= \parG{\uv \in
    \Lsp^2(\srfdomflat;\,\R^2): \divergt \uv \in \Lsp^2(\srfdomflat)},
\\ 
&\CaSpB^{2} && = \Lsp^2(\srfdomflat)&& = \parG{u:
    \int_{\srfdomflat} u^2 < \infty}. 
\end{alignedat}$$ Here, $\gradt$ and $\divergt$ are the standard gradient and divergence in Cartesian coordinates $$\gradt u = \begin{bmatrix}\pd{\rho}u\\ \pd{z}u\end{bmatrix}~, \qquad
\divergt\parT{\begin{bmatrix}u_\rho\\u_z\end{bmatrix}} =
\pd{\rho}{u_\rho} + \pd{z}{u_z}$$ and the scalar curl is defined as $$\curltsc\parT{\begin{bmatrix}u_\rho\\u_z\end{bmatrix}} =
\pd{\rho}{u_z} - \pd{z}{u_\rho}~.$$ For functions in $\CaSpB^{0}$, also the following operator is well-defined: $$\renewcommand\arraystretch{1.2}
\rott(u) = \begin{bmatrix}\phantom{-}\pd{z}{u} \\
-\pd{\rho}{u}\end{bmatrix} = \mat{P} \parT{\gradt(u)}~,$$ where $$\mat{P} = \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}~.$$ We will use the word rotor, or perpendicular gradient, to indicate $\rott$ and to avoid the confusion with the three-dimensional $\curlt$. For error estimates we will also need the the spaces $\Hsp^s(\curlt; \, S)$, which is the space of functions in $\Hsp^s(S;\, \R^3)$ such that their curl belongs to $\Hsp^s(S;\,\R^3)$ and $\Hsp^s(\divergt; \, S)$, which is the space of functions in $\Hsp^s(S;\, \R^3)$ such that their divergence belongs to $\Hsp^s(S)$.

We assume that $\CaSpB_h^0$, $\CaSpB_h^1$, $\CaSpB_h^{1*}$ and $\CaSpB_h^2$ are conforming discretizations of  and that, with a sequence of $\Lsp^2$-stable projectors, make the following diagrams commutative: $$\label{eq:2D_curl_comp}
\begin{split}
\tikzsetnextfilename{de_rham/de_rham_2d_cart_curl}
\begin{tikzpicture}[commutative diagrams/every diagram,
arr/.style={commutative diagrams/.cd,
    every arrow,
    every label}]
\def\minW{25mm}
\def\minH{15mm}
\def\hShift{10mm}
\def\vShiftA{20mm}
\def\vShiftB{4mm}
\def\auxA{1mm}
\def\auxB{4mm}
