Recently, I met an equation about the integration by parts and surface integrals. It says:$$
\int\_{|\xi|\geq\epsilon}D\_i\Gamma(\xi){\partial\over\partial{\xi\_i}}f(\xi)d\xi=-\int\_{|\xi|=\epsilon}D\_i\Gamma(\xi)f(\xi){\xi\_i\over{|\xi|}}dS$$
here $\Gamma$ satisfy $\triangle\Gamma(\xi)=0$ and $f\in C\_0^\infty(R^n)$. So, is there someone that could explain it in detail?



the relationship between integration by parts and surface integrals
This follows immediately from Green's formula, which says for $X$ vector field,
$\Omega$ open set,
$n$ the unit exterior normal to the boundary $\partial \Omega$
$$
\int\_{\Omega}div X\ dx=\int\_{\partial \Omega}X\cdot n d\sigma.
$$
Apply this to the vector field $X=u\nabla v$ and you get
$$
\int\_{\Omega}(\nabla u\cdot\nabla v+u\Delta v)dx=\int\_{\partial \Omega}u\frac{\partial v}{\partial n}d\sigma,
$$
so that for $v$ harmonic,
$
\int\_{\Omega}\nabla u\cdot\nabla vdx=\int\_{\partial \Omega}u\frac{\partial v}{\partial n}d\sigma$,
which exactly your formula, since the unit exterior normal to the set {$\vert \xi\vert\ge \epsilon$} is -$\xi/\vert\xi\vert$.


