Consider $$\int_{S_r} \left(\frac{\partial\phi}{\partial\eta}\right)^2 (x\cdot\eta)\ d\sigma $$ Where $S_r$ is the sphere of radius $r$ and centered at zero, $\phi\in L^2(\mathbb{R}^n)$ and $\eta$ is the outward unit vector to the sphere. Can anyone explain what does this integral means? And is it true that the integral tends to zero as $r$ tends to infinity?
Thanks in advance.