cylindrical and time, coordinate

chart coordinates: $x^\tilde{\mu} = \{t, r, \phi, z\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{t}}, e_{\tilde{r}}, e_{\tilde{\phi}}, e_{\tilde{z}}\}$
embedding coordinates: $u^I = \{t, x, y, z\}$
embedding basis $e_I = \{e_{t}, e_{x}, e_{y}, e_{z}\}$
flat metric: ${{{ \eta} _I} _J} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{cccc} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$


transform from basis to coordinate:
${{{ \tilde{e}} _A} ^a} = {\overset{A\downarrow a\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$


transform from coorinate to basis:
${{{ \tilde{e}} ^a} _A} = {\overset{a\downarrow A\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$


tensor index associated with coordinate $t$ has operator $e_{t}(\zeta) = $$\frac{\partial \zeta}{\partial t}$
tensor index associated with coordinate $r$ has operator $e_{r}(\zeta) = $$\frac{\partial \zeta}{\partial r}$
tensor index associated with coordinate $\phi$ has operator $e_{\phi}(\zeta) = $$\frac{\partial \zeta}{\partial \phi}$
tensor index associated with coordinate $z$ has operator $e_{z}(\zeta) = $$\frac{\partial \zeta}{\partial z}$

chart in embedded coordinates:
${u} = {\overset{I\downarrow}{\left[\begin{matrix} t \\ {{r}} {{\cos\left( \phi\right)}} \\ {{r}} {{\sin\left( \phi\right)}} \\ z\end{matrix}\right]}}$


basis operators applied to chart:
${{{ e} _u} ^I} = {{{ u} ^I} _{,u}}$
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \cos\left( \phi\right)& \sin\left( \phi\right)& 0\\ 0& -{{{r}} {{\sin\left( \phi\right)}}}& {{r}} {{\cos\left( \phi\right)}}& 0\\ 0& 0& 0& 1\end{array}\right]}}$

${{{ e} ^u} _I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \cos\left( \phi\right)& \sin\left( \phi\right)& 0\\ 0& -{{\frac{1}{r}} {\sin\left( \phi\right)}}& {\frac{1}{r}} {\cos\left( \phi\right)}& 0\\ 0& 0& 0& 1\end{array}\right]}}$

${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
basis determinant: ${det(e)} = {r}$
${{{{ c} _a} _b} ^c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$

${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& 1\end{array}\right]}}$

${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
metric determinant: ${det(g)} = {-{{r}^{2}}}$
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}{-{{{{ g} _b} _c} _{,a}}} + {{{{ c} _a} _b} _c} + {{{{ c} _a} _c} _b}{-{{{{ c} _c} _b} _a}}}\right)}}}$
commutation coefficients: ${c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$

metric: ${g} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& 1\end{array}\right]}}$

metric inverse: ${g} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& 0& 1\end{array}\right]}}$

metric derivative: ${{\partial g}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$

1st kind Christoffel: ${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{r}& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& r& 0\\ 0& r& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$

connection coefficients / 2nd kind Christoffel: ${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{r}& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{matrix}\right]}}$

connection coefficients derivative: ${{\partial \Gamma}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{1}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

connection coefficients squared: ${{(\Gamma^2)}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{1}& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{1}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{1}{{r}^{2}}& 0& 0\\ 0& 0& -{1}& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

Riemann curvature, $\sharp\flat\flat\flat$: ${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

Riemann curvature, $\sharp\sharp\flat\flat$: ${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$

Ricci curvature, $\sharp\flat$: ${R} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$

Gaussian curvature: $0$
trace-free Ricci, $\sharp\flat$: ${{(R^{TF})}} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$

Einstein / trace-reversed Ricci curvature, $\sharp\flat$: ${G} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$

Schouten, $\sharp\flat$: ${P} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$

Weyl, $\sharp\sharp\flat\flat$: ${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

Weyl, $\flat\flat\flat\flat$: ${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$

