Uniqueness theorem for Poisson's equation: Difference between revisions

Revision as of 17:09, 6 January 2021

Revision as of 17:14, 6 January 2021

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Let us first consider the case where [[Dirichlet boundary condition|Dirichlet boundary conditions]] are specified as <math>\varphi = 0</math> on the boundary of the region. These follow because the boundary conditions and the charge distributions are the same for both 'solutions'.

WeBy canthe nowapplication useof the [[Vector calculus identities#Divergence 2|vector differential identity]] we know that

:<math>\nabla \cdot (\varphi \, \nabla \varphi )= \, (\nabla \varphi )^2 + \varphi \, \nabla^2 \varphi.</math>

However, from <math>(1)</math> we also know that throughout the region <math>\nabla^2 \varphi = 0.</math> throughout the region soConsequently, the second term goes to zero.

:<math>\nabla \cdot (\varphi \, \nabla \varphi )= \, (\nabla \varphi )^2</math>