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# antipodal isothermic points

Assume that the momentary temperature on any great circle of a sphere varies continuously. Then there exist two diametral points (i.e. antipodal points, end points of a certain diametre) having the same temperature.

Proof. Denote by $x$ the distance of any point $P$ measured in a certain direction along the great circle from a fixed point and let $T(x)$ be the temperature in $P$. Then we have a continuous (and periodic) real function $T$ defined for $x\geqq 0$ satisfying $T(x\!+\!p)=T(x)$ where $p$ is the perimetre of the circle. Then also the function $f$ defined by

$f(x)\;:=\;T\left(x\!+\!\frac{p}{2}\right)-T(x),$ |

i.e. the temperature difference in two antipodic (diametral) points of the great circle, is continuous. We have

$\displaystyle f\left(\frac{p}{2}\right)\;=\;T(p)-T\left(\frac{p}{2}\right)=T(0% )-T\left(\frac{p}{2}\right)=-f(0).$ | (1) |

If $f$ happens to vanish in $x=0$, then the temperature is the same in $x=\frac{p}{2}$ and the assertion proved. But if $f(0)\neq 0$, then by (1), the values of $f$ in $x=0$ and in $x=\frac{p}{2}$ have opposite signs. Therefore, by Bolzano’s theorem, there exists a point $\xi$ between $0$ and $\frac{p}{2}$ such that $f(\xi)=0$. Thus the temperatures in $\xi$ and $\xi\!+\!\frac{\pi}{2}$ are the same.

Reference: Fråga Lund om matematik, 6 april 2006

## Mathematics Subject Classification

26A06*no label found*

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