Plebanski, $\sharp\sharp\flat\flat$: ${P} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _i} _j}} {{{ A} ^j}}}} = {{{{{A^{\hat{r}}}}} \cdot {{\frac{1}{r}}}} + {\frac{\partial {A^{\hat{\phi}}}}{\partial \phi}} + {\frac{\partial {A^{\hat{r}}}}{\partial r}} + {\frac{\partial {A^{\hat{t}}}}{\partial t}} + {\frac{\partial {A^{\hat{z}}}}{\partial z}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{\hat{t}} \\ \ddot{\hat{r}} \\ \ddot{\hat{\phi}} \\ \ddot{\hat{z}}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} 0 \\ {{r}} {{{\dot{\hat{\phi}}}^{2}}} \\ {{-2}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{r}}}} \cdot {{\frac{1}{r}}} \\ 0\end{matrix}\right]}}$

parallel propagators:

${{[\Gamma_t]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$

$\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

${ P} _t$ = $\exp\left( -{\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

${{ P} _t}^{-1}$ = $\exp\left({\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

${{[\Gamma_r]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& 0\end{array}\right]}$

$\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& 0\end{array}\right]}}d r$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \log\left( {{\frac{1}{{r_L}}} {{r_R}}}\right)& 0\\ 0& 0& 0& 0\end{array}\right]$

${ P} _r$ = $\exp\left( -{\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& 0\end{array}\right]}}d r}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]$

${{ P} _r}^{-1}$ = $\exp\left({\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& 0\end{array}\right]}}d r}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_L}}} {{r_R}}& 0\\ 0& 0& 0& 1\end{array}\right]$

${{[\Gamma_\phi]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$

$\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \phi$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {{r}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}& 0\\ 0& {\frac{1}{r}}{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

${ P} _{\phi}$ = $\exp\left( -{\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \phi}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]$

${{ P} _{\phi}}^{-1}$ = $\exp\left({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \phi}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{r}} {{\left({{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}} + {{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{-{{\phi_L}}} + {{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{\phi_R}}}\right)}}}{{{2}} {{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]$

${{[\Gamma_z]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$

$\int\limits_{{{z_L}}}^{{{z_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d z$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

${ P} _z$ = $\exp\left( -{\int\limits_{{{z_L}}}^{{{z_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d z}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

${{ P} _z}^{-1}$ = $\exp\left({\int\limits_{{{z_L}}}^{{{z_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d z}\right)$ = $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

propagator commutation:

[ ${ P} _t$ , ${ P} _r$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _t$ , ${ P} _{\phi}$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}} {{{r_L}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{{r_L}}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}} {{{r_L}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{{r_L}}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _t$ , ${ P} _z$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _r$ , ${ P} _{\phi}$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}} {{{r_L}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{{r_L}}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}} {{{r_R}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{{r_R}}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _r$ , ${ P} _z$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _{\phi}$ , ${ P} _z$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}} {{{r_L}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{{r_L}}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}} {{{r_L}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{{r_L}}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

propagator partials
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial z}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial z}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{\phi_R}}^{2}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{-{{\phi_L}}} + {{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{\phi_R}}}\right)}}}{{{2}} {{\left({{{{{{\phi_L}}^{2}}} {{{r}^{2}}}} + {{{{{\phi_R}}^{2}}} {{{r}^{2}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{r}^{2}}}}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial z}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\left({{{{\phi_L}}^{3}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{\phi_R}}^{3}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\left({{{{\phi_L}}^{3}}{-{{{\phi_R}}^{3}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}}}\right)}}}& \frac{{{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}}}& 0\\ 0& \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}} {{\left({{{\phi_L}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}{-{{\phi_R}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{r}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial z}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
volume element: ${{i}} {{r}}$
volume integral: ${{{\frac{1}{2}} {{{{\Delta (r^2)}}} \cdot {{\Delta t}} \cdot {{i}}}}} {{\Delta \phi}} \cdot {{\Delta z}}$
finite volume (0,0)-form:
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{\mathcal{V}(x_C)}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$

${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{{{\frac{1}{2}} {{{{\Delta (r^2)}}} \cdot {{\Delta t}} \cdot {{i}}}}} {{\Delta \phi}} \cdot {{\Delta z}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$

${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{{{\frac{1}{2}} {{{{\Delta (r^2)}}} \cdot {{\Delta t}} \cdot {{i}}}}} {{\Delta \phi}} \cdot {{\Delta z}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$

${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{{S(x_C)}}} \cdot {{\Delta t}}}}$