Editing Generalized Stokes theorem (section)

==Generalization to rough sets==
[[Image:Green's-theorem-simple-region.svg|thumb|upright|180px|A region (here called {{mvar|D}} instead of {{math|Ω}}) with piecewise smooth boundary.  This is a [[manifold with corners]], so its boundary is not a smooth manifold.]]

The formulation above, in which <math>\Omega</math> is a smooth manifold with boundary, does not suffice in many applications. For example, if the domain of integration is defined as the plane region between two <math>x</math>-coordinates and the graphs of two functions, it will often happen that the domain has corners.  In such a case, the corner points mean that <math>\Omega</math> is not a smooth manifold with boundary, and so the statement of Stokes' theorem given above does not apply.  Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true.  This is because <math>\Omega</math> and its boundary are well-behaved away from a small set of points (a [[measure zero]] set).

A version of Stokes' theorem that allows for roughness was proved by Whitney.<ref>Whitney, ''Geometric Integration Theory,'' III.14.</ref> Assume that <math>D</math> is a connected bounded open subset of <math>\R^n</math>.  Call <math>D</math> a ''standard domain'' if it satisfies the following property: there exists a subset <math>P</math> of <math>\partial D</math>, open in <math>\partial D</math>, whose complement in <math>\partial D</math> has [[Hausdorff measure|Hausdorff <math>(n-1)</math>-measure]] zero; and such that every point of <math>P</math> has a ''generalized normal vector''. This is a vector <math>\textbf{v}(x)</math> such that, if a coordinate system is chosen so that <math>\textbf{v}(x)</math> is the first basis vector, then, in an open neighborhood around <math>x</math>, there exists a smooth function <math>f(x_2,\dots,x_n)</math> such that <math>P</math> is the graph <math>\{x_1=f(x_2,\dots,x_n)\}</math> and <math>D</math> is the region <math>\{x_1:x_1<f(x_2,\dots,x_n)\}</math>. Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff <math>(n-1)</math>-measure and a finite or countable union of smooth <math>(n-1)</math>-manifolds, each of which has the domain on only one side. He then proves that if <math>D</math> is a standard domain in <math>\R^n</math>, <math>\omega</math> is an <math>(n-1)</math>-form which is defined, continuous, and bounded on <math>D\cup P</math>, smooth on <math>D</math>, integrable on <math>P</math>, and such that <math>d\omega</math> is integrable on <math>D</math>, then Stokes' theorem holds, that is,
<math display="block">\int_P \omega = \int_D d\omega\,.</math>

The study of measure-theoretic properties of rough sets leads to [[geometric measure theory]]. Even more general versions of Stokes' theorem have been proved by Federer and by Harrison.<ref>{{cite journal|last=Harrison|first=J.|title=Stokes' theorem for nonsmooth chains|journal=Bulletin of the American Mathematical Society |series=New Series|volume=29|issue=2|date=October 1993| pages=235–243|doi=10.1090/S0273-0979-1993-00429-4|arxiv=math/9310231|bibcode=1993math.....10231H|s2cid=17436511}}</ref>