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Electrostatic Field and Potential

I would appreciate help in completing the proof of the equation on the
second line of the attached file.

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What you need to prove is not that div[phi E] = 0, but rather that div[phi E] integrated over a volume approaches zero in the limit that the volume becomes infinitely large. If div[phi E] were a positive definite quantity then this would also imply that div[phi E] = 0 everywhere, but it isn't! div[phi E] can be negative or positive.

I'm now going to derive the result step by step.

Consider the volume integral:

Volume integral [rho phi].

Replacing rho = epsilon_0 div E ...

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