there exist two antipodal points on the equator that have the same temperature.
Argue that there exist at any time two antipodal points on the equator that have the same temperature.
The temperature function can be assumed to be continuous.
I am supposed to use the mean value theorem, but I don't see how. Thanks in advance.
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$\begingroup$Okay, so lets just consider points along the equator, and let $ t:\left [ 0,2\pi \right ]\rightarrow \mathbb{R}$ be the temperature at a point with angle $\theta$ from some predetermined point on the equator.
Now, we are given that t is continuous in $\theta$ on $\left [ 0,2\pi \right ]$, and we see that t is $2\pi$ periodic.
Define $T:\left [ 0,2\pi \right ]\rightarrow \mathbb{R}$ to be the antipodal difference in temperature, that is $T := t(\theta + \pi) - t(\theta)$
Then T is also continuous on $\left [ 0,2\pi \right ]$, and we have that:
$T(0) = t(\pi) - t(0)$ and $T(\pi) = t(2\pi) - t(\pi)$
So as t is $2\pi$ periodic, we get that $T(0) = -T(\pi)$
If $T(0) = 0$ then we are done and we have our antipodal points with equal temperature, otherwise if $T(0) \neq 0$, then as $T$ is continuous on $\left [ 0,\pi \right ] \subset$ $\left [ 0,2\pi \right ]$ and without loss of generality $T(0) < 0 < T(\pi)$, then $\exists \alpha \in \left [ 0,\pi \right ]$ such that $T(\alpha) = 0$.
And then $t(\alpha) = t(\alpha + \pi)$ so we have found our antipodal points with the same temperature.
This is really just a specific case of a really interesting theorem called the Borsuk–Ulam Theorem, which makes similar sorts of statements for n-dimensional spheres mapping to n-dimensional planes. Here we have a 2 dimensional sphere mapping to a 1 dimensional plane, but we considered a 1 dimensional subsphere (our equator), and the Borsuk–Ulam Theorem says on any continuous mapping of an n-dimensional sphere to an n-dimensional plane, there will be two antipodal points who get mapped to the same point.
$\endgroup$ $\begingroup$Let $\theta$ denote longitude; it varies from $+180$ to $-180$. Define $f(\theta)$ to be the temperature at longitude $\theta$, minus the temperature at the antipodal point to longitude $\theta$. If ever $f(\theta)=0$, you are done. However, $f(\theta)=-f(\theta')$, where $\theta'$ is antipodal to $\theta$. So, continuously vary $\theta$ to get $\theta'$, and apply IVT(Intermediate Value Theorem).
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