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?


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$.

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  • $\begingroup$ Thank you! I just don't familiar with the Green's formula in n-dimentions, is there any excellent book about this topic? $\endgroup$ – Weeson Dorne Jul 26 '12 at 10:40
  • $\begingroup$ @WeesonDorne: From what I know "Callahan - Advanced Calculus, A Geometric View" seems to be a good book. The only unfortunate thing is that it treats only the 3D case, which is of course no different to the n-dimensional one. $\endgroup$ – Wizard Sep 12 '14 at 18:05

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