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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|
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.
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