Assume stationary currents in vacuum, $\text{curl}\textbf{B} = \mu_0\textbf{J}$. With $\text{curl}\textbf{A}=\textbf{B}$ and $\text{div}\textbf{A} =0$ the vector potential $\textbf{A}$ can be written explicitly as a spatial integral over all space of the current density $\textbf{J}$, (see for example Jackson, section 5.4): $$\textbf{A(x)} = \frac{\mu_0}{4\pi}\int \textbf{J(x')}\frac{d^3x'}{\textbf{|x-x'|}} \tag{1}$$ So far so good, but what is puzzling about this integral is that the current density $\textbf{J}$ is associated with a 2-form and we are integrating it with differential 3D volume elements not 2D surface elements as I would have naively expected it.
Compare (1) with a similar formula for the scalar potential $\phi$ where $\textbf{E}=-\text{grad} \phi $, and now the charge density $\rho$ is given: $$\phi{(\textbf{x})} = \frac{1}{4\pi\epsilon_0 }\int \rho{(\textbf{x}')}\frac{d^3x'}{\textbf{|x-x'|}} \tag{2}$$ Charge density $\rho$ is a spatial density and as such is associated with a 3-form, and as I would expect it is integrated with a 3D differential volume element.
My question is what does this all mean, what happens when something that is apparently a natural 2-form is integrated if it was something else?