Okay, so this is a step in the proof to Liouville's theorem, and it comes from Fritz John'sPartial Differential Equations4th ed (p109, sec. 4.3). The problem is, it requires computation of a surface integral (I think), and I don't know how to do that.

Here's the setup:

Let $\displaystyle u$ be harmonic in $\displaystyle \mathbb{R}^n$ with $\displaystyle \xi$ in the ball of radius $\displaystyle a>0$ about the origin. Let $\displaystyle \omega_n=2\pi^{n/2}/\Gamma(n/2)$ be the surface area of the unit sphere $\displaystyle S^{n-1}\subset\mathbb{R}^n$ (i.e. $\displaystyle \omega_2=2\pi$, $\displaystyle \omega_3=4\pi$, and so on.)

We are also given the following identity:

$\displaystyle u_{\xi_i}(0)=\frac{n}{\omega_n a^{n+1}}\int_{|x|=a}x_i u(x)\;dS_x$

Frankly, I'm not sure what $\displaystyle dS_x$ is supposed to denote. I assume it's the "surface integral" with respect to x, but then I'm not sure what the definition of a surface integral is, much less how to evaluate them routinely.

Anyway, we have to use the above information to prove the following identity:

$\displaystyle |u_{\xi_i}(0)|\leq\frac{n}{a}\max_{|x|=a}|u(x)|$

Any ideas on how to do this? Any help would be much appreciated!