I am stuck at proving the mean value property of harmonic functions for an open domain $X \subset \mathbb{R^n}$: If $u \in C^2(X)$ is harmonic in $X$ and $\overline{B_r(y)} \subset X$, then:
$$u(y)=\frac{1}{\overline{B_1(0)}r^{n-1}}\int_{\partial B_r(y)} u \, ds$$
where $ds$ is the surface element.
Here's the proof I am going through:
$$0= \int_{B_r(y)} \Delta u\, dx=\int_{B_r(y)}\nabla \cdot \nabla u \,dx=\int_{\partial B_r(y)} \nabla u \cdot n \,ds=\int_{\partial B_r(y)} \nabla u\cdot \frac{x-y}{r} \,ds $$
In my notes, this is followed by the substitution $w=\frac{x-y}{r}$, apparently giving us: $$ =r^{n-1}\int_{|w|=1} \nabla u(y+rw)\cdot w dw=\cdots $$
Can anyone please tell me how did we get that expression? How did our surface integral changed to a standard integral? What exactly is the Jacobian here?
Thank you!