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edited Jul 21, 2023 at 19:03

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Ampére Ampère-Maxwell law from differntialdifferential forms

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

I wannewant to do this by inegratingintegrating the second Maxwell equation, not by figuring out the differntialdifferential equations. I already achieved this for Gauss'sGauß's law by integrtingintegrating $d*F$$\mathrm d\!\star\!F$ in a volume, but i couldn't figure it out in the case of AmpereAmpère, scincesince it looklooks 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 becuasebecause of the duality between 1-forms, and 2 forms-forms, but in 4d4D, 2-forms are dual to 2-forms.

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

Ampére-Maxwell law from differntial forms

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.

Ampère-Maxwell law from differential forms

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.