Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: [[single-patch scalar integral over a compact manifold]] Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Properties:: [[linearity of scalar integral over compact manifold]], [[computing scalar integrals over compact manifolds]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- - Suppose $f: M \to \rr$ is [[continuous]], where $M$ is a [[compact]] [[differentiable Euclidean submanifold (with or without boundary)|differentiable k-manifold (with or without boundary)]] in $\rrn$. > [!definition] Definition. ([[scalar integral over a compact Euclidean submanifold]]) > Define $\int _{M} f \, \d V := \sum_{i=1}^{\ell} \int _{M} (\phi_{i} f) = \sum_{i=1}^{\ell} \int _{\UU _{i}} \big( (\phi_{i}f) \circ \alpha \big) V(D\alpha) \, .$ The middle equality references the [[single-patch scalar integral over a compact manifold]]; the rightmost makes this reference explicit. The $\phi_{i}$ are a [[Partition of Unity on a manifold (deprecated)|partition of unity on the manifold]] $M$. ![[single-patch scalar integral over a compact manifold#^67c4d0]] > [!justification] > Munkres proves the following: > - If $\text{supp }f$ lies in a single [[coordinate patch]], then the result should equal that for [[single-patch scalar integral over a compact manifold]]. > - $\int _{M} f \d V \,$ is independent of the choice of [[partition of unity]] for $M$. ![[MOC scalar integral over a compact 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 > ```