Poincaré lemma
The Poincaré lemma states that every closed differential form is locally exact (http://planetmath.org/ExactDifferentialForm).
Theorem.
(Poincaré Lemma) [1] Suppose $X$ is a smooth manifold^{}, ${\mathrm{\Omega}}^{k}\mathit{}\mathrm{(}X\mathrm{)}$ is the set of smooth differential $k$forms on $X$, and suppose $\omega $ is a closed form in ${\mathrm{\Omega}}^{k}\mathit{}\mathrm{(}X\mathrm{)}$ for some $k\mathrm{>}\mathrm{0}$.

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Then for every $x\in X$ there is a neighbourhood $U\subset X$, and a $(k1)$form $\eta \in {\mathrm{\Omega}}^{k1}(U)$, such that
$$d\eta ={\iota}^{\ast}\omega ,$$ where $\iota $ is the inclusion $\iota :U\hookrightarrow X$.

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If $X$ is contractible, this $\eta $ exists globally; there exists a $(k1)$form $\eta \in {\mathrm{\Omega}}^{k1}(X)$ such that
$$d\eta =\omega .$$
Notes
Despite the name, the Poincaré lemma is an extremely important result. For instance, in algebraic topology, the definition of the $k$th de Rham cohomology group
$${H}^{k}(X)=\frac{\mathrm{Ker}\{d:{\mathrm{\Omega}}^{k}(X)\to {\mathrm{\Omega}}^{k+1}(X)\}}{\mathrm{Im}\{d:{\mathrm{\Omega}}^{k1}(X)\to {\mathrm{\Omega}}^{k}(X)\}}$$ 
can be seen as a measure of the degree in which the Poincaré lemma fails. If ${H}^{k}(X)=0$, then every $k$ form is exact, but if ${H}^{k}(X)$ is nonzero, then $X$ has a nontrivial topology (or “holes”) such that $k$forms are not globally exact. For instance, in $X={\mathbb{R}}^{2}\setminus \{0\}$ with polar coordinates^{} $(r,\varphi )$, the $1$form $\omega =d\varphi $ is not globally exact.
References
 1 L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
Title  Poincaré lemma 

Canonical name  PoincareLemma 
Date of creation  20130322 14:06:28 
Last modified on  20130322 14:06:28 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  12 
Author  matte (1858) 
Entry type  Theorem 
Classification  msc 5300 
Related topic  ExactDifferentialForm 
Related topic  ClosedDifferentialFormsOnASimpleConnectedDomain 
Related topic  LaminarField 