---- > [!definition] Definition. ([[coordinate chart]]) >Let $M$ be a $n$-[[topological manifold]]. A **coordinate chart** on $M$ is a pair $(U, \varphi)$, where $U$ is an open subset of $M$ and $\varphi: U \to \widehat{U}$ is a [[homeomorphism]] from $U$ to an open subset $\widehat{U}=\varphi(U) \subset \mathbb{R}^{n}$. > >Also see the related notion of [[coordinate patch]]. > >The component functions $(x^{1},\dots,x^{n})$ are called **local coordinates on $U$**. > [!definition] Definition for Euclidean Manifolds. ([[coordinate chart]]) > Let $k \leq n$ and let $M \subset \rrn$.A **coordinate chart** $\varphi: V \to \mathcal{U}$ on a (Euclidean) [[manifold]] $M$ is a [[homeomorphism]] between an a set $V \subset M$ [[subspace topology|open in]] $M$ and an [[open set]] $\mathcal{U} \subset$ $\rrk$. > \ We call $\varphi ^{-1}$ a [[coordinate patch]], often denoted $\alpha:\mathcal{U} \to V$. >\ >Sometimes written as the pair $(\varphi, V)$. >\ ![[CleanShot 2023-01-10 at 18.13.32 1.jpg]] ---- #### ---- #### 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 > ```