could you help me with the following problem please:
Verify the following identity $\displaystyle \iint_{\partial{W}} f \frac{\partial{f}}{\partial{\hat{n}}} \,dS=\iiint_{W} \left \| \nabla f \right \| ^2\,dx\,dy\,dz$ if $f$ is harmonic
My main question is, what do you mean $\displaystyle \frac{\partial{f}}{\partial{\hat{n}}}$ or if the problem is badly posed, thank you very much in advance
From the divergence theorem and using $\displaystyle\frac{\partial f}{\partial n}=\hat n\cdot \nabla f$, we assert that
$$\begin{align} \oint_{\partial W}f\frac{\partial f}{\partial n}\,dS&=\int_W \nabla \cdot (f\nabla f)\,dV\\\\ &=\int_W (\nabla f\cdot \nabla f+f\nabla^2 f)\,dV\tag1 \end{align}$$
Since $f$ is harmonic, $\nabla^2f=0$. Using the notation $\nabla f\cdot \nabla f=||\nabla f||^2$ in $(1)$, we find that
$$\oint_{\partial W}f\frac{\partial f}{\partial n}\,dS=\int_V ||\nabla f||^2\,dV$$
as was to be shown!