Editing Brouwer fixed-point theorem (section)

===A proof using Stokes' theorem===
As in the proof of Brouwer's fixed-point theorem for continuous maps using homology, it is reduced to proving that there is no continuous retraction {{mvar|''F''}} from the ball {{mvar|''B''}} onto its boundary ∂{{mvar|''B''}}. In that case it can be assumed that {{mvar|''F''}} is smooth, since it can be approximated using the [[Weierstrass approximation theorem]] or by [[convolution|convolving]] with non-negative smooth [[bump function]]s of sufficiently small support and integral one (i.e. [[mollifier|mollifying]]). If {{mvar|ω}} is a [[volume form]] on the boundary then by [[Stokes' theorem]],
:<math>0<\int_{\partial B}\omega = \int_{\partial B}F^*(\omega) = \int_BdF^*(\omega)= \int_BF^*(d\omega)=\int_BF^*(0) = 0,</math>
giving a contradiction.<ref>{{harvnb|Boothby|1971}}</ref><ref>{{harvnb|Boothby|1986}}</ref>

More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifold {{mvar|''M''}} onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the form {{mvar|ω}} generates the [[De Rham cohomology|de Rham cohomology group]] {{mvar|''H''<sup>''n''-1</sup>}}(∂{{mvar|''M''}}) which is isomorphic to the homology group {{mvar|''H''<sub>''n''-1</sub>}}(∂{{mvar|''M''}}) by [[De Rham cohomology#De Rham's theorem|de Rham's theorem]].<ref>{{harvnb|Dieudonné|1982}}</ref>