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 pointsPlanetmathPlanetmath of a certain diametre ( having the same temperature.

Proof.  Denote by x the distanceMathworldPlanetmath 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 ( real function T defined for  x0  satisfying  T(x+p)=T(x)  where p is the perimetre of the circle.  Then also the function f defined by


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

f(p2)=T(p)-T(p2)=T(0)-T(p2)=-f(0). (1)

If f happens to vanish in  x=0,  then the temperature is the same in  x=p2  and the assertion proved.  But if  f(0)0,  then by (1), the values of f in  x=0  and in  x=p2  have opposite signs.  Therefore, by Bolzano’s theorem, there exists a point ξ between 0 and p2 such that  f(ξ)=0.  Thus the temperatures in ξ and ξ+π2 are the same.

Reference:åga 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