I am studying the mean value property of harmonic function. In the book,

Introducing radial and angular coordinates $r=|x-y|$, $w=\frac{x-y}{r}$ and writing $u(x)=u(y+rw),$ we have $$ \int_{\partial B_\rho}\frac{\partial u}{\partial \nu}ds=\int_{\partial B_\rho}\frac{\partial u}{\partial r}(y+\rho w)ds=\rho^{n-1}\int_{|w|=1}\frac{\partial u}{\partial r}(y+\rho w)dw=\cdots, $$ where $\nu$ is outer unit normal vector and $ds$ indicates the $(n-1)$-dimensional area element.

It seems that a change of variable is used in the second equality above. I don't know how to do it exactly. Actually, I don't know about polar coordinates( or surface integral) in higher dimensional space.

Would you give any comment for the second equality? Thanks in advance!


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