---- > [!definition] Definition. ([[surface integral]]) > Let $S \subset \mathbb{R}^{3}$ be a [[differentiable Euclidean submanifold|regular surface]]; take a [[coordinate chart|chart]] $\varphi=(u,v):R \subset S \to \mathbb{R}^{2}$, $u,v:S \to \mathbb{R}$. Let $\alpha$ be the corresponding [[coordinate patch|patch]]. > > Suppose $f$ is a function/field on $S$. In this case the [[single-patch scalar integral over a compact manifold#^67c4d0|canonical]] [[differential form|volume form]] ([[area form]]) $\mathrm{d}V=: \mathrm{d}A$ is manifestly $\sqrt{ \det g } \ du \wedge dv$, where $g=\begin{bmatrix}E(u,v) & F(u,v) \\F(u,v) & G(u,v)\end{bmatrix}$ is the [[first fundamental form]] expressed in $\alpha$.[^1] Since $\det g=\|D_{u} \alpha \|^{2} \|D_{v} \alpha\|^{2} - (D_{u}\alpha \cdot D_{v} \alpha)^{2} = \|D_{u}\alpha \times D_{v} \alpha\|^{2},$ > ([[cross product|cf. #4]]), we have $\mathrm{d}A=\sqrt{ \det g } \ du \wedge dv= \|D_{u}\alpha \times D_{v}\alpha\| \ du \wedge dv.$ > Hence the [[scalar integral over a compact Euclidean submanifold|scalar]] [[integration of a compactly supported volume form on an oriented smooth manifold|integral]] $\int _{R} f\, \mathrm{d}V$ is $\int _{R} f \, \mathrm{d}A = \int _{R} f \ \|D_{u} \alpha \times D_{v}\alpha\| \ du \wedge dv = \int _{\varphi(R)} (f \circ \alpha) \|D_{u}\alpha \times D_{v}\alpha\|. $ > In classical calculus notation, one writes $\alpha=: \boldsymbol r$, $D_{u}\alpha=\boldsymbol r_{u}$, $D_{v}\alpha=\boldsymbol r_{v}$ and invokes the suggestive but meaningless notation $du \ dv$, writing: $\int _{R} f \ \mathrm{d}A =: \iint_{R} f\big( \boldsymbol r(u, v) \big) \ \|\boldsymbol r_{u} \times \boldsymbol r_{v}\| \, du \ dv .$ > We call $\int _{S} f \ \mathrm{d}A$ the **(scalar) surface integral of $f$ over the region $R$ of the regular surface $S$**. > [^1]: Put differently, $g$ is the [[Riemannian manifold|Riemannian metric]] on $S$ induced by pulling back the [[dot product]] on $\mathbb{R}^{3}$, and $\mathrm{d}A$ is the consequent [[Riemannian volume form]]. model off of ![[line integral]] ---- #### ---- #### 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 > ```