---- > [!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|,]] $\nabla=(du)^{\sharp}$. ---- #### ---- #### 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 > ```