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\section*{Notations}
\begin{longtable}[c]{|r|l|l|}
\hline
\Rheolef & mathematics
& description
\\ \hline \hline \endhead
\code{d} & $d\in\{1,2,3\}$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& dimension of the physical space
\\ \hline
\code{interpolate}(\code{Vh},\emph{expr})
& $\pi_{V_h} (\mbox{\emph{expr}})$
& interpolation in the space $V_h$
\\ \hline
\code{integrate}(\code{omega},\emph{expr})
& $\displaystyle \int_\Omega \mbox{\emph{expr}}\;\mathrm{d}x$
& integration in $\Omega\subset\mathbb{R}^d$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
\\ \hline
\begin{tabular}{l}
\code{integrate}(omega, \\
\ \ \ \code{on_local_sides}(\emph{expr}))
\end{tabular}
& $\displaystyle \sum_{K\in\mathscr{T}_h} \int_{\partial K} \mbox{\emph{expr}}\;\mathrm{d}s$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& integration on the local element sides
\\ \hline
\hline
\code{dot(u,v)}
& ${\bf u}.{\bf v}={\displaystyle \sum_{i=0}^{d-1} u_iv_i}$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& vector scalar product
\\ \hline
\code{ddot(sigma,tau)}
& $\sigma:\tau={\displaystyle \sum_{i,j=0}^{d-1} \sigma_{i,j}\tau_{i,j}}$
& tensor scalar product
\\ \hline
\code{tr(sigma)}
& ${ {\rm tr}(\sigma) = \displaystyle \sum_{i=0}^{d-1} \sigma_{i,i} }$
& trace of a tensor
\\ \hline
\code{trans(sigma)}
& $\sigma^T$
& tensor transposition
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
\\ \hline
% TODO
% \code{otimes(u,v)}
% & ${\bf u}\otimes{\bf v}={\displaystyle \left( u_iv_j \right)_{0\leq i,j<d} }$
% \phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
% & tensorial product of two vectors
% \\ \hline
\begin{tabular}{r}
\code{sqr(phi)} \\
\code{norm2(phi)}
\end{tabular} & $\phi^2$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& square of a scalar
\\ \hline
\code{norm2(u)}
& $|{\bf u}|^2={\displaystyle \sum_{i=0}^{d-1} u_i^2}$
& square of the vector norm
\\ \hline
\code{norm2(sigma)}
& $|\sigma|^2={\displaystyle \sum_{i,j=0}^{d-1} \sigma_{i,j}^2}$
& square of the tensor norm
\\ \hline
\begin{tabular}{r}
\code{abs(phi)} \\
\code{norm(phi)}
\end{tabular} & $|\phi|$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& absolute value of a scalar
\\ \hline
\code{norm(u)}
& $|{\bf u}|={\displaystyle \left( \sum_{i=0}^{d-1} u_i^2 \right)^{1/2} }$
& vector norm
\\ \hline
\code{norm(sigma)}
& $|\sigma|={\displaystyle \left( \sum_{i,j=0}^{d-1} \sigma_{i,j}^2 \right)^{1/2} }$
& tensor norm
\\ \hline
\code{grad(phi)}
& $\nabla\phi={\displaystyle \left( \frac{\partial \phi}{\partial x_i} \right)_{0\leq i<d} }$
& gradient of a scalar field % in $\Omega\subset\mathbb{R}^d$
\\ \hline
\code{grad(u)}
& $\nabla{\bf u}={\displaystyle \left( \frac{\partial u_i}{\partial x_j} \right)_{0\leq i,j<d} }$
& gradient of a vector field
\\ \hline
\code{div(u)}
& ${\rm div}({\bf u})={\rm tr}(\nabla{\bf u})={\displaystyle \sum_{i=0}^{d-1} \frac{\partial u_i}{\partial x_i}}$
& divergence of a vector field
\\ \hline
\code{D(u)} & $D({\bf u})=\left( \nabla{\bf u} + \nabla{\bf u}^T \right)/2$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& \begin{tabular}{l}
symmetric part of \\
the gradient of a vector field
\end{tabular}
\\ \hline
\code{curl(u)}
& ${\bf curl}({\bf u})=\nabla \wedge {\bf u}$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& curl of a vector field, when $d=3$
\\ \hline
\code{curl(phi)}
& ${\bf curl}(\phi)= {\displaystyle \left(\frac{\partial \phi}{\partial x_1},
- \frac{\partial \phi}{\partial x_0} \right) }$
& curl of a scalar field, when $d=2$
\\ \hline
\code{curl(u)}
& ${\rm curl}({\bf u})= {\displaystyle \frac{\partial u_1}{\partial x_0}
- \frac{\partial u_0}{\partial x_1} }$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& curl of a vector field, when $d=2$
\\ \hline
