The temperature T ( x ) at each point x on the surface of Mars (a sphere) is a continuous function. Show that there is a point x on the surface such that T ( x ) = T ( − x )(Hint: Represent the surface of Mars as { x ∈ R 3 : | | x | | = 1 }.)Consider the function f ( x ) = T ( x ) − T ( − x )So.....I consider an unit sphere is locating at the origin of a x y z-plane.As | | x | | = 1, I can say with r a d i u s = 1 = x 2 + y 2 + z 2 To find there is a point T ( x ) = T ( − x ), we use the formula f ( x ) = T ( x ) − T ( − x ) and show somehow f ( x ) will equal to 0 ???It will be a point in the upper hemisphere and another point with the exactly opposite vector (if using i j k plane) on the lower hemisphere

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The temperature   T  (  x  ) at each point   x on the surface of Mars (a sphere) is a continuous function. Show that there is a point   x on the surface such that   T  (  x  )  =  T  (  −  x  )(Hint: Represent the surface of Mars as   {  x  ∈                    R            3        :      |        |    x      |        |    =  1  }.)Consider the function   f  (  x  )  =  T  (  x  )  −  T  (  −  x  )So.....I consider an unit sphere is locating at the origin of a   x  y  z-plane.As       |        |    x      |        |    =  1, I can say with   r  a  d  i  u  s  =  1  =            x      2        +          y      2        +          z      2      To find there is a point   T  (  x  )  =  T  (  −  x  ), we use the formula   f  (  x  )  =  T  (  x  )  −  T  (  −  x  ) and show somehow   f  (  x  ) will equal to 0 ???It will be a point in the upper hemisphere and another point with the exactly opposite vector (if using   i  j  k plane) on the lower hemisphere

tykoyz · Accepted Answer

Hint: Pick some point   x. If you are lucky,   f  (  x  )  =  0, but probably this does not occur. In the other case, what can you say about the relation between   f  (  x  ) and   f  (  −  x  )? Now connect   x and   −  x using a path along the sphere...

Bernard Boyer · Accepted Answer

If   f  (  x  )  =  T  (  x  )  −  T  (  −  x  ) is zero, we are done. Assume there does not exist an   x that makes   f  (  x  ) zero. Then   f  (  x  )  &amp;gt;  0 or   &amp;lt;  0 for all   x. If say   f  (  x  )  &amp;gt;  0 is never satisfied, then replacing   x by   −  x we get a contradiction. Similarly if   f  (  x  )  &amp;lt;  0 never holds. So there must be points for which   f  (  x  )  &amp;gt;  0 and points for which   f  (  x  )  &amp;lt;  0. The intermediate value theorem then implies there must exist a point   y at which   f  (  y  )  =  0 (contradiction).

The temperature T(x) at each point x on the surface of Mars (a sphere) is a continuous function. Show that there is a point x on the surface such that T(x)=T(−x)

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