Uniqueness theorem for Poisson's equation: Difference between revisions

Revision as of 18:39, 6 January 2021

Revision as of 00:17, 7 January 2021

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By taking the volume integral over the region <math>V</math>, we find that

:<math>\int_V \mathbf{\nabla}\cdot(\varphi \, \mathbf{\nabla}\varphi) \, \mathrm{d}V= \int_V \mathbf{(\nabla}\varphi)\cdot( \mathbf{\nabla}\varphi) \, \mathrm{d}V= \int_V (\mathbf{\nabla}\varphi)^2 \, \mathrm{d}V.</math>

By applying the [[divergence theorem]], we rewrite the expression above as