Editing Generalized Stokes theorem (section)

===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>.
}}