---- - Suppose $f: M \to \rr$, where $M$ is a [[compact]] [[differentiable Euclidean submanifold (with or without boundary)|differentiable k-manifold (with or without boundary)]] in $\rrn$, is [[continuous]]. - Suppose there is a [[coordinate patch]] $\alpha: \widehat{U} \subset \rrk \to U \subset \rrn$ such that [[support|supp]] $f \subset U$.[^1] > [!definition] Definition. ([[single-patch scalar integral over a compact manifold]]) >Define (but see [[single-patch scalar integral over a compact manifold#^67c4d0|below]]) $\int _{M}f \, \d V := \int _{\text{int } \widehat{U} \, } (f \circ \alpha )V(D \alpha) \, ,$ >where $V(D\alpha)$ is [[volume of a parallelopiped|as defined here]]. **Remark.** This wasn't proven in class, but Munkres demonstrates that the RHS is indeed ordinary [[integral over a bounded set|integrable]] and that the value of $\int _{M} f \ \d V \,$ does not depend on the choice of $\alpha$. > [!generalization] > While the notation $\int _{M} f \ \mathrm{d} V$ arose above as a matter of definition, it is *not* merely notation when considered in the more general context of integrating $k$-forms over smooth $k$-manifolds. > > Specifically, from [[differentiable Euclidean submanifold|the rank condition in the embedded Euclidean submanifold definition]] ([[columns linearly independent iff gram matrix is invertible|plus this]]) we observe $V(D \alpha): (\widehat{U} \subset \mathbb{R}^{k}) \to \mathbb{R}_{\geq 0}$ is nonvanishing. This canonically defines an [[orientation of a smooth manifold|orientation form]] $\d V=(V(D \alpha) \circ \alpha ^{-1}) \ dx^{1} \wedge \dots \wedge dx^{k}$, which we may integrate against using [[integral of a form over a compact oriented Euclidean submanifold|this]] or [[integration of a compactly supported volume form on an oriented smooth manifold|this]] definition. Then, writing out these more general definitions, one *deduces* $\begin{align} > \int _{M} f \ \mathrm{d} V &= \int _{M} f \cdot (V(D\alpha) \circ \alpha ^{-1}) \ dx^{1} \wedge \dots \wedge dx^{k} \\ > &= \int _{\text{int }\widehat{U}} f \cdot (V(D\alpha) \circ \alpha ^{-1}) \circ \alpha \\ > &= \int _{\text{int }\widehat{U}} (f \circ \alpha) \ V(D_{\alpha}) > \end{align}$ > where $f \cdot (V(\alpha) \circ \alpha ^{-1})$ is a *product* of functions $U \subset M \to \mathbb{R}$. > ^67c4d0 ![[CleanShot 2023-01-20 at 20.18.54.jpg]] > [!basicexample] > ![[CleanShot 2023-01-20 at 20.34.39 1.jpg]] ![[MOC single-patch scalar integral over a compact manifold]] [^1]: Note that since $\alpha ^{-1}(\text{supp }f)$ is [[compact]] ([[Heine-Borel theorem|hence bounded]]) in $\rrk$, we may choose $\mathcal{U}$ to be [[bounded set|bounded]]. ---- #### ---- #### 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 > ```