My professor in the real analysis class had state the following in class but forgot to put the reference of this formula in the power point slide.
The formula for integration by parts can be extended to functions of several variables. Instead of an interval one needs to integrate over an n-dimensional set. Also, one replaces the derivative with a partial derivative.
More specifically, suppose $\Omega$ 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
$\int_{\Omega} \frac{\partial u}{\partial x_i} v \,d\Omega = \int_{\Gamma} u v \, \hat\nu_i \,d\Gamma – \int_{\Omega} u \frac{\partial v}{\partial x_i} \, d\Omega$
where $\hat{\mathbf{\nu}}$ is the outward unit surface normal to $\Gamma$, $\hat\nu_i$ is its $i$-th component, and $i$ ranges from $1$ to $n$.
Find a rigorous reference that prove the following integration by parts formula in higher dimension?
Best Answer
Look for it in the book Multidimensional Real Analysis, volume II. The authors are Duistermaat and Kolk.
Look for section 7.6 Integration of a total derivative, Corollary 7.6.2 (Integration by parts in $\textbf{R}^n$.