---- - Let $C$ be an [[orientation of a Euclidean submanifold|oriented]] $1$-[[differentiable Euclidean submanifold (with or without boundary)|manifold]] in $\rrn$; - Suppose there exists $\gamma=\boldsymbol r: (a,b) \to \rrn$, a [[coordinate patch]] belonging to the [[orientation of a Euclidean submanifold|orientation]] s.t. $M - \gamma\big( (a,b) \big)$ has [[measure zero in manifold|measure zero]] in $C$ (i.e. $\operatorname{im }\gamma$ [[cover|covers]] $C$ up to a subset of [[measure zero]]). Write $t$ for the [[parameterized curve|parameter]] [[coordinate chart|charting]] $\operatorname{im }\gamma$. - Let $\omega$ be a 1-[[differential form|form]] on $\mathbb{R}^{n}$, $\omega=F_{i} \ dy^{i} \in \Omega^{1}(\mathbb{R}^{n})$, $F_{i} \in C^{1}(\mathbb{R}^{n})$. [[musical isomorphism induced by a nondegenerate bilinear form|Musically identify]] $\omega$ with the [[vector field]] $\boldsymbol F=(F_{1},\dots,F_{n}) \in \mathscr{V}(\mathbb{R}^{n})$. > [!theorem] Theorem. ([[circulation integral]]) > ![[CleanShot 2023-03-18 at 14.24.53.jpg]] > Then $\int _{C} \omega = \int _{C} (\boldsymbol F \cdot \boldsymbol T) \ \d s = \int_{a}^{b} (\boldsymbol F \circ \boldsymbol r) \cdot \boldsymbol r' \overbrace{ = }^{ \text{notation} } \int _{a}^{b} \boldsymbol F\big( \boldsymbol r(t) \big) \frac{d\boldsymbol r}{dt}dt \overbrace{ = }^{ d\boldsymbol r:= \frac{d \boldsymbol r}{dt} dt }\int _{C} \boldsymbol F \cdot d\boldsymbol r, $ > where: > - $\int _{C} \omega$ denotes the [[integration of a compactly supported volume form on an oriented smooth manifold|integral]] ([[integral of a form over a compact oriented Euclidean submanifold|or]]) of the $1$-form $\omega$ over the [[smooth manifold|manifold]] $C$ > - $\boldsymbol T(t)=\frac{\gamma'(t)}{\|\gamma'(t)\|} \in \mathbb{R}^{n}$ is the normalized [[velocity vector of a parameterized curve|tangent]] > - $\mathrm{d}s=\mathrm{d}V$ is the [[single-patch scalar integral over a compact manifold#^67c4d0|canonical]] [[orientation of a smooth manifold|orientation]] [[differential form|volume form]] ([[arc length of a path|arc length]]), which in this case is $\mathrm{d}V=\|\gamma'\| \ dt$ > The first three equalities are earnest (though straightforward) deductions; the latter two are the 'classical calculus' way of notating them. > We call this value the **circulation integral of $\omega$ over $C$** or **line integral over $C$ with respect to the vector field $\boldsymbol F$**. > When the curve $C$ is closed, the symbol $\oint$ is sometimes used. > [!proof]- Proof. ([[circulation integral]]) > On the one hand, using the second specialization in [[integration of a compactly supported volume form on an oriented smooth manifold]] and [[computation of dual transform of forms]], > $\begin{align} > \int _{C} \omega &= \int _{a}^{b} \gamma^{*} \omega \\ > &= \int _{a}^{b} \gamma^{*}\left( \sum_{i} F_{i} dy^{i} \right) \\ > &= \sum_{i} \int _{a}^{b} (F_{i} \circ \gamma) \ d \gamma^{i} \\ > &= \sum_{i} \int _{a}^{b} (F_{i} \circ \gamma) \ \gamma'(t) dt \text{ (this is the integral of a 1-form)} \\ > &= \sum_{i} \int _{a}^{b} (F_{i} \circ \gamma ) \gamma'(t) \text{ (this is an ordinary Riemann integral)} > \end{align}$ > . On the other hand, $\begin{align} > \int _{C} (\boldsymbol F \cdot \boldsymbol T) \ \mathrm{d}V &= \boldsymbol \int _{C} (\boldsymbol F \cdot \boldsymbol T) \ \|\gamma'\| \ dt \\ > &= \int _{a}^{b} \left( \boldsymbol F \cdot \underbrace{\boldsymbol{ T}}_{\frac{\gamma'}{\cancel{ \|\gamma'\| }}} \right) \circ \gamma \ \cancel{ \|\gamma'\| } \\ > &= \int _{a}^{b} \sum_{i} (F_{i} \circ \gamma) \ \gamma'_{i} > \end{align}$ > > > > (Redo the below) > > Write > $\begin{align} \int _{M} \sum_{i=1}^{n} F_{i} dx_{i} = & \int_{a}^{b} \alpha^{*} \left( \sum_{i=1}^{n} F_{i} dx_{i} \right) \\ > = & \sum_{i=1}^{n} \int _{a} ^b \alpha^{*}(F_{i}) \alpha^{*}( dx_{i}) \\ > = & \sum_{i=1}^{n} \int_{a}^{b} (F_{i} \circ \alpha) d \alpha _{i} \\ > = & \sum_{i=1}^{n} \int_{a}^{b} (F_{i} \circ \alpha) \alpha _i '(t) \ dt \text{ (this is an integral of a form)}\\ > = & \sum_{i=1}^{n} \int_{a}^{b} (F_{i} (\alpha(t)) \alpha _i '(t) \text{ (this is an integral of a function)}\\ \\ > = & \int _{a}^{b} F(\alpha(t)) \cdot D\alpha (t) , \text{ where } F=(F_{1},\dots,F_{n}) \\ > = & \int _{a} ^{b} F\big( \alpha (t) \big) \cdot \frac{D\alpha (t)}{\|D \alpha (t)\|} \|D \alpha (t)\| \text{ (the UTF appears)} \\ > = & \int _{a} ^{b} \big(F \cdot T\big) (\alpha(t)) \|D\alpha (t)\| \\ > = & \int _{a} ^{b} (F \cdot T) \circ \alpha \ V(D\alpha) \\ > =: & \int _{M}(F \cdot T) \ \d V. > \end{align}$ > (Where we get $\|D\alpha(t)\|=V(D\alpha)$ because $V(D\alpha)= \det \sqrt{ D\alpha^{\top}D\alpha }=\det \sqrt{ D\alpha \cdot D\alpha } = \det \|D\alpha(t)\|$) > [!intuition] > ![[CleanShot 2023-03-18 at 14.46.59.jpg]] ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```