HELP!!!CURVES AND SURFACES!! HELP!!!

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2 total messages Started by Martin Gillstedt Mon, 29 Jun 1998 00:00

#73

Author: Martin Gillstedt
Date: Mon, 29 Jun 1998 00:00

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Can anyone give me an example of 2 regular (i.e. the derivative is never
zero) simple closed curves in the plane, which are infinitely many times
differentiable everywhere (absolutely NO corners or edges in derivatives
of ANY order), and who are not similar to each other, such that their
isoperimetric quotients are equal?
  The isoperimetric quotient is defined as =4*pi*A / l^2, where A is the
area that the curve bounds, and l is the arc length of the curve.

  An analogous problem is in space: Does there exist 2 closed surfaces
in space, which bound a volume, V, and has an area, S, an is infinitely
many times differentiable without exception, not similar to each other,
whose isoperimetric quotients are equal?
  The isoperimetric quotient is here = 36*pi*V^2 / S^3.

Re: HELP!!!CURVES AND SURFACES!! HELP!!!

#74

Author: Virgil
Date: Mon, 29 Jun 1998 00:00

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In article <3596E7AD.13026B32@hotmail.com>, Martin Gillstedt
<m_gillstedt@hotmail.com> wrote:

> Can anyone give me an example of 2 regular (i.e. the derivative is never
> zero) simple closed curves in the plane, which are infinitely many times
> differentiable everywhere (absolutely NO corners or edges in derivatives
> of ANY order), and who are not similar to each other, such that their
> isoperimetric quotients are equal?

In the book "Counterexamples in Analysis" a method is shown for joining any two 
"infinitely smooth" curves which cross so that the join is also
"infinitely smooth".

Sorry, but I can't remember author, publisher, etc. 
I believe the book was published in the 50s.

-- 
vmhjrx@friix.comx
de-x to reply

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