Editing Generalized Stokes theorem (section)

==Special cases==
The general form of the Stokes theorem using differential forms is more powerful and easier to use than the special cases. The traditional versions can be formulated using [[Cartesian coordinates]] without the machinery of differential geometry, and thus are more accessible. Further, they are older and their names are more familiar as a result. The traditional forms are often considered more convenient by practicing scientists and engineers but the non-naturalness of the traditional formulation becomes apparent when using other coordinate systems, even familiar ones like spherical or cylindrical coordinates. There is potential for confusion in the way names are applied, and the use of dual formulations.

===Classical (vector calculus) case===
{{Main|Stokes' theorem}}

[[Image:Stokes' Theorem.svg|thumb|right|An illustration of the vector-calculus Stokes theorem, with surface <math>\Sigma</math>, its boundary <math>\partial\Sigma</math> and the "normal" vector {{mvar|n}}.]]

This is a (dualized) (1&nbsp;+&nbsp;1)-dimensional case, for a 1-form (dualized because it is a statement about [[vector field]]s). This special case is often just referred to as ''Stokes' theorem'' in many introductory university vector calculus courses and is used in physics and engineering. It is also sometimes known as the '''[[Curl (mathematics)|curl]]''' theorem.

The classical Stokes' theorem relates the [[surface integral]] of the [[Curl (mathematics)|curl]] of a [[vector field]] over a surface <math>\Sigma</math> in Euclidean three-space to the [[line integral]] of the vector field over its boundary. It is a special case of the general Stokes theorem (with <math>n=2</math>) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral, <math>\partial\Sigma</math>, must have positive [[curve orientation|orientation]], meaning that <math>\partial\Sigma</math> points counterclockwise when the [[normal (geometry)|surface normal]], <math>n</math>, points toward the viewer.

One consequence of this theorem is that the [[field line]]s of a vector field with zero curl cannot be closed contours. The formula can be rewritten as:{{clear}}
{{math theorem
| math_statement = Suppose <math>\textbf{F}=\big(P(x,y,z),Q(x,y,z),R(x,y,z)\big)</math> is defined in a region with smooth surface <math>\Sigma</math> and has continuous first-order [[partial derivatives]]. Then
<math display="block">
 \iint_\Sigma \Biggl(\left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) dy \, dz + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) dz\,dx + \left (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dx \, dy\Biggr)
  = \oint_{\partial\Sigma} \Big(P\,dx + Q\,dy + R\,dz\Big)\,,
</math>
where <math>P,Q</math> and <math>R</math> are the components of <math>\textbf{F}</math>, and <math>\partial\Sigma</math> is the boundary of the region <math>\Sigma</math>.
}}

===Green's theorem===
[[Green's theorem]] is immediately recognizable as the third integrand of both sides in the integral in terms of {{mvar|P}}, {{mvar|Q}}, and {{mvar|R}} cited above.

====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.

===Divergence theorem===
Likewise, the [[divergence theorem]]
<math display="block">\int_\mathrm{Vol} \nabla \cdot \mathbf{F} \, d_\mathrm{Vol} = \oint_{\partial \operatorname{Vol}} \mathbf{F} \cdot d\boldsymbol{\Sigma}</math>
is a special case if we identify a vector field with the <math>(n-1)</math>-form obtained by contracting the vector field with the Euclidean volume form. An application of this is the case <math>\textbf{F}=f\vec{c}</math> where <math>\vec{c}</math> is an arbitrary constant vector. Working out the divergence of the product gives
<math display="block">\vec{c} \cdot \int_\mathrm{Vol} \nabla f \, d_\mathrm{Vol} = \vec{c} \cdot \oint_{\partial \mathrm{Vol}} f\, d\boldsymbol{\Sigma}\,.</math>
Since this holds for all <math>\vec{c}</math> we find
<math display="block">\int_\mathrm{Vol} \nabla f \, d_\mathrm{Vol} = \oint_{\partial \mathrm{Vol}} f\, d\boldsymbol{\Sigma}\,.</math>

===Volume integral of gradient of scalar field===
Let <math>f : \Omega \to \mathbb{R}</math> be a [[scalar field]]. Then
<math display="block">\int_\Omega \vec{\nabla} f = \int_{\partial \Omega} \vec{n} f</math>
where <math>\vec{n}</math> is the [[normal vector]] to the surface <math>\partial \Omega</math> at a given point.

Proof:
Let <math>\vec{c}</math> be a vector. Then
<math display="block">
\begin{align}
0
&= \int_\Omega \vec{\nabla} \cdot \vec{c} f - \int_{\partial \Omega} \vec{n} \cdot \vec{c} f & \text{by the divergence theorem} \\
&= \int_\Omega \vec{c} \cdot \vec{\nabla} f - \int_{\partial \Omega} \vec{c} \cdot \vec{n} f \\
&= \vec{c} \cdot \int_\Omega \vec{\nabla} f - \vec{c} \cdot \int_{\partial \Omega} \vec{n} f \\
&= \vec{c} \cdot \left( \int_\Omega \vec{\nabla} f - \int_{\partial \Omega} \vec{n} f \right)
\end{align}
</math>
Since this holds for any <math>\vec{c}</math> (in particular, for every [[basis vector]]), the result follows.