Editing Differential form (section)

===Stokes's theorem===
{{main|Stokes's theorem}}
The fundamental relationship between the exterior derivative and integration is given by the [[Stokes' theorem]]: If {{math|''ω''}} is an ({{math|''n'' − 1}})-form with compact support on {{math|''M''}} and {{math|''∂M''}} denotes the [[manifold#Manifold with boundary|boundary]] of {{math|''M''}} with its induced [[Orientation (mathematics)|orientation]], then
<math display="block">\int_M d\omega = \int_{\partial M} \omega.</math>

A key consequence of this is that "the integral of a closed form over homologous chains is equal": If {{math|''ω''}} is a closed {{math|''k''}}-form and {{math|''M''}} and {{math|''N''}} are {{math|''k''}}-chains that are homologous (such that {{math|''M'' − ''N''}} is the boundary of a {{math|(''k'' + 1)}}-chain {{math|''W''}}), then <math>\textstyle{\int_M \omega = \int_N \omega}</math>, since the difference is the integral <math>\textstyle\int_W d\omega = \int_W 0 = 0</math>.

For example, if {{math|1=''ω'' = ''df''}} is the derivative of a potential function on the plane or {{math|'''R'''<sup>''n''</sup>}}, then the integral of {{math|''ω''}} over a path from {{math|''a''}} to {{math|''b''}} does not depend on the choice of path (the integral is {{math|''f''(''b'') − ''f''(''a'')}}), since different paths with given endpoints are [[homotopic]], hence homologous (a weaker condition). This case is called the [[gradient theorem]], and generalizes the [[fundamental theorem of calculus]]. This path independence is very useful in [[contour integration]].

This theorem also underlies the duality between [[de Rham cohomology]] and the [[homology (mathematics)|homology]] of chains.