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where <math>S</math> is the boundary surface specified by the boundary conditions.
If the Dirichlet boundary condition is satisified by both solutions (ie, <math>\varphi = 0</math> on the boundary) then the left-hand side of <math>(2)</math> is zero thus
:<math>\int_V (\mathbf{\nabla}\varphi)^2 \, \mathrm{d}V = 0</math> However, because this is the volume integral of a positive quantity (due to the squared term), we must have :<math>\nabla \varphi = 0</math> at all points. Finally, because the gradient of <math>\varphi</math> is everywhere zero and <math>\varphi</math> is zero on the boundary, <math>\varphi</math> must be zero throughout the whole region. This proves <math>\varphi_1 = \varphi_2</math> and the solutions are identical.
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