Wednesday, March 15, 2006

Oh the joys of continuity

Kevan Choset posts one of the better problems from analysis: given a continuous surface, like the earth, show that there are opposite points (antipodal points) with identical temperatures (the best solution in comments is this one).

While you can treat it as a surface, I don't actually have the math to do so. Taking an arbitrary circumference, like the equator, will accomplish the same thing and be much clearer (and will also show that there uncountably many such pairs of antipodal points with identical temperature).

You have a circle with continuous temperature. Temperature is bounded (finite), so you will have a maximum and a minimum. Now create a function of the difference in temperature at opposite points on the circle. Because this new function is simply the composition of a continuous bounded function (one that assigned temperature to points on the circle) it will also be continuous and bounded. You have a continuous bounded function giving the differences in temperature values at antipodal points on some circumference of the earth.

Right about now we'd like to invoke the intermediate value theorem and say, aha, the function will be zero at some point. Yet this requires knowing that the distance function is positive somewhere and negative somewhere else. To do this note that there was a minimum value on the circumference. Then the difference from a point opposite it on the circle will be positive (or negative). Similarly, we have a maximum. So the difference from a point opposite on the circle will be negative (or positive). Hence we have a positive and negative value in our distance function.

By the intermediate value theorem, because this a continuous function that has a positive value somewhere and a negative value somewhere else, it must take on the value 0 somewhere in between. At this point, then, we have our antipodal points with identical temperatures.

Because this was for an arbitrary circumference, we've shown that on any circumference around the earth (or even just a part of the earth) there are antipodal points with identical temperature.

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