documentation of operator terms updated

doc/operatorTerms.tex

@@ -28,13 +28,13 @@
 \hline
 \multicolumn{2}{c}{\scriptsize Zero-Order-Terms}\\
 \hline
-\langle c\;\phi,\psi\rangle & \texttt{Simple\_ZOT}($c\in\mathbb{R}$) \\
+\langle c\;\phi,\psi\rangle & \texttt{Simple\_ZOT}($[c\in\mathbb{R}]$) \\
 \langle f(\vec{x})\;\phi,\psi\rangle & \texttt{CoordsAtQP\_ZOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
-\langle f(v)\;\phi,\psi\rangle & \texttt{VecAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$) \\
+\langle f(v)\;\phi,\psi\rangle & \texttt{VecAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $[f:\mathbb{R}\rightarrow\mathbb{R}]$) \\
 \langle f(v, \vec{x})\; \phi,\psi\rangle & \texttt{VecAndCoordsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\ 
 \langle f(v)\;g(w)\;\phi,\psi\rangle & \texttt{MultVecAtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $g:\mathbb{R}\rightarrow\mathbb{R}$) \\
-\langle f(v, w)\;\phi,\psi\rangle & \texttt{Vec2AtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$) \\
-\langle f(v_1, v_2, v_3)\;\phi,\psi\rangle & \texttt{Vec3AtQP\_ZOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$) \\ 
+\langle f(v, w)\;\phi,\psi\rangle & \texttt{Vec2AtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $[f:(\mathbb{R})^2\rightarrow\mathbb{R}]$) \\
+\langle f(v_1, v_2, v_3)\;\phi,\psi\rangle & \texttt{Vec3AtQP\_ZOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $[f:(\mathbb{R})^3\rightarrow\mathbb{R}]$) \\ 
 \langle f(\nabla v)\;\phi,\psi\rangle & \texttt{FctGradient\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
 \langle f(\nabla v, \vec{x})\;\phi,\psi\rangle & \texttt{FctGradientCoords\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\ 
 \langle f(v, \nabla v)\;\phi,\psi\rangle & \texttt{VecAndGradAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
@@ -44,38 +44,39 @@
 \langle f(v_1, v_2 \nabla v_3)\;\phi,\psi\rangle & \texttt{Vec2AndGradVecAtQP\_ZOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
 \langle f(v, \nabla w_1, \nabla w_2)\;\phi,\psi\rangle & \texttt{VecAndGradVec2AtQP\_ZOT}($v,w_1,w_2\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
 \langle f(v,w, \nabla v, \nabla w)\;\phi,\psi\rangle & \texttt{Vec2AndGrad2AtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
-\langle f(\{v_i\}_i)\;\phi,\psi\rangle & \texttt{VecOfDOFVecsAtQP\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle \rangle$, $f:$\small{vector}$\langle\mathbb{R}\rangle\rightarrow\mathbb{R}$) \\
-\langle f(\{\nabla v_i\}_i)\;\phi,\psi\rangle & \texttt{VecOfGradientsAtQP\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
-\langle f(v, \{\nabla w_i\}_i)\;\phi,\psi\rangle & \texttt{VecAndVecOfGradientsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
+\langle f(\{v_i\}_i)\;\phi,\psi\rangle & \texttt{VecOfDOFVecsAtQP\_ZOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle \rangle$, $f:$\small{vec}$\langle\mathbb{R}\rangle\rightarrow\mathbb{R}$) \\
+\langle f(\{\nabla v_i\}_i)\;\phi,\psi\rangle & \texttt{VecOfGradientsAtQP\_ZOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
+\langle f(v, \{\nabla w_i\}_i)\;\phi,\psi\rangle & \texttt{VecAndVecOfGradientsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
 \langle \partial_1 v_1\,[+\partial_2 v_2 + \partial_3 v_3]\;\phi,\psi\rangle & \texttt{VecDivergence\_ZOT}($v_1\,[,v_2,v_3]\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$) \\
-\langle f(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x})\;\phi,\psi\rangle & \texttt{General\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}^n\times$\small{vector}$\langle\mathbb{R}\rangle\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
