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I was just reading about De Rham theorem :

Let $\Omega$ be an open set of $\mathbb R^n$. Let pose $\mathcal V := \{u=(u_1,...,u_N)\in \mathcal D^N (\Omega)\}$, And $f=(f_1,...,f_N) \in (\mathcal D' (\Omega))^N $; So The two properties are equivalent:

1) $\exists P \in \mathcal D' (\Omega)$ such that: $f = \nabla P . \\ $

2) $\left \langle f,v \right \rangle = 0$, $\forall v \in \mathcal V .$

I want to prove that $2\Rightarrow 1 $, ($1\Rightarrow 2$ is OKEY).

I need a reference for learning the proof. I will be very happy if someone could help me to prove it.

Thanks!

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I know it's a bit late, but you can find the original proof in

De Rham, G.: Differentiable Manifolds 1984.

at page 114, theorem 17.

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