Editing Generalized Stokes theorem (section)

==Topological preliminaries; integration over chains==

Let {{mvar|M}} be a [[smooth manifold]]. A (smooth) singular [[Simplex|{{mvar|k}}-simplex]] in {{mvar|M}} is defined as a [[smooth map]] from the standard simplex in {{math|'''R'''<sup>''k''</sup>}} to {{mvar|M}}. The group {{math|''C''<sub>''k''</sub>(''M'', '''Z''')}} of singular {{mvar|k}}-[[chain (algebraic topology)|chains]] on {{mvar|M}} is defined to be the [[free abelian group]] on the set of singular {{mvar|k}}-simplices in {{mvar|M}}. These groups, together with the boundary map, {{math|∂}}, define a [[chain complex]]. The corresponding homology (resp. cohomology) group is isomorphic to the usual [[singular homology]] group {{math|''H''<sub>''k''</sub>(''M'', '''Z''')}} (resp. the [[singular cohomology]] group {{math|''H''<sup>''k''</sup>(''M'', '''Z''')}}), defined using continuous rather than smooth simplices in {{mvar|M}}.

On the other hand, the differential forms, with exterior derivative, {{mvar|d}}, as the connecting map, form a cochain complex, which defines the [[de Rham cohomology]] groups <math>H_{dR}^k(M, \mathbf{R})</math>.

Differential {{mvar|k}}-forms can be integrated over a {{mvar|k}}-simplex in a natural way, by pulling back to {{math|'''R'''<sup>''k''</sup>}}. Extending by linearity allows one to integrate over chains. This gives a linear map from the space of {{mvar|k}}-forms to the {{mvar|k}}th group of singular cochains, {{math|''C<sup>k</sup>''(''M'', '''Z''')}}, the linear functionals on {{math|''C<sub>k</sub>''(''M'', '''Z''')}}. In other words, a {{mvar|k}}-form {{mvar|ω}} defines a functional
<math display="block">I(\omega)(c) = \oint_c \omega.</math>
on the {{mvar|k}}-chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, {{mvar|d}}, behaves like the ''dual'' of {{math|∂}} on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means:

#closed forms, i.e., {{math|1=''dω'' = 0}}, have zero integral over ''boundaries'', i.e. over manifolds that can be written as {{math|∂Σ<sub>''c''</sub> ''M<sub>c</sub>''}}, and
#exact forms, i.e., {{math|1=''ω'' = ''dσ''}}, have zero integral over ''cycles'', i.e. if the boundaries sum up to the empty set: {{math|1=∂Σ<sub>''c''</sub> ''M<sub>c</sub>'' = ∅}}.

[[De Rham's theorem]] shows that this homomorphism is in fact an [[isomorphism]]. So the converse to 1 and 2 above hold true. In other words, if {{math|{''c<sub>i</sub>''} }} are cycles generating the {{mvar|k}}th homology group, then for any corresponding real numbers, {{math|{''a<sub>i</sub>''} }}, there exist a closed form, {{mvar|ω}}, such that
<math display="block">\oint_{c_i} \omega = a_i\,,</math>
and this form is unique up to exact forms.

Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa.<ref>{{Cite book|title=Manifolds, Tensors, and Forms|last=Renteln|first=Paul|publisher=Cambridge University Press|year=2014| isbn=9781107324893|location=Cambridge, UK|pages=158–175}}</ref>  Formally stated, the latter reads:<ref>{{Cite book | title=Introduction to Smooth Manifolds|last=Lee|first=John M.|publisher=Springer|year=2013|isbn=9781441999818|location=New York |pages=481}}</ref>

{{math theorem
| note = ''Stokes' theorem for chains''
| math_statement = If {{mvar|c}} is a smooth {{mvar|k}}-chain in a smooth manifold {{mvar|M}}, and {{mvar|ω}} is a smooth {{math|(''k'' − 1)}}-form on {{mvar|M}}, then
<math display="block">\int_{\partial c}\omega = \int_c d\omega.</math>
}}