0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.