+\langle f(\vec{x}, \{v_i\}, \{\nabla w_j\})\;\phi,\psi\rangle & \texttt{General\_ZOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
+\langle f(\vec{x}, \vec{n}, \{v_i\}, \{\nabla w_j\})\;\phi,\psi\rangle & \texttt{GeneralParametric\_ZOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}^n\times\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
 \hline
 \end{longtable}
-\begin{longtable}{>{\begin{math}}p{0.275\textwidth}<{\end{math}}|>{\begin{math}}p{0.275\textwidth}<{\end{math}}|p{.7\textwidth}}
+\begin{longtable}{>{\begin{math}}p{0.28\textwidth}<{\end{math}}|>{\begin{math}}p{0.28\textwidth}<{\end{math}}|p{.69\textwidth}}
 \hline
 %==============================================
 \multicolumn{3}{c}{\scriptsize First-Order-Terms}\\
 \hline
 \text{\scriptsize GRD\_PHI} & \text{\scriptsize GRD\_PSI} & \\
 \hline
-\langle \vec{1} \cdot \nabla \phi,\psi\rangle & \langle \vec{1} \phi,\nabla \psi\rangle & \texttt{Simple\_FOT}() \\
-\langle c\,\vec{1} \cdot \nabla \phi,\psi\rangle & \langle c\,\vec{1} \phi,\nabla \psi\rangle & \texttt{FactorSimple\_FOT}($c\in\mathbb{R}$) \\
-\langle \vec{b} \cdot \nabla \phi,\psi\rangle & \langle \vec{b} \phi,\nabla \psi\rangle & \texttt{Vector\_FOT}($b\in\mathbb{R}^n$) \\
-\langle v\cdot w\cdot\vec{b}\cdot\nabla \phi,\psi\rangle & \langle v\cdot w\cdot\vec{b} \phi,\nabla \psi\rangle & \texttt{Vec2AtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $b\in\mathbb{R}^n$) \\
-\langle f(v)\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(v)\cdot \vec{b} \phi,\nabla \psi\rangle &\texttt{VecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
-\langle f(\vec{x})\,\vec{1} \cdot \nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot \vec{1} \phi,\nabla \psi\rangle &\texttt{CoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
-\langle f(\vec{x})\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot \vec{b} \phi,\nabla \psi\rangle &\texttt{VecCoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
-\langle f(\vec{x})\cdot v\cdot\vec{b}\cdot\nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot v\cdot\vec{b} \phi,\nabla \psi\rangle &\texttt{FctVecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$,$f:\mathbb{R}^n\rightarrow\mathbb{R}$,$b\in\mathbb{R}^n$) \\
-\langle v_1\cdot f(v_2,v_3)\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle v_1\cdot f(v_2,v_3)\cdot\vec{b} \phi,\nabla \psi\rangle &\texttt{Vec3FctAtQP\_FOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
-\langle f(v,w,\nabla v)\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(v,w,\nabla v)\cdot\vec{b}  \phi,\nabla \psi\rangle &\texttt{Vec2AndGradAtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
+\langle c\,\vec{1} \cdot \nabla \phi,\psi\rangle & \langle c\,\vec{1} \phi,\nabla \psi\rangle & \texttt{Simple\_FOT}($[c\in\mathbb{R}]$) \\
+\langle c\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle c\,\vec{b} \phi,\nabla \psi\rangle & \texttt{Vector\_FOT}($\{b\in\mathbb{R}^n\,|\,i\in\mathbb{N}\}$, $[c\in\mathbb{R}]$) \\
+\langle f(v)\cdot\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(v)\cdot \vec{b} \phi,\nabla \psi\rangle &\texttt{VecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $\{b\in\mathbb{R}^n\,|\,i\in\mathbb{N}\}$) \\
+\langle f(v, w)\cdot\vec{b}\cdot\nabla \phi,\psi\rangle & \langle f(v, w)\cdot\vec{b} \phi,\nabla \psi\rangle & \texttt{Vec2AtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:(\mathbb{R})^2\rightarrow\mathbb{R}$, $\{b\in\mathbb{R}^n\,|\,i\in\mathbb{N}\}$) \\
