# Integral related to harmonic functios

It suppose to be a easy task. But I couldn't solve it (I guess I can't learn much analysis). If $u$ is harmonic, in the middle of my problem, I need to prove that the integral $\int _ {\partial \Omega } \displaystyle\frac{\partial u}{\partial v} d s_x =0$ THis $v$ denotes the unit exterior normal.

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$$\int_{\partial\Omega}\frac{\partial u}{\partial \nu}\,ds = \int_{\Omega} (\Delta u + \nabla 1 \cdot \nabla u)\,dV = 0,$$ since $\Delta u = 0$ by assumption and $\nabla 1 = \mathbf{0}$.
It is just a consequence of Divergence theorem, isn't it? You have that u is harmonic, so $$\Delta u =\nabla\cdot \nabla u=0.$$ Now integrate over $\Omega$ and apply the divergence theorem.