\code{grad_s(phi)}
& \begin{tabular}{l}
$\nabla_s\phi=P\nabla\phi$ \\
\hspace{1cm} where $P=I-{\bf n}\otimes{\bf n}$
\end{tabular}
& tangential gradient of a scalar
\\ \hline
\code{grad_s(u)}
& $\nabla_s{\bf u}=\nabla{\bf u}P$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& tangential gradient of a vector
\\ \hline
\code{Ds(u)}
& $D_s({\bf u})=P D({\bf u}) P$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& symmetrized tangential gradient
\\ \hline
\code{div_s(u)}
& ${\rm div}_s({\bf u})={\rm tr}(D_s({\bf u}))$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& tangential divergence
\\ \hline
& & unit outward normal on $\Gamma=\partial\Omega$ \\
\code{normal()}
& ${\bf n}$
& or on an oriented surface $\Omega$ \\
& & or on an internal oriented side $S$
\\ \hline
&&\\
\code{jump(phi)}
& $\jump{\phi}=\phi_{|K_0}-\phi_{|K_1}$
& jump accros inter-element side \\
& & $S=\partial K_0\cap K_1$
\\ \hline
\code{average(phi)}
& $\average{\phi}=(\phi_{|K_0}+\phi_{|K_1})/2$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& average across $S$
\\ \hline
\code{inner(phi)}
& $\phi_{|K_0}$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& inner trace on $S$
\\ \hline
\code{outer(phi)}
& $\phi_{|K_1}$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& outer trace on $S$
\\ \hline
\code{h_local()}
&
$ h_K = {\rm meas}(K)^{1/d}$
& length scale on an element $K$
\\ \hline
&&\\
\code{penalty()}
&
$ \varpi_s = {\rm max}\left(
\Frac{{\rm meas}(\partial K_0)}{{\rm meas}(K_0)}, \
\Frac{{\rm meas}(\partial K_1)}{{\rm meas}(K_1)}
\right)$
& penalty coefficient on $S$
\\ \hline
&&\\
\code{grad_h(phi)}
& $(\nabla_h\phi)_{|K}=\nabla(\phi_{|K}), \forall K\in\mathcal{T}_h$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& broken gradient
\\ \hline
\code{div_h(u)}
& $({\rm div}_h{\bf u})_{|K}={\rm div}({\bf u}_{|K}), \forall K\in\mathcal{T}_h$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& broken divergence of a vector field
\\ \hline
&&\\
\code{Dh(u)}
& $(D_h({\bf u}))_{|K}=D({\bf u}_{|K}), \forall K\in\mathcal{T}_h$
& broken symmetric part of \\
&& the gradient of a vector field
\\ \hline
&&\\
\code{sin(phi)} & $\sin(\phi)$
& standard mathematical functions
\\
\code{cos(phi)} & $\cos(\phi)$
& extended to scalar fields
\\
\code{tan(phi)} & $\tan(\phi)$ & \\
\code{acos(phi)} & $\cos^{-1}(\phi)$ & \\
\code{asin(phi)} & $\sin^{-1}(\phi)$ & \\
\code{atan(phi)} & $\tan^{-1}(\phi)$ & \\
\code{cosh(phi)} & $\cosh(\phi)$ & \\
\code{sinh(phi)} & $\sinh(\phi)$ & \\
\code{tanh(phi)} & $\tanh(\phi)$ & \\
\code{exp(phi)} & $\exp(\phi)$ & \\
\code{log(phi)} & $\log(\phi)$ & \\
\code{log10(phi)} & $\log10(\phi)$ & \\
\code{floor(phi)} & $\lfloor \phi \rfloor$
& largest integral not greater than $\phi$
\\
\code{ceil(phi)} & $\lceil \phi \rceil$
& smallest integral not less than $\phi$
\\
\code{min(phi,psi)} & $\min(\phi,\psi)$ & \\
\code{max(phi,psi)} & $\max(\phi,\psi)$ & \\
\code{pow(phi,psi)} & $\phi^\psi$ & \\
\code{atan2(phi,psi)} & $\tan^{-1}(\psi/\phi)$ & \\
\code{fmod(phi,psi)} & $\phi-\lfloor \phi/\psi+1/2\rfloor\,\psi$
& floating point remainder
\\
&& \\ \hline
\code{compose(f,phi)}
& $f\circ\phi = f(\phi)$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& applies an unary function $f$
\\ \hline
% \code{compose(f,phi,psi)}
% & $f(\phi,\psi)$
% \phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
% & applies a binary function $f$
% \\ \hline
\code{compose(f,phi1,\ldots,phin)}
& $f(\phi_1,\ldots,\phi_n)$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& applies a $n$-ary function $f$, \ $n\geq 1$
\\ \hline
\code{compose(phi,X)}
& $\phi\circ X$, \ $X(x)=x+{\bf d}(x)$
\phantom{${\displaystyle \sum_{i,j=0}^{d-1}}$}
& composition with a characteristic
\\ \hline
\end{longtable}
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