Definition:Smooth Mapping

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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 if and only if:

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:
$\ds \xi \circ \phi \circ \kappa^{-1}: \map \kappa U \subseteq \R^m \to \map \xi {V \cap \map \phi U} \subseteq \R^n$
is smooth.