Assume that (varphi) is harmonic. Show that (operatorname{div}(varphi abla varphi)=|abla varphi|^{2}) and conclude that [ iint_{mathcal{S}} varphi

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Assume that $\varphi$ is harmonic. Show that $\operatorname{div}(\varphi abla \varphi)=\|abla \varphi\|^{2}$ and conclude that
\[
\iint_{\mathcal{S}} \varphi D_{\mathbf{n}} \varphi d S=\iiint_{\mathcal{W}}\|abla \varphi\|^{2} d V
\]

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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Question Posted: Jan 28, 2024 12:00 AM

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Assume that (varphi) is harmonic. Show that (operatorname{div}(varphi abla varphi)=|abla varphi|^{2}) and conclude that [ iint_{mathcal{S}} varphi

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Calculus

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