Editing Maxwell's equations (section)

=== Flux and divergence ===

[[File:Divergence theorem in EM.svg|thumb|Volume {{math|Ω}} and its closed boundary {{math|∂Ω}}, containing (respectively enclosing) a source {{math|(+)}} and sink {{math|(−)}} of a vector field {{math|'''F'''}}. Here, {{math|'''F'''}} could be the {{math|'''E'''}} field with source electric charges, but ''not'' the {{math|'''B'''}} field, which has no magnetic charges as shown. The outward [[unit normal]] is '''n'''.]]

According to the (purely mathematical) [[divergence theorem|Gauss divergence theorem]], the [[electric flux]] through the 
[[homology (mathematics)|boundary surface]] {{math|∂Ω}} can be rewritten as
: {{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega} \mathbf{E}\cdot\mathrm{d}\mathbf{S}=\iiint_{\Omega} \nabla\cdot\mathbf{E}\, \mathrm{d}V</math>
The integral version of Gauss's equation can thus be rewritten as 
<math display="block"> \iiint_{\Omega} \left(\nabla \cdot \mathbf{E} - \frac{\rho}{\varepsilon_0}\right) \, \mathrm{d}V = 0</math>
Since {{math|Ω}} is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied [[if and only if]] the integrand is zero everywhere. This is 
the differential equations formulation of Gauss equation up to a trivial rearrangement.

Similarly rewriting the [[magnetic flux]] in Gauss's law for magnetism in integral form gives  
: {{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega} \mathbf{B}\cdot\mathrm{d}\mathbf{S} = \iiint_{\Omega} \nabla \cdot \mathbf{B}\, \mathrm{d}V = 0.</math>
which is satisfied for all {{math|Ω}} if and only if <math> \nabla \cdot \mathbf{B} = 0</math> everywhere.