Suppose $u\in C^2(\Omega)$ and $x\in \Omega\subset \mathbb R^N$. I am trying to prove that $$ \Delta u(x)=\lim_{r\to 0} \frac{2N}{r^2} \left[\frac{1}{\alpha(N)} \int_{\partial B(0,1)}u(x+ry)d\sigma(y)-u(x)\right] $$
The problem gives the hint: "consider the second order Taylor polynomial of $u$ about $x$. By symmetry consideration, $\int_{\partial B(0,1)} x_j=\int_{\partial B(0,1)}x_jx_k=0$ for $j\neq k$ and $\int_{\partial B(0,1)}x_k^2=N^{-1}\int_{\partial B(0,1)} \sum_{j=1}^N x_j^2=N^{-1}\int_{\partial B(0,1)}1$"
I tried with hint but I don't really get any progress... Any help is really welcome!