Stokes theorem for smooth differential forms is well-known. If $\alpha$ is a smooth differential $n$-form defined on an $(n+1)$-dimensional compact oriented manifold with boundary, then we have $$\int_Md\alpha=\int_{\partial M}\alpha.$$
In physics, one considers a generalized form of this equation, where the manifold need not be compact, and the differential form need not be smooth everywhere; it may have poles.
For example, in electrostatics one sees a 2-form field generated by a point charge $Q$ $$E=\dfrac{Q}{4\pi\epsilon_0r^3}\left(x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy\right)=\dfrac{Q}{4\pi\epsilon_0r^2}d\Omega_{S^2},$$ where $d\Omega_{S^2}$ is the volume form of the sphere. I suppose this must be understood as a distribution-valued form, because the physicist computes
$$dE=\frac{Q}{4\pi\epsilon_0}\delta^3(r)\,d\text{vol}_{\mathbb{R}^3},$$
where $\delta^3$ is the Dirac delta function in $\mathbb{R}^3$. Perhaps this is a weak exterior derivative?
The physics discussions I have seen justify this computation by saying it is necessary to make the Gauss's law hold. But where I come from, we don't conspire to ensure theorems hold because we like them, but rather prove from the definitions that theorems follow logically. Could you say how this theory of distribution valued differential forms looks, and state a version of Stokes' theorem for it? I would also accept a reference.