---- > [!definition] Definition. ([[exact form]]) > Let $M$ be a [[smooth manifold]] of dimension $n$. > A [[differential form]] $\alpha \in \Omega^{r}(M)$ is called **exact** if it is the [[exterior derivative|exterior differential]] of something else: $\alpha \in \text{im }d_{r-1}$, i.e., $\alpha=d_{r-1}\beta$ for some $\beta \in \Omega^{r-1}(M)$. > When $r=1$, $\beta=u$ is a function $M \to \mathbb{R}$. We call $u$ a **potential function for the $1$-form $\alpha$**. In the presence of a [[metric tensor|metric]] [[musical isomorphism induced by a nondegenerate bilinear form|identifying]] $\Omega^{1}(M) \xrightarrow[\cong]{\sharp} \Gamma(TM)$, $\alpha \mapsto X$, $u$ is called a **potential function for the [[vector field]]** $X$, $X=\nabla u$[[gradient|,]] $\nabla=(du)^{\sharp}$. Note that the function $u$ is unique up to additive constants. A [[vector field]] $X$ which may be written as the gradient of a [[scalar field]] $u$ is said to be **conservative**. > [!equivalence] Equivalence when $r=1$. > > By [[The Fundamental Theorem of Calculus]]/[[Generalized Stokes' Theorem|GST]], a [[vector field]] $X \in \Gamma(TM)$ is conservative if and only if for every pair of points $a,b \in M$, $\gamma^{*}X^{\flat}$ has the same [[circulation integral|circulation]] for any (piecewise smooth) [[parameterized curve|path]] $\gamma:[t_{0}, t_{1}] \to M$ from $a$ to $b$. > > (In classical mechanics, this equivalence implies that a [[force]] $F$ is conservative if and only if the work done by $F$ on a particle moving from $a$ to $b$ in the [[configuration space]] $M$ is the same regardless of which path from $a$ to $b$ the particle takes.) > Equivalently, $X$ is conservative iff the circulation around any [[parameterized curve|loop]] in $M$ is zero,[^1] $\oint_{\gamma}X^{\flat}=0 \text{ for every loop } \gamma.$ [^1]: Indeed, if $X$ is conservative then any loop must have the same circulation as the trivial loop, and the trivial loop of course has zero circulation; conversely... > [!basicproperties] >- If $X$ is conservative and $M \subset \mathbb{R}^{3}$, [[curl|then]] $\nabla \times X=0$. The converse statement, that curl-free implies conservative for all vector fields, holds [[de Rham cohomology|iff]] $H^{1}(M; \mathbb{R})=(0)$, e.g. when $M$ is [[simply connected]]. > > > [!proof] Proof. > > Suppose $X$ is [[exact form|conservative]], so that $\oint_{\gamma} X \cdot dr=0$ for every [[parameterized curve|loop]] $\gamma$ in $M$. By [[Classical Stokes' Theorem]], this implies $0=\int_{S} (\nabla \times X) \cdot n=0$ for every [[surface]] $S$, whence the [[fundamental lemma of the calculus of variations]] gives $\nabla \times X=0$. Conversely, [[singular cohomology|suppose]] $H^{1}(M; \mathbb{R})=(0)$ [[de Rham's theorem|and]] $\nabla \times X=0$. Since $H^{1}=0$, every [[closed form]] is exact, in particular $X^{\flat}$ is exact, so $X$ is conservative. If $H^{1}(M;\mathbb{R}) \neq (0)$, then there exists a [[closed form]] $\alpha$ which is not exact, and so $\alpha^{\sharp}$ is not a conservative [[vector field]]. > > [!basicexample] > - [[Newtonian gravity]] is conservative > - [[Hooke's Law]] is conservative > - More generally, any [[central vector field|central force]] is conservative ^basic-example ---- #### ---- #### 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 > ```