+\langle f(v_1, v_2, v_3)\cdot\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(v_1, v_2, v_3)\cdot\vec{b} \phi,\nabla \psi\rangle &\texttt{Vec3AtQP\_FOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:(\mathbb{R})^3\rightarrow\mathbb{R}$, $\{b^*\in\mathbb{R}^n\,|\,i\in\mathbb{N}\}$) \\
+\langle f(\vec{x})\cdot\vec{1} \cdot \nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot \vec{1} \phi,\nabla \psi\rangle &\texttt{CoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
+\langle f(\vec{x})\cdot\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot \vec{b} \phi,\nabla \psi\rangle &\texttt{VecCoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$, $b^*\in\mathbb{R}^n$) \\
+\langle f(\vec{x}, v)\cdot\vec{b}\cdot\nabla \phi,\psi\rangle & \langle f(\vec{x}, v)\cdot\vec{b} \phi,\nabla \psi\rangle &\texttt{FctVecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}$, $b^*\in\mathbb{R}^n$) \\
+\langle f(v,w,\nabla v)\cdot\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(v,w,\nabla v)\cdot\vec{b}  \phi,\nabla \psi\rangle &\texttt{Vec2AndGradAtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:(\mathbb{R})^2\times\mathbb{R}^n\rightarrow\mathbb{R}$, $b^*\in\mathbb{R}^n$) \\
+\langle F(\vec{x}) \cdot \nabla \phi,\psi\rangle & \langle F(\vec{x}) \, \phi,\nabla \psi\rangle &\texttt{VecFctAtQP\_FOT}($F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
 \langle F(v) \cdot \nabla \phi,\psi\rangle & \langle F(v)\,\phi, \nabla \psi\rangle &\texttt{VectorFct\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\rightarrow\mathbb{R}^n$) \\
 \langle F(\nabla v) \cdot \nabla \phi,\psi\rangle & \langle F(\nabla v)\,\phi, \nabla \psi\rangle &\texttt{VectorGradient\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
-\langle F(\vec{x}) \cdot \nabla \phi,\psi\rangle & \langle F(\vec{x}) \, \phi,\nabla \psi\rangle &\texttt{VecFctAtQP\_FOT}($F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
 \langle F(v, \nabla w) \cdot \nabla \phi,\psi\rangle & \langle F(v, \nabla w) \, \phi,\nabla \psi\rangle &\texttt{VecGrad\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
-\langle F(\nabla v, \nabla w) \cdot \nabla \phi,\psi\rangle & \langle F(\nabla v, \nabla w) \, \phi,\nabla \psi\rangle &\texttt{FctGrad2\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
+\langle F(\nabla v, \nabla w) \cdot \nabla \phi,\psi\rangle & \langle F(\nabla v, \nabla w) \, \phi,\nabla \psi\rangle &\texttt{FctGrad2\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:(\mathbb{R}^n)^2\rightarrow\mathbb{R}^n$) \\
 %\langle F(v_1, v_2,\nabla v_3) \cdot \nabla \phi,\psi\rangle & \langle F(v_1, v_2,\nabla v_3) \, \phi,\nabla \psi\rangle &\texttt{Vec2Grad\_FOT\footnote[1]{* available on request}}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
 %\langle F(\vec{v}) \cdot \nabla \phi,\psi\rangle & \langle F(\vec{v}) \, \phi,\nabla \psi\rangle &\texttt{WorldVecFct\_FOT\footnotemark[1]}($\vec{v}\in${\scriptsize WorldVector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
-\langle F(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x}) \cdot \nabla \phi,\psi\rangle & \langle F(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x}) \, \phi,\nabla \psi\rangle &\texttt{General\_FOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, \\ & & $F:\mathbb{R}^n\times$\small{vector}$\langle\mathbb{R}\rangle\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^n$) \\
