Let \(M\) and \(N\) denote [smooth manifolds](smooth manifold.md). A map \(F\colon M\to N\) is smooth if it is continuous and if for every \(p\in M\), there exists a smooth chart \((U,\varphi)\) for \(M\) with \(p\in U\) and a smooth chart \((V,\psi)\) with \(F(p)\in V\) such that \(\psi\circ F\circ \varphi^{-1}\) is smooth from \(\varphi(U\cap F^{-1}(V))\) to \(\psi(V)\).
Remarks
- \(\hat{F}=\psi\circ F\circ \varphi^{-1}\) is called the coordinate representation of \(F\) with respect to the given coordinates.
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- The identity map \(\id_M\) is smooth.