---- > [!definition] Definition. ([[smooth maps between manifolds]]) > Let $M$ and $N$ be [[smooth manifold|smooth manifolds]] and let $F:M \to N$ be [[continuous]]. We say $F$ is a **smooth map at $p \in M$** if there exists [[coordinate chart|charts]] $(\varphi,U)$ about $p$ on $M$, $(\psi, V)$ about $F(p)$ on $N$, such that $\psi \circ f \circ \varphi ^{-1}$ is [[continuously differentiable|smooth]] where defined. > ^d1c773 > [!justification] > We need to make sure this definition does not depend on the choice of chart (coordinates); so let $(\tilde{\varphi}, \tilde{U})$ and $(\tilde{\psi}, \tilde{V})$ be two other charts; then $\tilde{\psi} \circ f \circ \tilde{\varphi}^{-1} = (\tilde{\psi} \circ \psi ^{-1}) \circ (\psi \circ f \circ \varphi ^{-1}) \circ (\varphi \circ \tilde{\varphi} ^{-1}) $ is [[smooth]] because the [[transition map|transition maps]] are and $\psi \circ f \circ \varphi ^{-1}$ is smooth by assumption. ^justification > [!basicexample] > - [[restrictions of smooth maps to regular surfaces are smooth]] > - More generally: [[TODO]] > - A map $f: \Omega \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$ is smooth as a manifold map iff it is smooth in the usual sense > - Every [[coordinate chart]] is a smooth map between manifolds (indeed, a [[diffeomorphism]]) > - A composition of smooth maps is, of course, smooth ^abdee3 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```