Editing Maxwell's equations (section)

=== Key to the notation ===
<!---please do not make this list much longer – we used to have a gigantic table of all the constants, variables, terminology, and units, which was converted into prose as it probably should for an encyclopedia (see the history in pre-2013). Editors (by all means in good faith) may add the units, alternative names and symbols, etc. to the list and make it longer and denser, then eventually there would be a good reason to resurrect the big table format again...--->

Symbols in '''bold''' represent [[Vector (geometric)|vector]] quantities, and symbols in ''italics'' represent [[scalar (physics)|scalar]] quantities, unless otherwise indicated.
The equations introduce the [[electric field]], {{math|'''E'''}}, a [[vector field]], and the [[magnetic field]], {{math|'''B'''}}, a [[pseudovector]] field, each generally having a time and location dependence.
The sources are
* the total electric [[charge density]] (total charge per unit volume), {{math|''ρ''}}, and
* the total electric [[current density]] (total current per unit area), {{math|'''J'''}}.

The [[universal constant]]s appearing in the equations (the first two ones explicitly only in the SI formulation) are:
* the [[permittivity of free space]], {{math|''ε''<sub>0</sub>}}, and
* the [[permeability of free space]], {{math|''μ''<sub>0</sub>}}, and
* the [[speed of light]], <math>c = ({\varepsilon_0\mu_0})^{-1/2}</math>

==== Differential equations ====
In the differential equations, 
*the [[nabla symbol]], {{math|∇}}, denotes the three-dimensional [[gradient]] operator, [[del]],
*the {{math|∇⋅}} symbol (pronounced "del dot") denotes the [[divergence]] operator,
*the {{math|∇×}} symbol (pronounced "del cross") denotes the [[curl (mathematics)|curl]] operator.

==== Integral equations ====
In the integral equations, 
* {{math|Ω}} is any volume with closed [[boundary (topology)|boundary]] surface {{math|∂Ω}}, and
* {{math|Σ}} is any surface with closed boundary curve {{math|∂Σ}},
The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the [[differentiation under the integral sign]] in Faraday's law:
<math display="block"> \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} = \iint_{\Sigma}  \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{S}\,,</math>
Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss' and Stokes' theorems as appropriate. 
* {{oiint
   | intsubscpt=
   | integrand=}}<math>{\vphantom{\int}}_{\scriptstyle\partial \Omega}</math> is a [[surface integral]] over the boundary surface {{math|∂Ω}}, with the loop indicating the surface is closed
* <math>\iiint_\Omega</math> is a [[volume integral]] over the volume {{math|Ω}},
* <math>\oint_{\partial \Sigma}</math> is a [[line integral]] around the boundary curve {{math|∂Σ}}, with the loop indicating the curve is closed.
* <math>\iint_\Sigma</math> is a [[surface integral]] over the surface {{math|Σ}},
* The ''total'' [[electric charge]] {{math|''Q''}} enclosed in {{math|Ω}} is the [[volume integral]] over {{math|Ω}} of the [[charge density]] {{math|''ρ''}} (see the "macroscopic formulation" section below): <math display="block">Q = \iiint_\Omega \rho \ \mathrm{d}V,</math> where {{math|d''V''}} is the [[volume element]].
* The ''net'' [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} is the [[surface integral]] of the magnetic field {{math|'''B'''}} passing through a fixed surface, {{math|Σ}}: <math display="block">\Phi_B = \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d} \mathbf{S},</math>
* The ''net'' [[electric flux]] {{math|Φ<sub>''E''</sub>}} is the surface integral of the electric field {{math|'''E'''}} passing through {{math|Σ}}: <math display="block">\Phi_E = \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d} \mathbf{S},</math>
* The ''net'' [[electric current]] {{math|''I''}} is the surface integral of the [[electric current density]] {{math|'''J'''}} passing through {{math|Σ}}: <math display="block">I = \iint_{\Sigma} \mathbf{J} \cdot \mathrm{d} \mathbf{S},</math> where {{math|d'''S'''}} denotes the differential [[vector area|vector element]] of surface area {{math|''S''}}, [[Normal (geometry)|normal]] to surface {{math|Σ}}. (Vector area is sometimes denoted by {{math|'''A'''}} rather than {{math|'''S'''}}, but this conflicts with the notation for [[magnetic vector potential]]).