Properties:: [[volume of a parameterized manifold|definition of volume of a parameterized manifold]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- - Let $k \leq n$; - Let $A \subset \rr ^{k}$ be [[open set|open in]] $\rr ^{k}$; - Let $\alpha: \rr ^{k} \to \rr ^{n}$ be [[continuously differentiable]]. > [!definition] Definition. ([[parameterized Euclidean manifold]]) > Set $Y=\alpha(A)$. The pair $Y_\alpha := (Y,\alpha )$ is called a **parameterized (Euclidian) manifold** of dimension $k$. > [!intuition]- > Examples include [[line|lines]] and surfaces in $\rr ^{3}$. > ###### 1. (Parameterized Curve) > Define $\alpha : (0, 3 \pi) \subset \rr \to \rr ^{2}$ given by $\alpha(t)=(2 \cos t, 2\sin t).$ > ![[CleanShot 2023-01-08 at 22.56.48.jpg]] > > ###### 2. (Parameterized Surface) > Define $\alpha : (0, \pi) \subset \rr^{2} \to \rr ^{3}$ by $\alpha(\theta, \varphi)=(2\cos \theta \sin \varphi , 2\sin \theta \sin \varphi, 2\cos \varphi).$ > Compare to the [[spherical coordinates]] map. > ![[CleanShot 2023-01-08 at [email protected]]] > > ###### 3. ([[graph]] of a [[continuously differentiable]] function) > Suppose $\Omega \subset \rrn$ is [[open set|open in]] in $\rrn$ and $g: \Omega \to \rr$ a [[continuously differentiable]] function. ![[CleanShot 2023-01-08 at 23.08.07.jpg]] > We can make the [[graph]] of $g$ *into* a [[parameterized Euclidean manifold]] by taking $\alpha(\v x) = \big(\v x, g(x)\big) \in \rrn \times \rr. $ > Then $Y = \alpha(\Omega)$ is the [[graph]] of $g$. ---- #### ---- #### 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 > ```