---- > [!definition] Definition. ([[transition map]]) > Let $M$ be an $n$-[[topological manifold]]. If $(U, \varphi)$ and $(V, \psi)$ are two [[coordinate chart|coordinate charts]] such that $U \cap V \neq \emptyset$, the composite $\psi \circ \varphi ^{-1}: \varphi(U \cap V) \to \psi (U \cap V)$ is called a **transition map from $\varphi$ to $\psi$**. It is a composition of [[homeomorphism|homeomorphisms]], and hence itself a [[homeomorphism]]. > Since $\psi \circ \varphi ^{-1}$ is a map between open subsets of $\mathbb{R}^{n}$, it makes sense to discuss [[derivative|differentiation]]. If $\psi \circ \varphi ^{-1}$ is in fact a $C^{\infty}$ [[Euclidean diffeomorphism]], or $U \cap V = \emptyset$, then we say the charts $\varphi$ and $\psi$ are **smoothly compatible**. ---- #### ---- #### 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 > ```