I'm having trouble proving the following integration by parts formula for a bounded domain $\Omega \subset \mathbb{R}^n$ with $\partial \Omega$ being at least $C^2$:
$$ \int_{\partial \Omega} \psi \delta_j \phi = - \int_{\partial \Omega} \phi \delta_j \psi + \int_{\partial \Omega} \psi \phi H \eta^j $$
where $\delta_j = \sum_{i=1}^n( \delta_{ij} - \eta^i \eta^j) D_i $ is the $j^{th}$ component of the tangential derivative, and $\eta = Dd$ is the inward pointing normal from the boundary (which exists since the boundary is at least $C^2$) and $H = \Delta d|_{\partial \Omega}$ for $d$ the distance function for $\partial \Omega$ in a sufficiently small strip near the boundary.
The text claims:
These integration by parts formulae are easily proved. We just the divergence theorem over $\Omega$ to evaluate each side of the identity $\int_{\Omega} (d_k \phi)_{ki} = \int_{\Omega} (d_k \phi)_{ik}$, (where subscripts denote partial derivatives in the indicated variable), and this gives the required identity after replacing $\phi$ by $\phi \psi$.
I have no idea really how to interpret the hint and use it. The identity above seems to simply be a trivial consequence of the distance function being $C^2$, and I don't know what useful information I'm supposed to gather from it. Moreover, I truly don't know how to use the divergence theorem to "evaluate both sides" of that identity - since I have no idea what that quantity is the divergence of, so I don't know how to apply the divergence theorem.
If someone could tell me: How to "apply the divergence theorem" and why the identity above is useful for the problem, I would very much appreciate it.
Thanks