u}_\rho^2~,\\ 
&\norm{\uv}_{\Hsp_\rho(\curlm;\,\srfdomflat)}^2 &&= \norm{\uv}_\rho^2
+ \norm{\curlm u}_\rho^2~,\\ 
&\norm{\uv}_{\Hsp_\rho(\divergm;\,\srfdomflat)}^2 &&=
\norm{\uv}_\rho^2 + \norm{\divergm u}_\rho^2~. \end{aligned}$$ The spaces  arise naturally in the resolution of electromagnetic problems in cylindrical coordinates by means of Galerkin formulations. For example, for $m\neq0$, the formulation associated to  with boundary conditions reads

\[pb:weak\_curl\_curl\] Find the eigenpair $\parTs{\omega^2, \, \Ev^{(m)}} \in \R_+ \times \CySpT_0^{m, \, 1}$, with $\Ev \neq \vec{0}$ and where the subscript $0$ in the space denotes the vanishing tangential component on $\Gamma$, such that $$\label{eq:axis_weak_eig_pb2}
    \begin{split}
    &\int_\srfdomflat \parTs{\pmeab^{-1}\curltm \Ev^{(m)}}\cdot\curltm \vv\,\rho\,\dd\rho \dd z \\
    &\hspace{30mm}= \omega^2 \int_\srfdomflat \pmitt \Ev^{(m)} \cdot \vv\,\rho\,\dd\rho \dd z
    \end{split}~,\quad \vv \in \CySpT_0^{m, \, 1}~.$$

The Maxwell equations , through the differential operators –, relate quantities belonging to different spaces. A representation of the resulting structure is given by the sequence $$\label{eq:3d_cyl_seq}
\begin{split}
\tikzsetnextfilename{de_rham/de_rham_3d_cyl}
\begin{tikzpicture}[commutative diagrams/every diagram,
arr/.style={commutative diagrams/.cd,
    every arrow,
    every label}]
\def\minW{25mm}
\def\minH{20mm}
\def\hShift{10mm}
\def\vShiftA{20mm}
\def\vShiftB{4mm}
\def\auxA{1mm}
\def\auxB{4mm}
%
% Nodes
\node (X0) [] at (0, 0) {$\CySpT^{m, \, 0}$};
\node (X1) [xshift=1.1*\minW] at (X0) {$\CySpT^{m, \, 1}$};
\node (X2) [xshift=1.1*\minW] at (X1) {$\CySpT^{m, \, 2}$};
\node (X3) [xshift=1.1*\minW] at (X2) {$\CySpT^{m, \, 3}$};
%
\node (R) [xshift=-0.6*\minW] at (X0) {$0$};
\node (Z) [xshift=0.6*\minW] at (X3) {$0$};
% Arrows
\path[draw, arr] 