# How to apply integration by parts or the divergence theorem to a quantity involving the derivative of a vector field?

Is there a way to apply integration by parts or the divergence theorem to the integral:

$$\int_{\Omega} \langle F'(x)n(x),n(x)\rangle dx$$

where

• F: $$\mathbb{R}^m\rightarrow\mathbb{R}^m$$
• n: $$\mathbb{R}^m\rightarrow\mathbb{R}^m$$
• $$F^\prime(x) = \begin{bmatrix}\nabla F_1(x)^T\\\vdots\\\nabla F_m(x)^T\end{bmatrix}$$
• $$\langle x,y\rangle = x^Ty$$

To write it another way, I'd like to rewrite the integral

$$\int_{\Omega} n(x)^T F'(x)n(x) dx$$

as a surface integral on $$\partial \Omega$$ where we shed the derivative off of $$F$$. Please assume $$\Omega$$ to be as smooth as necessary and other properties on $$F$$ to get us as close as possible. Likely, $$m$$ will be 3, but I'd like a result for a general $$m$$ if possible. Thanks!

This can be done using index notation (and also vector notation). We have $$\begin{multline} \left = \mathbf{n}\cdot(\mathbf{n}\cdot\boldsymbol\nabla\mathbf{F})= n_in_j\partial_iF_j = \partial_i(n_in_jF_j) - \partial_i(n_in_j)F_j \\= \boldsymbol\nabla \cdot [\mathbf{n}(\mathbf{n}\cdot \mathbf{F})] - [\boldsymbol \nabla\cdot(\mathbf{n}\mathbf{n})]\cdot\mathbf{F} \end{multline}$$ Divergence theorem then gives $$\int_{\Omega}\mathbf{n}\cdot(\mathbf{n}\cdot\boldsymbol\nabla\mathbf{F})d^mV = \int_{\partial\Omega} (\mathbf{n}\cdot\mathbf{F})(\mathbf{n}\cdot d^{m-1}\mathbf{A}) - \int_\Omega[\boldsymbol\nabla\cdot(\mathbf{n}\mathbf{n})]\cdot\mathbf{F}d^mV$$ where $$d^m V$$ is the $$m$$-dimensional volume element and $$d^{m-1}\mathbf{A}$$ is the directed $$m-1$$-dimensional area element of the $$m-1$$-dimensional surface $$\partial\Omega$$.