Editing Generalized Stokes theorem (section)

====In electromagnetism====
Two of the four [[Maxwell equations]] involve curls of 3-D vector fields, and their differential and integral forms are related by the special 3-dimensional (vector calculus) case of [[Stokes' theorem]]. Caution must be taken to avoid cases with moving boundaries: the partial time derivatives are intended to exclude such cases. If moving boundaries are included, interchange of integration and differentiation introduces terms related to boundary motion not included in the results below (see [[Differentiation under the integral sign]]):

{| class="wikitable" border="1"
! Name
! [[Partial differential equation|Differential]] form
! [[Integral]] form (using three-dimensional Stokes theorem plus relativistic invariance, <math>\textstyle\int\tfrac{\partial}{\partial t}\dots\to\tfrac{d}{dt}\textstyle\int\cdots</math>)
|- valign="center"
| Maxwell–Faraday equation<br> [[Faraday's law of induction]]:
| style="text-align: center;" | <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
| style="text-align: center;" | <math>\begin{align}
\oint_C \mathbf{E} \cdot d\mathbf{l} &= \iint_S  \nabla \times \mathbf{E} \cdot d\mathbf{A} \\
&= -\iint_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A}
\end{align} </math>
(with {{mvar|C}} and {{mvar|S}} not necessarily stationary)
|- valign="center"
| [[Ampère's law]]<br /> (with Maxwell's extension):
| style="text-align: center;" | <math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>
| style="text-align: center;" | <math>\begin{align}
\oint_C \mathbf{H} \cdot d\mathbf{l} &= \iint_S \nabla \times \mathbf{H} \cdot d\mathbf{A}\\ &= \iint_S \mathbf{J} \cdot d\mathbf{A} + \iint_S \frac{\partial \mathbf{D}}{\partial t} \cdot d\mathbf{A}
\end{align} </math>
(with {{mvar|C}} and {{mvar|S}} not necessarily stationary)
|}
The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in [[SI units]]. In other systems of units, such as [[Maxwell's equations#CGS units|CGS]] or [[Gaussian units]], the scaling factors for the terms differ. For example, in Gaussian units, Faraday's law of induction and Ampère's law take the forms:<ref>{{cite book|first=J.&nbsp;D.| last=Jackson |title=Classical Electrodynamics|url=https://archive.org/details/classicalelectro00jack_0|url-access=registration |edition=2nd|publisher=Wiley|location=New York, NY|date=1975| isbn=9780471431329 }}</ref><ref>{{cite book|first1=M.|last1=Born|first2=E.|last2=Wolf| title=[[Principles of Optics]]|edition=6th|publisher=Cambridge University Press|location=Cambridge, England|date=1980}}</ref>
<math display="block">\begin{align}
\nabla \times \mathbf{E} &= -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}\,, \\
\nabla \times \mathbf{H} &= \frac{1}{c} \frac{\partial \mathbf{D}} {\partial t} + \frac{4\pi}{c} \mathbf{J}\,,
\end{align}</math>
respectively, where {{mvar|c}} is the [[speed of light]] in vacuum.