Proof of the uniqueness theorem
Suppose there were two functions V1(x,y,z) and V2(x,y,z), each of them satisfying the Laplace equation and having the same values on the boundary S but being different inside V.
We have: ; ; V1(S) = V2(S)
Define: V3(x,y,z) = V2(x,y,z) - V1(x,y,z)
because of the linearity of Laplace's equation.
Since V1(S) = V2(S) is valid on the boundary: V3(S) = 0
Since the maxima and minima of a harmonic function lie on the boundary, V3(x,y,z) = 0 everywhere inside V.
This means that V2(x,y,z) = V1(x,y,z) everywhere in V.
In other words: For given boundary conditions the solution of the Laplace equation is unique.
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