Smoothness proof for harmonic function

Question

I was reading the proof of theorem 6 in Evans PDE. I do not understand last 2 steps in the proof. Please, can anyone help me to understand?

Any Help will be appreciated

Answer 1

The second-to-last equality follows directly from the mean-value property. The last one comes from recognizing that \begin{align*}\frac{1}{\epsilon^n}u(x)\int\limits_0^\epsilon\eta\left(\frac{r}{\epsilon}\right)n\alpha(n)r^{n-1}\, dr&=u(x)\int\limits_0^\epsilon\eta_\epsilon(r)n\alpha(n)r^{n-1}\, dr\\ &=u(x)\int\limits_{S^{n-1}}\int\limits_0^\epsilon\eta_\epsilon(r)r^{n-1}\, drd\sigma(\omega)\\ &=u(x)\int\limits_{B(0,\epsilon)} \eta_\epsilon\, dy, \end{align*} where $d\sigma(\omega)$ is the surface measure on the sphere. In particular, the last inequality just comes from polar coordinates: $$\int\limits_{B(0,s)} f(x)\, dx=\int\limits_{S^{n-1}}\int\limits_0^s f(r\omega)r^{n-1}\, drd\sigma(\omega).$$ In our case, the integrand is purely radial, so the integral over $S^{n-1}$ just gives the surface area of $S^{n-1}.$ It's perhaps easiest to understand by starting at the end and reading backwards to the beginning.