The [[hairy ball theorem]] states that on the unit sphere {{mvar|''S''}} in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field {{mvar|'''w'''}} on {{mvar|''S''}}. (The tangency condition means that {{mvar|'''w'''('''x''') ⋅ '''x'''}} = 0 for every unit vector {{mvar|'''x'''}}.) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in {{harvtxt|Milnor|1978}}.
In fact, suppose first that {{mvar|'''w'''}} is '''''continuously differentiable'''''. By scaling, it can be assumed that {{mvar|'''w'''}} is a continuously differentiable unit tangent vector on {{mvar|'''S'''}}. It can be extended radially to a small spherical shell {{mvar|''A''}} of {{mvar|''S''}}. For {{mvar|''t''}} sufficiently small, a routine computation shows that the mapping {{mvar|'''f'''<sub>''t''</sub>}}({{mvar|'''x'''}}) = {{mvar|''t'' '''x'''}} + {{mvar|'''w'''('''x''')}} is a [[contraction mapping]] on {{mvar|''A''}} and that the volume of its image is a polynomial in {{mvar|''t''}}. On the other hand, as a contraction mapping, {{mvar|'''f'''<sub>''t''</sub>}} must restrict to a homeomorphism of {{mvar|''S''}} onto (1 + {{mvar|''t''<sup>2</sup>}})<sup>½</sup> {{mvar|''S''}} and {{mvar|''A''}} onto (1 + {{mvar|''t''<sup>2</sup>}})<sup>½</sup> {{mvar|''A''}}. This gives a contradiction, because, if the dimension {{mvar|''n''}} of the Euclidean space is odd, (1 + {{mvar|''t''<sup>2</sup>}})<sup>{{mvar|''n''}}/2</sup> is not a polynomial.
If {{mvar|'''w'''}} is only a '''''continuous''''' unit tangent vector on {{mvar|''S''}}, by the [[Weierstrass approximation theorem]], it can be uniformly approximated by a polynomial map {{mvar|'''u'''}} of {{mvar|''A''}} into Euclidean space. The orthogonal projection on to the tangent space is given by {{mvar|'''v'''}}({{mvar|'''x'''}}) = {{mvar|'''u'''}}({{mvar|'''x'''}}) - {{mvar|'''u'''}}({{mvar|'''x'''}}) ⋅ {{mvar|'''x'''}}. Thus {{mvar|'''v'''}} is polynomial and nowhere vanishing on {{mvar|''A''}}; by construction {{mvar|'''v'''}}/||{{mvar|'''v'''}}|| is a smooth unit tangent vector field on {{mvar|''S''}}, a contradiction.
The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that {{mvar|''n''}} is odd. If there were a fixed-point-free continuous self-mapping {{mvar|'''f'''}} of the closed unit ball {{mvar|''B''}} of the {{mvar|''n''}}-dimensional Euclidean space {{mvar|''V''}}, set
