1

Definition
$f$ is differentiable at $p$ for every pair of charts $\struct {U, \phi}$ and $\struct {V, \psi}$ of $M$ and $N$ with $p \in U$ and $\map f p \in V$:
 * $\psi \circ f \circ \phi^{-1}: \map \phi {U \cap \map {f^{-1} } V} \to \map \psi V$

is differentiable at $\map \phi p$.