How can the integral form of Gauss's law for magnetism be described as a version of general Stokes' theorem? How does it follow?

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How can the integral form of Gauss&#039;s law for magnetism be described as a version of general Stokes&#039; theorem? How does it follow?

Brianna Sims · Accepted Answer

Maxwell&#039;s equations in curved spacetime are written in the form                              ∇          a                          F                      a            b                                              =        −        4        π                  J          b                ,                                      ∇                      [            a                                    F                      b            c            ]                                              =        0        ,            with   F the Faraday two-form,       J    a   the current four-vector,   ∇ the covariant derivative and [] denotes antisymmetrization of the indices. In terms of exterior calculus they become:                    d        ⋆        F                            =        4        π        ⋆        J                            d        F                            =        0        ,            with   ⋆ the Hodge dual, which sends p-forms to   4  −  p-forms in dimension 4. If we integrate the left side of the first equation over a space-like hypersurface of dimension 3,   Σ, with normal time-like vector       t    a  , then Stokes&#039; theorem yields      ∫    Σ    d  ⋆  F  =      ∫    S    ⋆  F  ,with S the boundary of   Σ with normal       n    a  . Since   Σ is space-like and   (  ⋆  F      )          c      d        =      1    2        F          a      b            ϵ          a      b      c      d      , S is also space-like and one component of F must be time-like, therefore   ⋆  F  =      F          a      b            t    a        n    b    d  S. This is easy to see also if one takes the restriction of the dual on S in local coordinates, all the 2-forms are space-like. But we know that       E    a    =      F          a      b            t    b  , hence      ∫    S    ⋆  F  =  −      ∫    S        F          b      a            t    a        n    b    d  S  =  −      ∫    S        E    b        n    b    d  S  .Now we integrate the right side,      ∫    Σ    ⋆  J  =      ∫    Σ        J    a        ϵ          a      b      c      d        =      ∫    Σ        J    a        t    a    d  Σ  =  −  q  ,and after combining this and the previous one, we obtain:      ∫    S        E    a        n    a    d  S  =  4  π  q  ,Gauss&#039;s law applies also in curved spacetime. Note that       ϵ          a      b      c      d       is the Levi-Civita (volume) tensor, not the symbol. In local coordinates its components are the product of the symbol with             |        det                  g                  μ          ν                            |      For the case of the magnetic field,       B    a    =  −      1    2        ϵ          a      b      c      d            F          c      d            t    b  , only the space-like components of       F          a      b       are used, and the magnetic field part of the Faraday tensor is  F  =      F          12        d      x    1    ∧  d      x    2    +      F          23        d      x    2    ∧  d      x    3    +      F          31        d      x    3    ∧  d      x    1    ,and the components of the field are       B    1    =  −      |    det      g          μ      ν                  |              −      1              /            2            F          23       etc, therefore the integrals become  0  =      ∫    S    F  =  −      ∫    S              |        det          g              μ        ν                    |        (      B    1    d      x    2    ∧  d      x    3    +      B    2    d      x    3    ∧  d      x    1    +      B    3    d      x    1    ∧  d      x    2    )  =  −      ∫    S        B    a        n    a    d  S  .

How can the integral form of Gauss's law for magnetism

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