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antipodal isothermic points
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(Application)
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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
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(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
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"antipodal isothermic points" is owned by pahio.
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Cross-references: reference, Bolzano's theorem, opposite, vanish, diametral, difference, function, circle, real function, continuous, point, distance, proof, end points, antipodal points, diametral points, sphere, great circle
This is version 3 of antipodal isothermic points, born on 2008-11-09, modified 2008-11-10.
Object id is 11247, canonical name is AntipodicIsothermicPoints.
Accessed 411 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
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Pending Errata and Addenda
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