Two opposite points with the same temperature

Question

Assume that the temperature varies continuously around the Earth's equator. Prove that there are always two points on the equator, exactly opposite each other, that have the same temperature at the same moment. Then say whether the same is true for two opposite points on the whole surface of the Earth, and what happens if you want two opposite points with the same temperature and the same pressure.

Accepted Answer

Yes. Define D(x) as the temperature at x minus the temperature at the point opposite x. Then D at the opposite point is exactly minus D(x). A continuous function that takes a value and its negative must take the value zero in between, and D = 0 means two opposite points share a temperature. The argument in full. Parametrise the equator by an angle x from 0 to 360 degrees, with temperature T(x), continuous. Let D(x) = T(x) minus T(x + 180). Then D(x + 180) = T(x + 180) minus T(x) = minus D(x). If D(0) = 0 we are done: the points at 0 and 180 degrees match. Otherwise D(0) and D(180) have opposi…

Two opposite points with the same temperature

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