---- > [!definition] Definition. ([[closed form]]) > Let $M$ be a [[smooth manifold]] of dimension $n$. A [[differential form]] $\alpha \in \Omega^{r}(M)$ is said to be **closed** if $\alpha \in \text{ker }d_{r}$, i.e., $d\alpha=0$. > > Every [[exact form]] is closed, since $d_{r} \circ d_{r-1} \equiv 0$. The converse is false. ^definition ---- #### ---- #### 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 > ```