Uniqueness theorem for Poisson's equation: Difference between revisions

Revision as of 17:14, 6 January 2021

Revision as of 17:15, 6 January 2021

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:<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> Consequently, the second term goes to zero. and we find that

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

Taking the volume integral over the region gives