1 dimensions
metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {\rho_1} & 0 \\ 0 & {\rho_2}\end{matrix} \right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{1}{{\rho_1}} & 0 \\ 0 & \frac{1}{{\rho_2}}\end{matrix} \right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} \partial_ {{t}}\left( {\rho_1}\right) & \partial_ {{x}}\left( {\rho_1}\right) \\ 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ \partial_ {{t}}\left( {\rho_2}\right) & \partial_ {{x}}\left( {\rho_2}\right)\end{matrix} \right]}\end{matrix} \right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}} {\partial_ {{t}}\left( {\rho_1}\right)} & {\frac{1}{2}} {\partial_ {{x}}\left( {\rho_1}\right)} \\ {\frac{1}{2}} {\partial_ {{x}}\left( {\rho_1}\right)} & -{{\frac{1}{2}} {\partial_ {{t}}\left( {\rho_2}\right)}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{{\frac{1}{2}} {\partial_ {{x}}\left( {\rho_1}\right)}} & {\frac{1}{2}} {\partial_ {{t}}\left( {\rho_2}\right)} \\ {\frac{1}{2}} {\partial_ {{t}}\left( {\rho_2}\right)} & {\frac{1}{2}} {\partial_ {{x}}\left( {\rho_2}\right)}\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} \frac{\partial_ {{t}}\left( {\rho_1}\right)}{{{2}} {{{\rho_1}}}} & \frac{\partial_ {{x}}\left( {\rho_1}\right)}{{{2}} {{{\rho_1}}}} \\ \frac{\partial_ {{x}}\left( {\rho_1}\right)}{{{2}} {{{\rho_1}}}} & -{\frac{\partial_ {{t}}\left( {\rho_2}\right)}{{{2}} {{{\rho_1}}}}}\end{matrix} \right] \\ \left[ \begin{matrix} -{\frac{\partial_ {{x}}\left( {\rho_1}\right)}{{{2}} {{{\rho_2}}}}} & \frac{\partial_ {{t}}\left( {\rho_2}\right)}{{{2}} {{{\rho_2}}}} \\ \frac{\partial_ {{t}}\left( {\rho_2}\right)}{{{2}} {{{\rho_2}}}} & \frac{\partial_ {{x}}\left( {\rho_2}\right)}{{{2}} {{{\rho_2}}}}\end{matrix} \right]\end{matrix} \right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{-{{\partial_ {{t}}\left( {\rho_1}\right)}^{2}}} + {{{{\rho_1}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}} & \frac{{-{{{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}} \\ \frac{{-{{{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}} & \frac{{-{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}} + {{{{\rho_1}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{-{{{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}} & \frac{{-{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}} + {{{{\rho_1}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}} \\ \frac{{{{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}} - {{{{\rho_1}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}} & \frac{{{{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}} - {{{{\rho_1}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}}}\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}} & \frac{{{{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}} \\ \frac{{-{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}} + {{{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}} & \frac{{-{{{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{-{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}} + {{{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}} & \frac{{-{{{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}} \\ \frac{{-{{{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_{{{t}}{{x}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}} & \frac{{-{{\partial_ {{x}}\left( {\rho_2}\right)}^{2}}} + {{{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_2}\right)}}}}{{{2}} {{{{\rho_2}}^{2}}}}\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{-{{{{\rho_1}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} + {{{{\rho_2}}} \cdot {{{\partial_ {{t}}\left( {\rho_1}\right)}^{2}}}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} & \frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} \\ \frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} & \frac{{-{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} & \frac{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} \\ \frac{{-{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} & -{\frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}}}\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} -{\frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}}} & \frac{{-{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} + {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}} \\ \frac{{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}} & \frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{-{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} + {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}} & \frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}} \\ \frac{{{\left({{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}}} + {{{{\rho_2}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}}}}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}} & \frac{{-{{{{\rho_2}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{\rho_1}}} \cdot {{{\partial_ {{x}}\left( {\rho_2}\right)}^{2}}}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}}\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} \\ \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}}}{{{4}} {{{\rho_2}}} \cdot {{{{\rho_1}}^{2}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}} \\ \frac{{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}{{{4}} {{{\rho_1}}} \cdot {{{{\rho_2}}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & \frac{{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} \\ \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} \\ \frac{{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]\end{matrix} \right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} & 0 \\ 0 & \frac{{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}}\end{matrix} \right]}}$
Gaussian curvature:
${R} = {\frac{{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}{{{2}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}}}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{-{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} & 0 \\ 0 & \frac{{{-{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}}\end{matrix} \right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} \\ \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} \\ \frac{{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}{{{4}} {{{{\rho_1}}^{2}}} {{{{\rho_2}}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & \frac{{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}}{{{4}} {{{\rho_1}}} \cdot {{{\rho_2}}}} \\ \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}}}{{{4}} {{{\rho_1}}} \cdot {{{\rho_2}}}} & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & \frac{{{-{{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}}}{{{4}} {{{\rho_1}}} \cdot {{{\rho_2}}}} \\ \frac{{{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_2}\right)}} {{\partial_ {{x}}\left( {\rho_1}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_2}\right)}} {{\partial_ {{t}}\left( {\rho_1}\right)}}}}{{{4}} {{{\rho_1}}} \cdot {{{\rho_2}}}} & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]\end{matrix} \right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{2}} {R} \\ \frac{{{R}} {{\left({{{-{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} - {{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} + {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} + {{{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}}}\right)}}}{{{2}} {{\left({{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}}\right)}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{{R}} {{\left({{{-{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} + {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} + {{{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} - {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}}}\right)}}}{{{2}} {{\left({{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}}\right)}}} \\ {\frac{1}{2}} {R} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}} {R} & 0 \\ 0 & {\frac{1}{2}} {R}\end{matrix} \right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{R}} {{\left({{{-{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} + {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} + {{{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} - {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}}}\right)}}}{{{2}} {{\left({{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}}\right)}}} & 0 \\ 0 & \frac{{{R}} {{\left({{{-{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}}} - {{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}}} + {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}} + {{{{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}} - {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}}} - {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}}}\right)}}}{{{2}} {{\left({{{{{\rho_1}}} \cdot {{{\partial_ {{t}}\left( {\rho_2}\right)}^{2}}}} + {{{{{{\rho_1}}} \cdot {{\partial_ {{x}}\left( {\rho_1}\right)}} {{\partial_ {{x}}\left( {\rho_2}\right)}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{x}}{{x}}}\left( {\rho_1}\right)}}}} - {{{2}} {{{\rho_1}}} \cdot {{{\rho_2}}} \cdot {{\partial_{{{t}}{{t}}}\left( {\rho_2}\right)}}}} + {{{{\rho_2}}} \cdot {{{\partial_ {{x}}\left( {\rho_1}\right)}^{2}}}} + {{{{\rho_2}}} \cdot {{\partial_ {{t}}\left( {\rho_1}\right)}} {{\partial_ {{t}}\left( {\rho_2}\right)}}}}\right)}}}\end{matrix} \right]}}$