I'm trying to understand E&M in the language of forms, and my goal is to get from the Maxwell equations: $dF=0$ and $d*F=*J$ to the integral form of Ampere-Maxwell law, namely something that looks like: $$\oint_{\partial S} \vec{B} \cdot d\vec{l}=\int_S \vec{J}\cdot d\vec{s}+\int_s \partial_t \vec{E}\cdot d\vec{s}  $$


I wanne do this by inegrating the second Maxwell equation, not by figuring out the differntial equations. I already achieved this for Gauss's law by integrting $d*F$ in a volume, but i couldn't figure it out in the case of Ampere, scince it look like on the left we have the integral of a 2-form in a curve, and on the right the integral of 1-form on a surface. In 3D this wouldn't be a problem becuase of the duality between 1-forms, and 2 forms, but in 4d 2-forms are dual to 2-forms.

I'm clearly going about it the wrong way, any help would be appreciated.