Editing Generalized Stokes theorem (section)

== Introduction ==
The [[fundamental theorem of calculus|second fundamental theorem of calculus]] states that the [[integral]] of a function <math>f</math> over the [[interval (mathematics)|interval]] <math>[a,b]</math> can be calculated by finding an [[antiderivative]] <math>F</math> of <math>f</math>:
<math display="block">\int_a^b f(x)\,dx = F(b) - F(a)\,.</math>

Stokes' theorem is a vast generalization of this theorem in the following sense. 
* By the choice of <math>F</math>, <math>\frac{dF}{dx}=f(x)</math>. In the parlance of [[differential form]]s, this is saying that <math>f(x)\,dx</math> is the [[exterior derivative]] of the 0-form, i.e. function, <math>F</math>: in other words, that <math>dF=f\,dx</math>. The general Stokes theorem applies to higher [[Degree_of_a_polynomial|degree]] differential forms <math>\omega</math> instead of just 0-forms such as <math>F</math>.
* A closed interval <math>[a,b]</math> is a simple example of a one-dimensional [[manifold with boundary]]. Its boundary is the set consisting of the two points <math>a</math> and <math>b</math>. Integrating <math>f</math> over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be [[orientable]], and the form has to be [[compact support|compactly supported]] in order to give a well-defined integral.
* The two points <math>a</math> and <math>b</math> form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds <math>M</math> with boundary. The boundary <math>\partial M</math> of <math>M</math> is itself a manifold and inherits a natural orientation from that of <math>M</math>. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, <math>a</math> inherits the opposite orientation as <math>b</math>, as they are at opposite ends of the interval. So, "integrating" <math>F</math> over two boundary points <math>a</math>, <math>b</math> is taking the difference <math>F(b)-F(a)</math>.
In even simpler terms, one can consider the points as boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral (<math>f\,dx=dF</math>) over a 1-dimensional manifold (<math>[a,b]</math>) by considering the anti-derivative (<math>F</math>) at the 0-dimensional boundaries (<math>\{a,b\}</math>), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (<math>d\omega</math>) over <math>n</math>-dimensional manifolds (<math>\Omega</math>) by considering the antiderivative (<math>\omega</math>) at the <math>(n-1)</math>-dimensional boundaries (<math>\partial\Omega</math>) of the manifold.

So the fundamental theorem reads:
<math display="block">\int_{[a, b]} f(x)\,dx = \int_{[a, b]} \,dF = \int_{\partial[a, b]} \,F = \int_{\{a\}^- \cup \{b\}^+} F = F(b) - F(a)\,.</math>