In this paper the author has an equation (30) which reads
$\int_\Omega \sigma |\nabla u|^2 dx + \sum_\ell \int_{e_\ell} z_\ell \left(\sigma \nabla u \cdot \hat{n} \right)^2 dS = \sum_\ell V_\ell I_\ell$.
Then the author "takes perturbations" $\sigma \rightarrow \sigma + \delta\sigma$, $u \rightarrow u+ \delta u$, and $V \rightarrow V + \delta V$, and from this gets
$ \int_\Omega \delta \sigma |\nabla u|^2 dx + 2 \int_\Omega \sigma \nabla u \cdot \nabla \delta u\ dx + 2 \sum_\ell \int_{e_\ell} z_\ell \left(\sigma \nabla u \cdot \hat{n} \right) \delta \left(\sigma \nabla u \cdot \hat{n} \right) \ dS = \sum_\ell I_\ell \delta V_\ell.$
I think the author took what Wikipedia calls the "total differential", and that the $\delta \sigma$, etc. are the "differentials" of the independent variables.
I would like to extend this to have $\sigma$, $V_\ell$, and $u$ be complex, but after a couple of tries, I can't figure out how to reproduce a complex version of the authors result. In this case, $\sigma: \Omega \subset \mathbb{R}^3 \rightarrow \mathbb{C}$ and $u:\Omega \subset \mathbb{R}^3 \rightarrow \mathbb{C}$ and $V_\ell$ is a complex scalar.
How does one correctly "take perturbations" (total differentials?) of complex things?