angular variable $\theta$ as the last one will considerably simplify the notation of the following considerations. The Cartesian coordinates are related to the cylindrical ones by $$\begin{bmatrix} x \\ y \\ z\end{bmatrix} =
\vec{g} \parT{\begin{bmatrix} \rho \\ z \\ \theta \end{bmatrix}}
= \begin{bmatrix} \rho \,\cos{\theta} \\ \rho \, \sin{\theta} \\
z\end{bmatrix}$$ and, conversely $$\rho = \sqrt{x^2 + y^2} \quad \text{and} \quad \theta =
\left\{
\begin{aligned}
&-\arccos \parT{\tfrac{x}{\rho}}~, && \text{if $y < 0$}~, \\
&\arccos \parT{\tfrac{x}{\rho}}~, && \text{if $y \geq 0$}~.
\end{aligned}
\right.$$ We describe the domain $\voldomaxis$ using its cross-section with respect to the and $\rho z$-plane $\srfdomflat \subset \R^+ \times
\R$. We assume that $\srfdomflat \subset \R^2$ is a bounded Lipschitz domain obtained through a diffeomorphism $\Fv$ of the unit square and that $\partial \srfdomflat \cap \parG{\rho = 0}$ is either empty or coincides with $\Fv(\parGs{0} \times \parQ{0, \, 1})$. Let $\Gamma_0 = \interior\parT{\partial \srfdomflat \cap \parG{\rho =
        0}}$ be the interior of the intersection of the boundary of $\srfdomflat$ with the $z$-axis and $\Gamma = \partial \srfdomflat \setminus \Gamma_0$. The volume $\voldomaxis$ is obtained by rotating the cross-section over the symmetry axis $z$ and adding $\Gamma_0$: $$\voldomaxis = \parG{ \xv \in \R^3: \xv = \vec{g}\parT{\parT{\rho, \,
            z, \, 