----- Any exact differential form is [[exterior derivative|manifestly closed]]. The Poincare Lemma gives a sufficient topological condition for the converse to hold. > [!proposition] Proposition. ([[Poincare lemma]]) > Assume $U \subset \mathbb{R}^{n}$ is an [[open ball]], $\alpha \in \Omega^{k}(U)$ a [[closed form|closed]] [[differential form|differential]] $k$-[[differential form|form]] ($k \geq 1$): $d \alpha=0$. Then $\alpha$ is [[exact form|exact]]: there exists $\beta \in \Omega^{k-1}(U)$ such that $d \beta=\alpha$. ^proposition > [!note] Note. > The Poincare lemma *does not follow* from the fact that the [[singular cohomology]] of a [[contractible]] [[topological space]] vanishes in positive degree, since the [[de Rham's theorem]] is proven *via* the Poincare lemma. ^note > [!proposition] Corollary. > The [[de Rham cohomology]] of an open ball $U$ in $\mathbb{R}^{n}$ is: $H_{\text{dR}}^{k}\big(N\mathbb{B}^{n}\big)=\begin{cases} 0& k > 0 \\ \mathbb{R} & k=0 \end{cases}.$ ^proposition > [!proof]- Proof. ([[Poincare lemma]]) > Not in this course. ----- #### ---- #### 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 > ```