Definition:Smooth Mapping

Definition
Let $M, N$ be smooth manifolds.

Denote $m := \dim M$ and $n := \dim N$.

Let $\phi: M \to N$ be a mapping.

Then $\phi$ is a smooth mapping :
 * for every chart $\struct {U, \kappa}$ on $M$ and every chart $\struct {V, \xi}$ on $N$ such that $V \cap \map \phi U \ne \O$, the mapping:
 * $\displaystyle \xi \circ \phi \circ \kappa^{-1}: \map \kappa U \subseteq \R^m \to \map \xi {V \cap \map \phi U} \subseteq \R^n$
 * is smooth.