the relationship between integration by parts and surface integrals

Question

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?

Answer 1

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$.