+\langle F(\vec{x}, \{v_i\}, \{\nabla w_j\}) \cdot \nabla \phi,\psi\rangle & \langle F(\vec{x}, \{v_i\}, \{\nabla w_j\}) \, \phi,\nabla \psi\rangle &\texttt{General\_FOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, \\ & & $F:\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^n$) \\
+\langle F(\vec{x}, \vec{n}, \{v_i\}, \{\nabla w_j\}) \cdot \nabla \phi,\psi\rangle & \langle F(\vec{x}, \vec{n}, \{v_i\}, \{\nabla w_j\}) \, \phi,\nabla \psi\rangle &\texttt{GeneralParametric\_FOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, \\ & & $F:\mathbb{R}^n\times\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^n$) \\
 \hline
 \end{longtable}
 \newpage
@@ -104,8 +105,8 @@
 \langle A(\nabla v) \nabla \phi,\nabla \psi\rangle & \texttt{MatrixGradient\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
 \langle A(v, \nabla v) \nabla \phi,\nabla \psi\rangle & \texttt{VecMatrixGradientAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
 \langle A(\nabla v, \vec{x}) \nabla \phi,\nabla \psi\rangle & \texttt{MatrixGradientAndCoords\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
-\langle A(\vec{x}, \{v_i\}_i, \{\nabla w_j\}_j) \nabla \phi,\nabla \psi\rangle & \texttt{General\_SOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $ A:\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
-\langle A(\vec{x}, \vec{n}, \{v_i\}_i, \{\nabla w_j\}_j) \nabla \phi,\nabla \psi\rangle & \texttt{GeneralParametric\_SOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $ A:\mathbb{R}^n\times\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
+\langle A(\vec{x}, \{v_i\}, \{\nabla w_j\}) \nabla \phi,\nabla \psi\rangle & \texttt{General\_SOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $ A:\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
+\langle A(\vec{x}, \vec{n}, \{v_i\}, \{\nabla w_j\}) \nabla \phi,\nabla \psi\rangle & \texttt{GeneralParametric\_SOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $ A:\mathbb{R}^n\times\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
 \hline
 \end{longtable}
 
@@ -118,8 +119,10 @@ $L_2$-Scalar product: $\langle\cdot,\cdot\rangle$, trialfunction: $\phi$, testfu
 \item $f, F, A$ can be implemented as \texttt{(*)AbstractFunction$\langle$ReturnType, InputType1, InputType2, ...$\rangle$}, where \texttt{(*)}$\in$\{$\emptyset$, \texttt{Binary}, \texttt{Tertiary}, \texttt{Quart}\} depending on the number of input arguments.
 \item The data-structure \texttt{DOFVector<*>} is always a pointer to a DOFVector.
 \item Optional arguments are depicted in square brackets $[*]$, where constants $c = 1$ by default, functions are \texttt{NULL}-pointers by default and are treated as identity functors or simple multiplication functors.
+\item Alternative argument of the form $\{b\in\mathbb{R}^n\,|\,i\in\mathbb{N}\}$ mean in the second case: $\vec{b}:=\vec{e}_i$.
+\item The expression $b^*\in\mathbb{R}^n$ means a pointer to a \texttt{WorldVector<double>}.
 \item The argument $div:=\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^n$ is only interesting for error estimators and optional. Is should implement the divergence of the matrix function in the operator.
 \item The argument \small{vec}$\langle*\rangle$ should be implemented as \texttt{std::vector$\langle*\rangle$}.
-\item In the last Second-Order-Operator \texttt{GeneralParametric\_SOT}, the second argument $\vec{n}$ to $A$ is the elementnormal, especially for surface meshes.
+\item In the last Second-Order-Operator \texttt{GeneralParametric\_*OT}, the second argument $\vec{n}$ to the functor is the elementnormal, especially for surface meshes.
 \end{itemize}
 \end{document}