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Unformatted text preview: V Writing = i i i i M m M L ) 3 / 2 ( Differentiating with respect to m j  = = j i i j j i i j m M m M m M m L / ) 3 / 2 ( ) 3 / 2 ( / / From solid geometry, we also have (the not particularly obvious relationship) that + = = j i i j j j m M m M M m V / ) 3 / 1 ( ) 3 / 1 ( / Or = j i i j m M m M / ) 2 / 1 ( Then, combing this we get =  = = j i i i j i i j i i j m M m m M m m M m L / } { / / / Since the derivative is in general not zero, this has a solution = i i m / The shape then only scales up or down as we vary , i.e. the shape is the same no matter what the size is. One caveat about this proof. Technically it only proves that the Wulff construction is a stationary point, not that it is a true minimum. Mathematicians worry about this, we dont need to....
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 Winter '10
 MatSci

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