I need to solve this exercise:
A continuous form $w$ of degree $r$ in the open $U\subset\mathbb{R}^m$ is regular $\iff$ for all open $V$ with compact closure $\overline{V} \subset U$, there is a sequence of forms $w_k$ of degree $r$, with differentiable coefficients, such that $w_k\to w$ and $dw_k \to \overline{w}$ uniformly in $V$ (that is, each coefficient converges uniformly in $V$). Conclude that the exterior product of two regular forms is regular and that $\overline{w_1\wedge w_2} = \overline{w}\wedge w_2 + (-1)^{r_1}w_1\wedge \overline{w_2}$
My book says that a regular form $w$ is when this happens (even though this definition isn't necessary for the exercise, I want to prove the exercise's $\iff$ part):
$w$ is a continuous form of degree $r$ in an open $U\subset\mathbb{R}^m $ when there exists, in $U$, a regular form $\overline{w}$, of degree $r+1$, such that $\int_M \overline{w} = \int_{ \partial M}w$ for all surfaces $M$ of class $C^2$ with boundary contained in $U$.
Even though the exercise does not ask me to prove the assertion it does about regularity and the condition of sequences of forms $w_k$ existing, I'm trying to find a proof of it, but there's not even mention about regular forms on the internet. Are they called another name in english?
Also, I see no connection between the existence of a sequence and the exterior product of these forms. Any idea?
Update:
I tried doing:
$w = \sum_I a_I d_I$ in open $U$ is regular, then for all open $V$ with compact closure $\overline{V}$, there is $w_k = \sum_I a_{Ik}d_i$, where $a_{Ik}\to a_I$. The exterior derivative of $w_k$, called $dw_k$ is given by
$$dw_k = \sum_{j,I} \frac{\partial a_{Ik}}{\partial x_j}dx_j\wedge dx_I$$
We have that $dw_k \to \overline{w}$, where $\overline{w}$ is uniquely determined. I don't know, however, how to find how $\overline{w}$ looks. I just know that the terms of $dw_k$ converge to the terms of $w_k$.
I also need to conclude that the product of two regular forms $w$ and $z$ are regular, probably using the $\iff$ that the exercise talks about. I looks like I need to use the formula for $d(w_1\wedge w_2) = dw_1\wedge w_2 + (-1)^r(w_1\wedge dw_2)$.