Implications of Theorem in Real World
The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or any other similar scalar quantity which varies continuously, there will always exist two antipodal points that share the same value for that variable.
Proof: Take f to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points A and B. Let d be defined by the difference f(A) − f(B). If the line is rotated 180 degrees, the value −d will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which d = 0, and as a consequence f(A) = f(B) at this angle.
This is a special case of a more general result called the Borsuk–Ulam theorem.
Another generalization for which this holds is for any closed convex n (n>1) dimensional shape. Specifically, for any continuous function whose domain is the given shape, and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same. The proof is identical to the one given above.
The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily-met constraints).
Read more about this topic: Intermediate Value Theorem
Famous quotes containing the words implications of, implications, theorem, real and/or world:
“The power to guess the unseen from the seen, to trace the implications of things, to judge the whole piece by the pattern, the condition of feeling life in general so completely that you are well on your way to knowing any particular corner of it—this cluster of gifts may almost be said to constitute experience.”
—Henry James (1843–1916)
“The power to guess the unseen from the seen, to trace the implications of things, to judge the whole piece by the pattern, the condition of feeling life in general so completely that you are well on your way to knowing any particular corner of it—this cluster of gifts may almost be said to constitute experience.”
—Henry James (1843–1916)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“When much intercourse with a friend has supplied us with a standard of excellence, and has increased our respect for the resources of God who thus sends a real person to outgo our ideal; when he has, moreover, become an object of thought, and, whilst his character retains all its unconscious effect, is converted in the mind into solid and sweet wisdom,—it is a sign to us that his office is closing, and he is commonly withdrawn from our sight in a short time.”
—Ralph Waldo Emerson (1803–1882)
“My mother protected me from the world and my father threatened me with it.”
—Quentin Crisp (b. 1908)