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Minimal SA?

  1. Dec 24, 2006 #1
    Could someone give a clue to how I could prove the sphere has minimal surface area for a given volume?

    Note this is not a homework problem. I saw in a chemistry textbook that water droplets tend to be spherical because each water molecule has a force directed inwards. In order to minimise the number of molecules on the boundary, the water droplet tend to form a sphere because the book claims it has the minimum surface area per volume. I would like to mathematically prove this fact but don't know how to start.
    Last edited: Dec 24, 2006
  2. jcsd
  3. Dec 25, 2006 #2

    D H

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    This is the isovolume problem, the 3D extension to the 2D isoperimetric problem. See http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml" [Broken] for an outline of a proof of the 2D problem.
    Last edited by a moderator: May 2, 2017
  4. Dec 25, 2006 #3

    It can be proven by some section in mathematics known as functional optimization (or analysis). It is involved with something called functional basis.

    Any way, you can do it by another way; by extending outwards an infinitesimal volume through certain point of the sphere and cutting another one inwards with an equal volume at another point. You will find that surface are is larger in that case. This proves that spherical surface is the local minima surface-function, but is not a proof yet that it is the smallest one.

    Amr Morsi.
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