Ampère-Maxwell law from differential forms

Question

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

I want to do this by integrating the second Maxwell equation, not by figuring out the differential equations. I already achieved this for Gauß's law by integrating $\mathrm d\!\star\!F$ in a volume, but i couldn't figure it out in the case of Ampère, since it looks 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 because 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.

Answer 1

It seems like you have too many misconceptions to get the answer.

You should be using the differential forms version, and the theorems in that framework, to relate everything together. You should be starting with [note sign change between Equation (1) and (2)] \begin{align} \tag1F=&E_x\,\mathrm dx\wedge\mathrm dt+E_y\,\mathrm dy\wedge\mathrm dt+E_z\,\mathrm dz\wedge\mathrm dt+B_x\,\mathrm dy\wedge\mathrm dz+B_y\,\mathrm dz\wedge\mathrm dx+B_z\,\mathrm dx\wedge\mathrm dy\\ \tag2\star F=&B_x\,\mathrm dt\wedge\mathrm dx+B_y\,\mathrm dt\wedge\mathrm dy+B_z\,\mathrm dt\wedge\mathrm dz+E_x\,\mathrm dy\wedge\mathrm dz+E_y\,\mathrm dz\wedge\mathrm dx+E_z\,\mathrm dx\wedge\mathrm dy\\ \mathrm d\!\star\!F=&+(\partial_xE_x+\partial_yE_y+\partial_zE_z)\,\mathrm dx\wedge\mathrm dy\wedge\mathrm dz-(\partial_yB_z-\partial_zB_y-\partial_tE_x)\,\mathrm dt\wedge\mathrm dy\wedge\mathrm dz\\ \tag3&-(\partial_zB_x-\partial_xB_z-\partial_tE_y)\,\mathrm dt\wedge\mathrm dz\wedge\mathrm dx-(\partial_xB_y-\partial_yB_x-\partial_tE_z)\,\mathrm dt\wedge\mathrm dx\wedge\mathrm dy\\ \tag4\star J=&-\rho\star\!\mathrm dt+j_x\star\!\mathrm dx+j_y\star\!\mathrm dy+j_z\star\!\mathrm dz\\ \tag5\therefore\ \rho=&\vec\nabla\cdot\vec E\qquad\bigwedge\qquad\vec J+\partial_t\vec E=\vec\nabla\times\vec B \end{align} where we have compared Equations (4) and (3) to get (5). Now, we can try to use $\int_\Omega\mathrm d\!\star\!F=\int_{\partial\Omega}\star F$ $$\tag{6, Gauß's Law}\int_\Omega\rho\,\mathrm dx\wedge\mathrm dy\wedge\mathrm dz=\oint_{\partial\Omega} E_x\,\mathrm dy\wedge\mathrm dz+E_y\,\mathrm dz\wedge\mathrm dx+E_z\,\mathrm dx\wedge\mathrm dy$$ $$\tag7\mathrm dt\wedge\oint_{\partial\Omega}B_x\,\mathrm dx+B_y\,\mathrm dy+B_z\,\mathrm dz=\mathrm dt\wedge\int_\Omega(j_x+\partial_tE_x)\,\mathrm dy\wedge\mathrm dz+(j_y+\partial_tE_y)\,\mathrm dz\wedge\mathrm dx\\+(j_z+\partial_tE_z)\,\mathrm dx\wedge\mathrm dy\qquad\text{(Ampère-Maxwell)}$$ That is, these are a mix of 3-forms and 2-forms, and 1-forms are not supposed to be appearing in them.