cylindrical, anholonomic, orthonormal
chart coordinates: $x^\tilde{\mu} = \{r, \theta, z\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{r}}, e_{\tilde{\theta}}, e_{\tilde{z}}\}$
embedding coordinates: $u^I = \{x, y, z\}$
embedding basis $e_I = \{e_{x}, e_{y}, e_{z}\}$
flat metric:
${{{ \eta} _I} _J} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$
transform from basis to coordinate:
${{{ \tilde{e}} _A} ^a} = {\overset{a\downarrow A\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& r& 0\\ 0& 0& 1\end{array}\right]}}$
transform from coorinate to basis:
${{{ \tilde{e}} ^a} _A} = {\overset{a\downarrow A\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& \frac{1}{r}& 0\\ 0& 0& 1\end{array}\right]}}$
tensor index associated with coordinate $r$ has operator $e_{r}(\zeta) = $$\frac{\partial \zeta}{\partial r}$
tensor index associated with coordinate $\theta$ has operator $e_{\theta}(\zeta) = $${\frac{1}{r}} {\frac{\partial \zeta}{\partial \theta}}$
tensor index associated with coordinate $z$ has operator $e_{z}(\zeta) = $$\frac{\partial \zeta}{\partial z}$
chart in embedded coordinates:
${u} = {\overset{I\downarrow}{\left[\begin{matrix} {{r}} {{\cos\left( \theta\right)}} \\ {{r}} {{\sin\left( \theta\right)}} \\ z\end{matrix}\right]}}$
basis operators applied to chart:
${{{ e} _u} ^I} = {{{ u} ^I} _{,u}}$
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{ccc} \cos\left( \theta\right)& \sin\left( \theta\right)& 0\\ -{\sin\left( \theta\right)}& \cos\left( \theta\right)& 0\\ 0& 0& 1\end{array}\right]}}$
${{{ e} ^u} _I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{ccc} \cos\left( \theta\right)& \sin\left( \theta\right)& 0\\ -{\sin\left( \theta\right)}& \cos\left( \theta\right)& 0\\ 0& 0& 1\end{array}\right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$
basis determinant: ${det(e)} = {1}$
${{{{ c} _a} _b} ^c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& -{\frac{1}{r}}& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& \frac{1}{r}& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
metric determinant: ${det(g)} = {1}$
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}{-{{{{ g} _b} _c} _{,a}}} + {{{{ c} _a} _b} _c} + {{{{ c} _a} _c} _b}{-{{{{ c} _c} _b} _a}}}\right)}}}$
commutation coefficients:
${c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& -{\frac{1}{r}}& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& \frac{1}{r}& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
metric:
${g} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$
metric inverse:
${g} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$
metric derivative:
${{\partial g}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& -{\frac{1}{r}}& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ \frac{1}{r}& 0& 0\\ 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{ccc} 0& 0& 0\\ 0& -{\frac{1}{r}}& 0\\ 0& 0& 0\end{array}\right] \\ \left[\begin{array}{ccc} 0& 0& 0\\ \frac{1}{r}& 0& 0\\ 0& 0& 0\end{array}\right] \\ \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\end{matrix}\right]}}$
connection coefficients derivative:
${{\partial \Gamma}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{ccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ \frac{1}{{r}^{2}}& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{array}\right]}}$
connection coefficients squared:
${{(\Gamma^2)}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{ccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{ccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{ccc} \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\\ \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\\ \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\end{array}\right]}}$
Ricci curvature, $\sharp\flat$:
${R} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}$
Gaussian curvature:
$0$
trace-free Ricci, $\sharp\flat$:
${{(R^{TF})}} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${G} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}$
Schouten, $\sharp\flat$:
${P} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{ccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Weyl, $\flat\flat\flat\flat$:
${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{ccc} \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\\ \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\\ \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]& \left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\end{array}\right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${P} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{ccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\end{array}\right]}}$
divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _i} _j}} {{{ A} ^j}}}} = {{{{{A^{\hat{r}}}}} \cdot {{\frac{1}{r}}}} + {{{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}} {{\frac{1}{r}}}} + {\frac{\partial {A^{\hat{r}}}}{\partial r}} + {\frac{\partial {A^{\hat{z}}}}{\partial z}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{\hat{r}} \\ \ddot{\hat{\theta}} \\ \ddot{\hat{z}}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{{\dot{\hat{\theta}}}^{2}}} {{\frac{1}{r}}} \\ {{-1}} {{\dot{\hat{\theta}}}} \cdot {{\dot{\hat{r}}}} \cdot {{\frac{1}{r}}} \\ 0\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_r]}} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}d r$
= $\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
${ P} _r$ = $\exp\left( -{\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}d r}\right)$
= $\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
${{ P} _r}^{-1}$ = $\exp\left({\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}d r}\right)$
= $\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
${{[\Gamma_\theta]}} = {\left[\begin{array}{ccc} 0& -{1}& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]}$
$\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{ccc} 0& -{1}& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]}}d \theta$
= $\left[\begin{array}{ccc} 0& {{\theta_L}}{-{{\theta_R}}}& 0\\ {-{{\theta_L}}} + {{\theta_R}}& 0& 0\\ 0& 0& 0\end{array}\right]$
${ P} _{\theta}$ = $\exp\left( -{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{ccc} 0& -{1}& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]}}d \theta}\right)$
= $\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]$
${{ P} _{\theta}}^{-1}$ = $\exp\left({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{ccc} 0& -{1}& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]}}d \theta}\right)$
= $\left[\begin{array}{ccc} \frac{{-{{\theta_L}}} + {{\theta_R}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{-{{\theta_L}}} + {{\theta_R}} + {{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}{-{{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{1}{-{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]$
${{[\Gamma_z]}} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
$\int\limits_{{{z_L}}}^{{{z_R}}}{{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}d z$
= $\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
${ P} _z$ = $\exp\left( -{\int\limits_{{{z_L}}}^{{{z_R}}}{{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}d z}\right)$
= $\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
${{ P} _z}^{-1}$ = $\exp\left({\int\limits_{{{z_L}}}^{{{z_R}}}{{\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}}d z}\right)$
= $\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
propagator commutation:
[ ${ P} _r$ , ${ P} _{\theta}$ ] = ${{{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
[ ${ P} _r$ , ${ P} _z$ ] = ${{{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
[ ${ P} _{\theta}$ , ${ P} _z$ ] = ${{{\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
propagator partials
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \theta}}\left({\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial z}}\left({\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \theta}}\left({\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial z}}\left({\left[\begin{array}{ccc} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\left({{-{{\theta_L}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{\theta_R}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}\right)}}}{{{2}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}\right)}}}& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \theta}}\left({\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial z}}\left({\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{ccc} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}$
volume element: $r$
volume integral: ${{{\frac{1}{2}} {{\Delta (r^2)}}}} {{\Delta \theta}} \cdot {{\Delta z}}$
finite volume (0,0)-form:
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{\mathcal{V}(x_C)}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{{{\frac{1}{2}} {{\Delta (r^2)}}}} {{\Delta \theta}} \cdot {{\Delta z}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{{{\frac{1}{2}} {{\Delta (r^2)}}}} {{\Delta \theta}} \cdot {{\Delta z}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{{S(x_C)}}} \cdot {{\Delta t}}}}$