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 $$ \int_{\partial B_\rho}\frac{\partial u}{\partial \nu}ds=\int_{\partial B_\rho}\frac{\partial u}{\partial r}(y+\rho w)ds \cdots, $$ where $\nu$ is outer normal vector and $ds$ indicates the $(n-1)$-dimensional area element.
It seems that the chain rule is used in the above. I don't know how to do it exactly.
$\frac{\partial u}{\partial \nu}=\nabla u\cdot \nu=\nabla u \cdot w$ and what does $\frac{\partial u}{\partial r}(y+\rho w)$ mean exactly?
Would you give any comment for it? Thanks in advance!
(1) Definition of directional derivative : $\frac{\partial u}{\partial \nu } = {\rm grad}\ u\cdot \nu $
(2) Here $\nu$ is unit out normal to $ \partial B(x_0,\rho)$ so that $$ \nu (x_0+\rho v) = v $$ where $|v|=1$
(3) Hence $$ \frac{d}{dt}\bigg|_{t=\rho} \ u(x_0 +tv ) = {\rm grad}\ u\cdot v ={\rm grad}\ u\cdot \nu =\frac{\partial u}{\partial \nu } $$