14.7
Example.
Two points
![$P,Q$](img552.gif)
on a sphere are called
antipodal points if
![$P$](img550.gif)
and
![$Q$](img551.gif)
are opposite ends of the same diameter
of the sphere. We will consider the surface of the earth to be a sphere
of radius
![$R$](img47.gif)
. At any fixed time, let
![$T(p)$](img3461.gif)
denote the temperature
of the earth at the point
![$p$](img246.gif)
on the surface of the earth. (More precisely,
let
![$T(p)$](img3461.gif)
be the number such that the temperature at
![$p$](img246.gif)
is
![$T(p)^{\circ}C$](img3462.gif)
).
We will show that there are two antipodal points
![$P,Q$](img552.gif)
on the surface of
the earth such that
![$T(P)=T(Q).$](img3463.gif)
In fact, we will show that there are two
antipodal points on the equator with the same temperature. We first introduce
a coordinate system so that the center of the earth is at the origin,
and the plane of the equator is the
![$x$](img35.gif)
-
![$y$](img1044.gif)
plane, and the point on the
equator passing through the Greenwich meridian is the point
![$(R,0)$](img3464.gif)
.
Then the points on the equator are the points
Define a function
![$f: [0,\pi] \to \mbox{{\bf R}}$](img3466.gif)
by
Thus
We suppose that
![$f$](img676.gif)
is a continuous function on
![$[0,\pi]$](img2542.gif)
.
If
![$f(0) = 0$](img3469.gif)
then
![$T(R,0) = T(-R,0)$](img3470.gif)
, so
![$(R,0)$](img3464.gif)
and
![$(-R,0)$](img3471.gif)
are a pair
of antipodal points with the same temperature. Now
so if
![$f(0) \neq 0$](img3473.gif)
then
![$f(0)$](img3474.gif)
and
![$f(\pi)$](img3475.gif)
have opposite signs. Hence by the
intermediate value property, there is a number
![$c \in(0,\pi)$](img3476.gif)
such that
![$f(c) = 0$](img3447.gif)
, i.e.
Then
![$(R\cos(c),R(\sin(c))$](img3478.gif)
and
![$(-R\cos(c),-R(\sin(c))$](img3479.gif)
are a pair of
antipodal points with the same temperature.
14.8
Example.
Let
where
![$a_0,a_1,a_2,$](img3481.gif)
and
![$a_3$](img3482.gif)
are real numbers, and
![$a_3 \neq 0$](img3483.gif)
.
Then there exists some number
![$r\in\mbox{{\bf R}}$](img2852.gif)
such that
![$P(r) = 0$](img3484.gif)
.
Proof: I will suppose that
for all
and
derive a contradiction. Let
Since
![$P(x) \neq 0$](img3487.gif)
for all
![$x \in \mbox{{\bf R}}$](img633.gif)
,
![$Q$](img551.gif)
is continuous on
R. We know that
Hence
![$Q(N) < 0$](img3489.gif)
for some
![$N\in\mbox{${\mbox{{\bf Z}}}^{+}$}$](img2202.gif)
. Then
![$P(N)$](img3490.gif)
and
![$P(-N)$](img3491.gif)
have
opposite signs, so by the intermediate value property there
is a number
![$r \in [-N,N]$](img3492.gif)
such that
![$P(r) = 0$](img3484.gif)
. This contradicts
our assumption that
![$P(t) \neq 0$](img3485.gif)
for all
![$t\in\mbox{{\bf R}}$](img836.gif)
.
14.10
Exercise.
A
Three wires
![$AC,BC,DC$](img3495.gif)
are joined at a common
point
![$C$](img37.gif)
.
Let
![$S$](img49.gif)
be the Y-shaped figure formed by the three wires.
Prove that at any time there are two points in
![$S$](img49.gif)
with the same temperature.
14.11
Exercise.
A
Six wires are joined to form the figure
![$F$](img162.gif)
shown in the diagram.
Show that at any time there are three points in
![$F$](img162.gif)
that have the same temperature. To simplify the problem, you may assume
that the temperatures at
![$A$](img6.gif)
,
![$B$](img145.gif)
,
![$C$](img37.gif)
, and
![$D$](img164.gif)
are all distinct.