Poincaré's Lemma states that on a contracting manifold, all closed forms are exact. \
Can anyone explain me the following statement using Poincaré's Lemma?
Consider a function $G(x):\mathbb{R}^n\rightarrow \mathbb{R}^m$, $m\leq n$. If $G(x)$ is integrable then Poincaré's Lemma ensures the existance of a function $\Gamma(x)$ such that $G(x)=\nabla_x\Gamma$.
where: $\nabla_xG=\dfrac{\partial G}{\partial x}$ and $x\in \mathbb{R}^n$ need not be scalar. For example of $G(x)=\begin{bmatrix} x_1x_2\\ \dfrac{1}{2}x_1^2\end{bmatrix}$. This will give $\Gamma=\dfrac{1}{2}x_1^2x_2$.