Definition:Smooth Mapping

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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 iff:

for every chart $\left({U, \kappa}\right)$ on $M$ and every chart $\left({V, \xi}\right)$ on $N$ such that $V \cap \phi \left({U}\right) \ne \varnothing$, the mapping:
$\displaystyle \xi \circ \phi \circ \kappa^{-1}: \kappa \left({U}\right) \subseteq \R^m \to \xi \left({V \cap \phi \left({U}\right)}\right) \subseteq \R^n$
is smooth.