Editing Maxwell's equations (section)

== Relationship between differential and integral formulations ==
<!---PLEASE NOTE: This section on the "relation between int/diff forms" is independent of units and should not be made a subsection or merged with the above section on SI units&nbsp;— it should stay in its own section, yet as close as possible to the first mention of the equations. Thanks. --->
The equivalence of the differential and integral formulations are a consequence of the [[divergence theorem|Gauss divergence theorem]] and the [[Kelvin–Stokes theorem]].

=== 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.

=== Circulation and curl ===
[[File:Curl theorem in EM.svg|thumb|Surface {{math|Σ}} with closed boundary {{math|∂Σ}}. {{math|'''F'''}} could be the {{math|'''E'''}} or {{math|'''B'''}} fields. Again, {{math|'''n'''}} is the [[unit normal]]. (The curl of a vector field does not literally look like the "circulations", this is a heuristic depiction.)]]

By the [[Stokes' theorem|Kelvin–Stokes theorem]] we can rewrite the [[line integral]]s of the fields around the closed boundary curve {{math|∂Σ}} to an integral of the "circulation of the fields" (i.e. their [[curl (mathematics)|curl]]s) over a surface it bounds, i.e.
<math display="block">\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \iint_\Sigma (\nabla \times \mathbf{B}) \cdot \mathrm{d}\mathbf{S},</math>
Hence the [[Ampère–Maxwell law]], the modified version of Ampère's circuital law, in integral form can be rewritten as 
<math display="block"> \iint_\Sigma  \left(\nabla \times \mathbf{B} - \mu_0 \left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)\right)\cdot \mathrm{d}\mathbf{S} = 0.</math>
Since {{math|Σ}} can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zero [[if and only if]] the Ampère–Maxwell law in differential equations form is satisfied.
The equivalence of Faraday's law in differential and integral form follows likewise.

The line integrals and curls are analogous to quantities in classical [[fluid dynamics]]: the [[circulation (fluid dynamics)|circulation]] of a fluid is the line integral of the fluid's [[flow velocity]] field around a closed loop, and the [[vorticity]] of the fluid is the curl of the velocity field.