Fourier orthogonal system in the angular variable $\theta$. Moreover, it is convenient to express each function on $\voldomaxis$ as the sum of a symmetric and an antisymmetric part with respect to the plane at $\theta=0$, so that, for a scalar function, we have that $u = u^s + u^a$, where $$\begin{aligned}
u^s & = u^\parTs{0} + \sum_{m=1}^{\infty} u^\m \,\cos{m\theta}~, \\
u^a & = \sum_{m=1}^{\infty}u^\mm \,\sin{m\theta}~.\end{aligned}$$ For a vector function we have instead that $\uv = \uv^s +
\uv^a$, where $$\begin{aligned}
\uv^s &= \begin{bmatrix}
u_\rho^\parTs{0} \\
u_z^\parTs{0}\\
0
\end{bmatrix}+\sum_{m=1}^{\infty}\begin{bmatrix}
u_\rho^\m \,\cos{m\theta}\\
u_z^\m\,\cos{m\theta}\\
u_\theta^\m\,\sin{m\theta}
\end{bmatrix}~, \\
\uv^a &= \begin{bmatrix}
0 \\
0\\
u_\theta^\parTs{0}
\end{bmatrix}+\sum_{m=1}^{\infty}\begin{bmatrix}
u_\rho^\mm \,\sin{m\theta}\\
u_z^\mm\,\sin{m\theta}\\
u_\theta^\mm\,\cos{m\theta}
\end{bmatrix}~.\end{aligned}$$ The Fourier coefficients $u^\m$, $\uv^\m$ are defined on the cross-section $\srfdomflat$. With this representation, the effect of the differential operators on each mode $m$ can be considered independently, leading to the definition of the following operators acting on the Fourier coefficients: $$\label{eq:cyl_gradm}
\renewcommand\arraystretch{1.5}
\gradm u^\m = \begin{bmatrix}
\pd{\rho}{u^\m} \\
\pd{z}{u^\m}\\
-\dfrac{m}{\rho}u^\m
\end{bmatrix}~,$$ $$\label{eq:cyl_curlm}
\renewcommand\arraystretch{1.5}
\curlm \uv^\m = \begin{bmatrix}
-\dfrac{m}{\rho}u^\m_z - \pd{z}{u^\m_\theta}\\
\dfrac{1}{\rho}\parT{\pd{\rho}\parT{\rho\,u^\m_\theta} +
    {m}\,{u^\m_\rho}}\\ 
\pd{z}{u^\m_\rho} - \pd{\rho}{u^\m_z}
\end{bmatrix}~,$$ $$\label{eq:cyl_divm}
\divergm \uv^\m = \dfrac{1}{\rho}\,\pd{\rho}\parT{\rho\, u^\m_\rho} -
\dfrac{m}{\rho}{u^\m_\theta} + \pd{z}{u_z}~.$$ In cylindrical coordinates, the integral of a function $f$ over a volume has the form 