---- > [!definition] Definition. ([[surface]]) > A **surface** is a [[topological manifold|topological]] $2$-manifold. > > [!specialization] > In the classical setting of calculus on manifolds in $\mathbb{R}^{3}$: > A subset $S$ of $\mathbb{R}^{3}$ is called a surface (without boundary) if it is locally [[homeomorphism]] to $\mathbb{R}^{2}$, in the sense that for each $p \in S$ there exists a > - A [[neighborhood]] $V \ni p$, [[subspace topology|(relatively) open]] in $S$; > - An open subset $\mathcal{U} \subset \mathbb{R}^{2}$; > - A [[coordinate chart]] $X: \mathcal{U} \to V$. ^specialization > [!generalization] > - [[topological manifold]] ^generalization > [!basicexample] >- [[plane|Affine planes]] in $\mathbb{R}^{3}$ are surfaces. Indeed, let $S:= \{ (x,y,z) \in \mathbb{R}^{3}: Ax+By+Cz = D \}.$We know $(A,B,C) \neq \b 0$ is perpindicular to the plane. WLOG $A \neq 0$. Then we can take $(-B,A,0)$ and $(-C,0,A)$ which are each orthogonal to $(A,B,C)$ and therefore in $S$, and which are [[linearly independent]] and therefore '[[submodule generated by a subset]]' $S$. In light of [[plane#Parameterization Equation]], given $p \in S$ we can construct a [[coordinate chart|chart]] $\begin{align} X: \mathbb{R}^{2} \to & S \\ (u,v) \mapsto & X(u,v) = (x_{0},y_{0},z_{0}) + u(-B,A,0) + v(-C,0,A). \end{align}$ $X$ is clearly [[continuous]]. It has inverse $\begin{align} X ^{-1} : S \to \mathbb{R}^{2} \\ (x,y,z) \mapsto X ^{-1}(x,y,z) = (\frac{y-y_{0}}{A}, \frac{z-z_{0}}{A}) \end{align}$ which is also [[continuous]]. So $X$ is a valid chart covering all of $S$. > The [[sphere]] $S_{r}(0)$ of radius $r>0$ in $\mathbb{R}^{3}$ (centered at origin) is a [[surface]]. The definition is$S_{r}= \{ (x,y,z) \in \mathbb{R}^{3}: x^{2}+y^{2}+z^{2} = r^{2} \}.$ > How should we cover it? We can rewrite the set as $S_{r}=\{ (x,y,z) \in \mathbb{R}^{3}: z^{2} = r^{2} - x^{2} - y ^{2} \}.$ > It is clear that we need $x^{2}+y^{2}<r$, i.e., we need $(x,y) \in B_{r}(0) \subset \mathbb{R}^{2}$. > > So we can start by covering the upper half-sphere thus: $\begin{align} > X_{\text{upper}}: B_{r}(0) \to & \mathbb{R}^{3}_{z > 0} \cap S_{r}(0) \\ > (u,v) \mapsto & (u,v, \sqrt{ r^{2} - u^{2} - v^{2} }), > \end{align}$ > and the lower half-sphere too: $\begin{align} > X_{\text{lower}}: B_{r}(0) \to & \mathbb{R}^{3}_{z < 0} \cap S_{r}(0)\\ > (u,v) \mapsto & (u,v, \sqrt{ r^{2} - u^{2} - v^{2} }). > \end{align}$ > (the inverses of both these functions is just $(u,v,\cdot) \mapsto (u,v)$, [[continuous]] as a [[projection function]]). > > We just have to cover the equator now. We can do a similar process but permuting the coordinates so that the images are $\mathbb{R}^{3}_{y>0}$ and $\mathbb{R}^{3}_{y<0}$. Same for $z$. Then, any $p \in S_{r}$ locates in one of the six charts given. So, $S_{r}$ is a [[surface]]. > >- [[the cone as a surface]] ---- #### ---- #### 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 > ```