Editing Maxwell's equations (section)

=== 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.