Let $X$ be a [[Set|set]] and $(U_i, \psi_i)_{i \in I}$ be an [[Atlas|atlas]] of class $C^p$. > [!definition] > > Let $U \subset X$ be [[Open Set|open]] set and $\psi: U \to U'$ be a [[Homeomorphism|homeomorphism]] into an open subset of $E$. The chart $(U, \psi)$ is **compatible** with the atlas $(U_i, \psi_i)_{i \in I}$ defined on a suitable intersection if each $\psi_i\psi^{-1}$ is a [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions)-isomorphism. > [!definition] > > Let $(V_j, \varphi_j)_{j \in J}$ be another atlas of class $C^p$. The atlases $(V_j, \varphi_j)_{j \in J}$ and $(U_i, \psi_i)_{i \in I}$ are **compatible** if each $\varphi_j$ is compatible with $(U_i, \psi_i)_{i \in I}$. > [!theorem] > > Let $(U, \varphi) \in X$ be a chart compatible with $(U_i, \psi_i)_{i \in I}$. Let $\widehat{U} = \varphi(U)$ and $\phi: \widehat{U} \to \widehat{U'}$ be a $C^p$-isomorphism, then $(U, \phi \circ \varphi)$ is also compatible with $(U_i, \psi_i)_{i \in I}$. > > *Proof*. Let $p \in X$ and $i \in I$ such that $x \in U_i$, then > $ > \psi_i \circ (\phi \circ \varphi)^{-1} = \underbrace{\psi_i \circ \varphi^{-1}}_{C^p} \circ \phi > $ > is a composition of $C^p$-isomorphisms, making it a $C^p$-isomorphism. > [!theorem] > > Define two atlases on $X$ to be equivalent if they are compatible, then this is an [[Equivalence Relation|equivalence relation]]. > > *Proof*. From the third axiom, every atlas is compatible with itself. > > Let $(U_i, \psi_i)_{i \in I}$, $(V_j, \varphi_j)_{j \in J}$, and $(W_k, \chi_k)_{k \in K}$ be atlases of class $C^p$. > > If $(U_i, \psi_i)_{i \in I}$ is compatible with $(V_j, \varphi_j)_{j \in J}$, then $\varphi_j \circ \psi_i^{-1}$ is a $C^p$ isomorphism for any $j \in J$ and $i \in I$. Therefore $\psi_i \circ \varphi_j^{-1}$ is also a $C^p$ isomorphism for each $j \in J$ and $i \in I$, and the relation is symmetric. > > Suppose that $(U_i, \psi_i)_{i \in I}$ is compatible with $(V_j, \varphi_j)_{j \in J}$ and $(V_j, \varphi_j)_{j \in J}$ is compatible with $(W_k, \chi_k)_{k \in K}$. Then the map $\varphi_j \circ \psi_i^{-1}$ is a $C^p$ isomorphism for all $j$, and $\chi_k \circ \varphi_j^{-1}$ is a $C^p$ isomorphism for all $k$. Therefore for any suitable $j$s, $\chi_k \circ \varphi_j \circ \varphi_j^{-1} \circ \psi_i^{-1}$ is equal to $\chi_k \circ \varphi_i^{-1}$ and is a $C^p$ isomorphism, making the relation transitive. # Final Topology > [!theorem] > > Let $(V_j, \varphi_j)_{j \in J}$ be another atlas of class $C^p$. If $(V_j, \varphi_j)_{j \in J}$ is compatible with $(U_i, \psi_i)_{i \in I}$, then they induce the same [[Topology on Manifold|topology]]. > > *Proof*. Let $i \in I$, $\psi_i: U_i \to E_i$ and $W \subset E_i$ be an open set, then $\psi_i^{-1}(W) = \bigcup_{j \in J}\psi_i^{-1}(W) \cap V_j$. Since the maps $\varphi_j \circ \psi_i^{-1}$ from $U_i \cap V_j$ are all $C^p$ isomorphisms, > $ > \psi_i^{-1}(W) = \bigcup_{j \in J}\varphi_j^{-1} \circ \underbrace{\varphi_ j\circ \psi_i^{-1}(W)}_{\text{open in }\varphi_j(V_j)} > $ > is a union of open sets in the final topology induced by $(V_j, \varphi_j)_{j \in J}$. Since this relation is symmetric, so is the argument, and the topologies are the same. # Unifying Model Space > [!theorem] > > Let $X$ be a set and $(U_i, \psi_i)_{i \in I}$ be an atlas with $\psi_i: U_i \to E_i$ for all $i$. If all the $E_i$s are [[Space of Toplinear Isomorphisms|toplinearly isomorphic]] to $E$, then there exists an equivalent $E$-atlas $(U_i, \varphi_i)_{i \in I}$. > > *Proof*. Let $T_i: E_i \to E$ be a toplinear isomorphism for each $i \in I$. Define $\varphi_i = T_i \circ \psi_i$, then $\varphi_i$ is still a $C^p$-isomorphism, and for any $i, j \in I$, $\psi_j \circ \varphi_i^{-1} = \psi_j \circ \psi_i^{-1} \circ T_i^{-1}$, which is a $C^p$-isomorphism whenever the domain is suitable. # Change of Coordinates > [!theorem] > > Let $X$ be a [[n-Manifold|n-manifold]], $p \in X$, and $(U, \varphi)$, $(V, \psi)$ be [[Atlas|charts]] at $p$. Denote > $ > \psi \circ \varphi^{-1}(x) = \paren{\tilde x^1(x), \cdots, \tilde x^n(x)} > $ > as the **coordinate representation** the translation map. This induces an isomorphism $d(\psi \circ \varphi^{-1})_{\hat p}: T_{p}\real^n \to T_{p}\real^n$, which has coordinate representation > $ > \ppi \bigg\vert_{\hat p} \mapsto \frac{\partial \tilde x^j}{\partial x^i}(\hat p) \cdot \frac{\partial}{\partial \tilde x^j}\bigg\vert_{\hat p} > $ > therefore > $ > \ppip = \frac{\partial \tilde x^j}{\partial x^i}(\hat p) \cdot \frac{\partial}{\partial \tilde x^j}\bigg\vert_p > $ > > *Proof*. > $ > \begin{align*} > \ppip &= d\varphi^{-1}_{\hat p} \cdot \ppi \bigg\vert_{\hat p} \\ > &= d\varphi^{-1}_{\hat p} \circ d(\psi \circ \varphi^{-1})_{\hat p}\ppi\cdot \bigg\vert_{\hat p} \\ > &= d\varphi^{-1}_{\hat p} \cdot \braks{\frac{\partial \tilde x^j}{\partial x^i}(\hat p) \cdot \frac{\partial}{\partial \tilde x^j}\bigg\vert_{\hat p}} \\ > &= \frac{\partial \tilde x^j}{\partial x^i}(\hat p) \cdot \frac{\partial}{\partial \tilde x^j}\bigg\vert_{p} > \end{align*} > $