Editing Generalized Stokes theorem (section)

== Classical vector analysis example ==
Let <math>\gamma:[a,b]\to\R^2</math> be a [[piecewise]] smooth [[Jordan curve|Jordan plane curve]]. The [[Jordan curve theorem]] implies that <math>\gamma</math> divides <math>\R^2</math> into two components, a [[compact space|compact]] one and another that is non-compact. Let <math>D</math> denote the compact part that is bounded by <math>\gamma</math> and suppose <math>\psi:D\to\R^3</math> is smooth, with <math>S=\psi(D)</math>. If <math>\Gamma</math> is the [[space curve]] defined by <math>\Gamma(t)=\psi(\gamma(t))</math><ref name="cgamma" group="note"><math>\gamma</math> and <math>\Gamma</math> are both loops, however, <math>\Gamma</math> is not necessarily a [[Jordan curve]]</ref> and <math>\textbf{F}</math> is a smooth vector field on <math>\R^3</math>, then:<ref name="Jame">{{cite book |last=Stewart |first=James |url={{Google books |plainurl=yes |id=btIhvKZCkTsC |page=786 }} |title=Essential Calculus: Early Transcendentals |publisher=Cole |year=2010}}</ref><ref name="bath">This proof is based on the Lecture Notes given by Prof. Robert Scheichl ([[University of Bath]], U.K) [http://www.maths.bath.ac.uk/~masrs/ma20010/], please refer the [http://www.maths.bath.ac.uk/~masrs/ma20010/stokesproofs.pdf]</ref><ref name="proofwik">{{cite web |title=This proof is also same to the proof shown in |url=http://www.proofwiki.org/wiki/Classical_Stokes'_Theorem}}</ref>
<math display="block">\oint_\Gamma \mathbf{F}\, \cdot\, d{\mathbf{\Gamma}}  = \iint_S \left( \nabla \times \mathbf{F} \right) \cdot\, d\mathbf{S} </math>

This classical statement is a special case of the general formulation after making an identification of vector field with a 1-form and its curl with a two form through 
<math display="block">\begin{pmatrix}
F_x \\ 
F_y \\ 
F_z \\
\end{pmatrix}\cdot d\Gamma \to F_x \,dx + F_y \,dy + F_z \,dz</math>
<math display="block">\begin{align}
&\nabla \times \begin{pmatrix} F_x \\ F_y \\ F_z \end{pmatrix} \cdot d\mathbf{S} = 
\begin{pmatrix}
\partial_y F_z - \partial_z F_y  \\ 
\partial_z F_x - \partial_x F_z \\ 
\partial_x F_y - \partial_y F_x  \\
\end{pmatrix} \cdot d\mathbf{S} \to \\[1.4ex]
&d(F_x \,dx + F_y \,dy + F_z \,dz) = \left(\partial_y F_z - \partial_z F_y\right) dy \wedge dz + \left(\partial_z F_x -\partial_x F_z\right) dz \wedge dx + \left(\partial_x F_y - \partial_y F_x\right) dx \wedge dy.
\end{align}</math>