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suppose the wind is blowing on the surface of the earth in a constant and continuous fashion. Suppose also that at every point on the equator, the wind is blowing directly east, so the wind doesnt blow anything from one hemisphere to another. Show that there must be some point in the northern hemisphere that blows anything dropped at that point back to that point in exactly 1 minute

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What have you tried? –  M Turgeon Aug 6 '12 at 21:37
    
I have a continuous function $f$ defined on $N$ which assigns $x$ to a point where the feather lands after 1 minute... other than that, I don't know where to start, other than this question might use fixed points –  Alex Aug 6 '12 at 21:40

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Consider only the northern hemisphere, sans the equator. The described motion of the wind describes a continuous function $f$ which maps every point of the northern hemisphere to a point in the northern hemisphere. Now define $h$ as an bijective continuous function from the northern hemisphere to a disk, and then apply Brower's fixed point theorem to the continuous function on the disk $h(f(h^{-1}(x)))$. Then there exists some fixed point $x_0$, that is, $h(f(h^{-1}(x_0)))=x_0$ and therefore $f(h^{-1}(x_0))=h^{-1}(x_0)$, and hence $f$ has a fixed point.

(Some details have been omitted purposely, to leave some blanks to fill in - for instance, can you find a function $h$? How can the stipulation of 1 minute be addressed?)

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