Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: [[arc length of a path]] Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* ---- - Let $k \leq n$; - Let $A \subset \rrk$ be [[open set|open in]] $\rrk$; - Let $\alpha: A \to \rrn$ be [[continuously differentiable]]; - Set $Y=\alpha(A)$ and define the [[parameterized Euclidean manifold]] $Y_ \alpha = (Y,\alpha).$ > [!definition] Definition. ([[volume of a parameterized manifold]]) > We define the **volume** of $Y _ \alpha$ as $v(Y _ \alpha)= \int _A V(D \alpha) \, \d x_{1} \dots \d x_{k}, $ where $V(D\alpha)$ is the [[volume of a parallelopiped|volume of the parallelopiped]] formed by the columns of $D \alpha$, $V (D \alpha ) = V ( \begin{pmatrix} | & | & | \\ \frac{ \partial \alpha }{ \partial x_{1} } & \dots & \frac{ \partial \alpha }{ \partial x_{k} } \\ | & | & | \end{pmatrix} ).$ > [!justification]- > We should check that this definition accords with the specializations we've considered in the past. For example: > ###### Arc Length Agreement > Let $I \subset \rr$ be an [[open set|open]] [[interval]] in $\rr$ and let $\alpha: I \to \rrn$ be a [[parameterized curve]]. Define the [[parameterized Euclidean manifold]] $Y_ \alpha = (\alpha(I),\alpha).$ We have at arbitrary $t \in I$ $V(D\alpha (t)) = \sqrt{ \det \big((D \alpha(t)^{\top} D \alpha (t)\big)}= \sqrt{ \|D\alpha(t)\| ^{2} }=\|D\alpha(t)\|;$ > therefore $V(Y_\alpha) = \int _I V(D\alpha) = \int _I \|D\alpha\| $ > and this is indeed the [[arc length of a path]] formula. > [!basicexample] > ###### 1. (Example of agreement with [[arc length of a path|arc length formula]]) Let $\alpha : [0,3\pi) \subset \rr \to \rr^{2}$ be given by $\alpha(t)= \big(2\cos t,2\sin t \big).$ Then in general $D\alpha = \begin{bmatrix} -2\sin t \\ 2 \cos t \end{bmatrix}$. Now, $V(D\alpha)= \sqrt{ (D \alpha)^{\top} D \alpha } = \sqrt{ \begin{bmatrix} -2\sin t \\ 2 \cos t \end{bmatrix} \cdot \begin{bmatrix} -2\sin t \\ 2 \cos t \end{bmatrix}} =\sqrt{ 4 }=2.$ So our answer is $\int_{0}^{3\pi} 2\, \d t = 6\pi$. Indeed, this is the 'length one would travel' if they traversed around a circle of radius 2 one-and-a-half-turns. \ Compare this to the [[arc length of a path|arc length]] of $\alpha$, $\text{arclength}_\alpha = \int_{a}^{b}\|\alpha'(t) \, \d t\| = \int_{0}^{3\pi } \, \sqrt{ \begin{bmatrix} -2\sin t \\ 2 \cos t \end{bmatrix} \cdot \begin{bmatrix} -2\sin t \\ 2 \cos t \end{bmatrix}} = \int_{0}^{3\pi} 2 \, \d t = 6\pi, $ as expected. > [!intuition] > ![[CleanShot 2023-01-09 at 12.14.43.jpg]] ![[CleanShot 2023-01-09 at 12.16.07.jpg]] ![[CleanShot 2023-01-09 at 12.16.29.jpg]] ![[CleanShot 2023-01-09 at 12.19.56.jpg]] ![[MOC volume of a parameterized manifold]] ---- #### ---- #### 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 > ```