text: Various proofs of the theorem have been given by Hilton, Liebman, Laue, Herring, and a rather extensive treatment by Cerf. The following is after the method of R. F. Strickland-Constable.
  We begin with the surface energy for a crystal
  which is the product of the surface energy per unit area times the area of each face, summed over all faces. This is minimized for a given volume when
  Surface free energy, being an intensive property, does not vary with volume. We then consider a small change in shape for a constant volume. If a crystal were nucleated to a thermodynamically unstable state, then the change it would undergo afterward to approach an equilibrium shape would be under the condition of constant volume. By definition of holding a variable constant, the change must be zero, . Then by expanding  in terms of the surface areas  and heights  of the crystal faces, one obtains
  which can be written, by applying the product rule, as
  The second term must be zero, that is,
  This is because, if the volume is to remain constant, the changes in the heights of the various faces must be such that when multiplied by their surface areas the sum is zero. If there were only two surfaces with appreciable area, as in a pancake-like crystal, then . In the pancake instance,  on premise. Then by the condition, . This is in agreement with a simple geometric argument considering the pancake to be a cylinder with very small aspect ratio. The general result is taken here without proof. This result imposes that the remaining sum also equal 0, 
  Again, the surface energy minimization condition is that
  These may be combined, employing a constant of proportionality  for generality, to yield 
  The change in shape  must be allowed to be arbitrary, which then requires that , which then proves the Gibbs-Wulff Theorem.