# antipodal isothermic points

Assume that the momentary temperature on any great circle of a sphere varies continuously (http://planetmath.org/Continuous^{}). Then there exist two diametral points (i.e. antipodal points, end points^{} of a certain diametre (http://planetmath.org/Diameter^{})) 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 and let $T(x)$ be the temperature in $P$. Then we have a continuous (and periodic (http://planetmath.org/PeriodicFunctions)) 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

$f\left({\displaystyle \frac{p}{2}}\right)=T(p)-T\left({\displaystyle \frac{p}{2}}\right)=T(0)-T\left({\displaystyle \frac{p}{2}}\right)=-f(0).$ | (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 $f(0)\ne 0$, 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: http://www.maths.lth.se/query/Fråga Lund om matematik, 6 april 2006

Title | antipodal isothermic points |
---|---|

Canonical name | AntipodalIsothermicPoints |

Date of creation | 2013-03-22 18:32:10 |

Last modified on | 2013-03-22 18:32:10 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Application |

Classification | msc 26A06 |