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Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad u_j\in \mathscr D'(\mathbb R^n), $$ and assume that $du=0$, i.e. $\partial u_j/\partial x_k=\partial u_k/\partial x_j$. I want to prove that there exists $a\in\mathscr D'(\mathbb R^n)$ such that $da=u$. The same question can be raised for tempered distributions and also for higher degrees.

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See page 20 of J P Demailly Complex Analytic geometry.The other source to look at is de Rham's book on differentiable manifolds.Yet another source to look at is Laurent Schwartz Theorie des Distributions pages 59 and 355 in 1978 edition .

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    $\begingroup$ de Rham's book is wonderful btw and still reads as very modern. $\endgroup$ – Tom Mrowka Feb 12 '16 at 21:58
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    $\begingroup$ @TomMrowka . Absolutely ,the presentation is extremely efficient . $\endgroup$ – Mohan Ramachandran Feb 14 '16 at 21:38
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The case of a 1-form is also given in Proposition 4.3.9 on page 334 in John Horváth's Topological Vector Spaces and Distributions.

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