Smooth function
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Definition
A smooth function on a smooth [ilmath]n[/ilmath]-manifold, [ilmath](M,\mathcal{A})[/ilmath], is a function[1] [ilmath]f:M\rightarrow\mathbb{R}^k[/ilmath] that satisfies:
[ilmath]\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A} [/ilmath] such that [ilmath]f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k [/ilmath] is smooth in the usual sense, of having continuous partial derivatives of all orders.
Any smoothly compatible map (so all in the atlas of the smooth manifold) will have a smooth transition function, by composition, the result will be smooth, so [ilmath]f[/ilmath] is still smooth.
See also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
