I'm looking for a concrete example of an application of integration by parts in higher dimensions.
The formula I'm looking at is from here, here, and here.
$\Omega$ is an open bounded subset of $\mathbb R^n$ with a piece-wise smooth boundary $\Gamma$. If $u$ and $v$ are two continuously differentiable functions on the closure of $\Omega$, then the formula for integration by parts is
$\int_{\Omega} \frac{\partial u}{\partial x_i} v \,d\Omega = \int_{\Gamma} u v \, \hat\nu_i \,d\Gamma – \int_{\Omega} u \frac{\partial v}{\partial x_i} \, d\Omega$
I would appreciate a simple example of $u(x_1,x_2)$ and $v(x_1,x_2)$ and how the formula works out explicitly.
Best Answer
Let $u=v=1$. Then your formula says $$ \int_{\Gamma}\nu_i\,\mathrm{d}\Gamma=0 $$ which is what you should expect: the average normal is 0, or in other words, for a constant vector field on the ambient $\mathbb{R}^2$, there are no net flux flowing in/out of $\Gamma$.
Next, try $u=x_1, v=1$. Then your formula says $$ \operatorname{Area}(\Omega) = \int_{\Gamma} x_1\nu_1\,\mathrm{d}\Gamma $$ (which is a special case of Green's theorem with $M=x$ and $L=0$).
In particular, if $\Omega$ is the unit disc, then $\nu_1=x_1$ and so $$ \int_\Gamma x_1^2\,\mathrm{d}\Gamma=\int_0^{2\pi} \cos^2s\,\mathrm{d}s=\pi. $$ which agrees with the area of $\Omega$.
With $u=x_1, v=x_2$: $$ \int_\Omega x_2\,\mathrm{d}\Omega = \int_\Gamma x_1x_2\nu_1\,\mathrm{d}\Gamma $$ which you can verify for the unit disc (a boring $0=0$).