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 the distance of any point measured in a certain direction along the great circle from a and let be the temperature in . Then we have a continuous (and periodic (http://planetmath.org/PeriodicFunctions)) real function defined for satisfying where is the perimetre of the circle. Then also the function defined by
i.e. the temperature difference in two antipodic (diametral) points of the great circle, is continuous. We have
If happens to vanish in , then the temperature is the same in and the assertion proved. But if , then by (1), the values of in and in have opposite signs. Therefore, by Bolzano’s theorem, there exists a point between and such that . Thus the temperatures in and are the same.
Reference: http://www.maths.lth.se/query/Fråga Lund om matematik, 6 april 2006
|Title||antipodal isothermic points|
|Date of creation||2013-03-22 18:32:10|
|Last modified on||2013-03-22 18:32:10|
|Last modified by||pahio (2872)|