Integration by parts - multivariate case

Question

As part of a larger problem, I realised I don't quite know how to do integration by parts in the multivariate case. I looked up some formulas, but I couldn't get them to apply. For example, how would you do integration by parts on:

$$\int_{0}^{1}\int_{0}^{1}u_{xx} v\;dxdy$$ where $u$ and $v$ are functions of $x$ and $y$ and we are integrating across a unit square. The subscripts denote partial differentiation with respect to that variable. Along the boundary of the square we have $u=v$.

Many thanks!

Answer 1

I don't know if this is what you're searching for, but this is sometimes called integration by part formula in higher dimension.

Consider a function of two variable $f$ such that $f(x,y)=g(x,y)h(x,y)$ and let $\Omega \subseteq \Bbb R^2$; by Gauss-Green theorem you know: $$\iint_{\Omega} f_x \,\mathrm{d}x\mathrm{d}y=\int_{\partial \Omega} fn_1\,\mathrm{d}s= \int_{\partial \Omega}ghn_1\mathrm{d}s$$ Where $n_1$ is the first component of the normal unitary vector. On the other hand, $f_x=g_xh+gh_x$: $$\iint_{\Omega} (g_xh+gh_x)\mathrm{d}x\mathrm{d}y=\iint_{\Omega} g_xh\mathrm{d}x\mathrm{d}y+\iint_{\Omega} gh_x\mathrm{d}x\mathrm{d}y $$ Combining the two results, you have: $$\iint_{\Omega} g_xh\mathrm{d}x\mathrm{d}y=\int_{\partial \Omega}ghn_1\mathrm{d}s-\iint_{\Omega} gh_x\mathrm{d}x\mathrm{d}y $$ You can have a similar expression deriving with respect to $y$.