Symbolic Lua Output

gauge vars
${Q} = {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{ K}_i}_l}}}$
primitive $\partial_t$ defs
${{ \alpha}_{,t}} = {{ {-{\alpha}} {{Q}}} + {{{{ \alpha}_{,i}}} {{{ \beta}^i}}}}$
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \alpha}_{,i}}} {{{ \beta}^i}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{{{ \gamma}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{ K}_i}_j}_{,t}} = {{-{{{ \alpha}_{,i}}_{,j}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{ \alpha}_{,k}}}} + {{{\alpha}} \cdot {{\left({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma}_i}_j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S}_i}_j}}}}\right)}}}}$
lapse vars
${{ f}_{,k}} = {{{f'}} \cdot {{\alpha}} \cdot {{{ a}_k}}}$
hyperbolic state variables
${{ a}_k} = {{\left( \log\left( \alpha\right)\right)}_{,k}}$
${{ a}_k} = {\frac{{ \alpha}_{,k}}{\alpha}}$
${{ \alpha}_{,k}} = {{{\alpha}} \cdot {{{ a}_k}}}$
${{{{ d}_k}_i}_j} = {{{\frac{1}{2}}} {{{{{ \gamma}_i}_j}_{,k}}}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{2}} {{{{{ d}_k}_i}_j}}}$
connections wrt aux vars
${{{{ \Gamma}^k}_i}_j} = {{{\frac{1}{2}}} {{{{ \gamma}^k}^l}} {{\left({{{{{{ \gamma}_l}_i}_{,j}} + {{{{ \gamma}_l}_j}_{,i}}} - {{{{ \gamma}_i}_j}_{,l}}}\right)}}}$
${{{{ \Gamma}_i}_j}_k} = {{{\frac{1}{2}}} {{\left({{{{{{ \gamma}_i}_j}_{,k}} + {{{{ \gamma}_i}_k}_{,j}}} - {{{{ \gamma}_j}_k}_{,i}}}\right)}}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{{{ \Gamma}_i}_j}_k} + {{{{ \Gamma}_j}_i}_k}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{{{{ \gamma}_i}_a}} {{{{{ \Gamma}^a}_j}_k}}} + {{{{{ \gamma}_j}_a}} {{{{{ \Gamma}^a}_i}_k}}}}$
${{{{ \Gamma}_i}_j}_k} = {-{\left({{{{{{ d}_i}_j}_k} - {{{{ d}_j}_i}_k}} - {{{{ d}_k}_i}_j}}\right)}}$
${{{{ \Gamma}^i}_j}_k} = {{{{{ \gamma}^i}^l}} {{\left({{{{{ d}_j}_l}_k} + {{{{{ d}_k}_l}_j} - {{{{ d}_l}_j}_k}}}\right)}}}$
${\gamma^{ij}}_{,k}$ wrt aux vars
${{{{ \gamma}^i}^j}_{,k}} = { {-{{{ \gamma}^i}^l}} {{{{{ \gamma}_l}_m}_{,k}}} {{{{ \gamma}^m}^j}}}$
${{{{ \gamma}^i}^j}_{,k}} = {-{{{2}} {{{{ \gamma}^i}^l}} {{{{ \gamma}^m}^j}} {{{{{ d}_k}_l}_m}}}}$
Ricci wrt aux vars
${{{ R}_i}_j} = {{{{{{{{ \Gamma}^k}_i}_j}_{,k}} - {{{{{ \Gamma}^k}_i}_k}_{,j}}} + {{{{{{ \Gamma}^k}_l}_k}} {{{{{ \Gamma}^l}_i}_j}}}} - {{{{{{ \Gamma}^k}_l}_j}} {{{{{ \Gamma}^l}_i}_k}}}}$
${{{ R}_i}_j} = {{{{{\left( {{{{ \gamma}^k}^a}} {{\left({{{{{ d}_i}_a}_j} + {{{{{ d}_j}_a}_i} - {{{{ d}_a}_i}_j}}}\right)}}\right)}_{,k}} - {{\left( {{{{ \gamma}^k}^a}} {{\left({{{{{ d}_i}_a}_k} + {{{{{ d}_k}_a}_i} - {{{{ d}_a}_i}_k}}}\right)}}\right)}_{,j}}} + {{{{{{{ \gamma}^k}^a}} {{\left({{{{{ d}_l}_a}_k} + {{{{{ d}_k}_a}_l} - {{{{ d}_a}_l}_k}}}\right)}}}} {{{{{{ \gamma}^l}^a}} {{\left({{{{{ d}_i}_a}_j} + {{{{{ d}_j}_a}_i} - {{{{ d}_a}_i}_j}}}\right)}}}}}} - {{{{{{{ \gamma}^k}^a}} {{\left({{{{{ d}_l}_a}_j} + {{{{{ d}_j}_a}_l} - {{{{ d}_a}_l}_j}}}\right)}}}} {{{{{{ \gamma}^l}^a}} {{\left({{{{{ d}_i}_a}_k} + {{{{{ d}_k}_a}_i} - {{{{ d}_a}_i}_k}}}\right)}}}}}}$
${{{ R}_i}_j} = {{{{{{{ \gamma}^k}^a}_{,k}}} {{{{{ d}_i}_a}_j}}} + {{{{{{{ \gamma}^k}^a}_{,k}}} {{{{{ d}_j}_a}_i}}} - {{{{{{ \gamma}^k}^a}_{,k}}} {{{{{ d}_a}_i}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_j}_{,k}}}} + {{{{{{{{ \gamma}^k}^a}} {{{{{{ d}_j}_a}_i}_{,k}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_j}_{,k}}}}} - {{{{{{ \gamma}^k}^a}_{,j}}} {{{{{ d}_i}_a}_k}}}} - {{{{{{ \gamma}^k}^a}_{,j}}} {{{{{ d}_k}_a}_i}}}} + {{{{{{{{ \gamma}^k}^a}_{,j}}} {{{{{ d}_a}_i}_k}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_k}_{,j}}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_k}_a}_i}_{,j}}}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_k}_{,j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_l}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_i}_a}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_l}_a}_k}}} + {{{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_j}_a}_i}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_l}_a}_k}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_k}_a}_l}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_l}_k}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_l}_a}_j}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_k}_a}_i}} {{{{{ d}_l}_a}_j}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_l}_a}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_a}_l}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_l}} {{{{{ d}_k}_a}_i}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_a}_l}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_i}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_k}_a}_i}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_a}_i}_k}}}}}$
${{{ R}_i}_j} = {{{-{{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_i}_a}_j}} {{{{{ d}_k}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_j}_a}_i}} {{{{{ d}_k}_b}_c}}}} + {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_a}_i}_j}} {{{{{ d}_k}_b}_c}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_j}_{,k}}}} + {{{{{{ \gamma}^k}^a}} {{{{{{ d}_j}_a}_i}_{,k}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_j}_{,k}}}}} + {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_b}_c}}} + {{{{{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_j}_b}_c}} {{{{{ d}_k}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_b}_c}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_k}_{,j}}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_k}_a}_i}_{,j}}}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_k}_{,j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_l}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_i}_a}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_l}_a}_k}}} + {{{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_j}_a}_i}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_l}_a}_k}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_k}_a}_l}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_l}_k}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_l}_a}_j}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_k}_a}_i}} {{{{{ d}_l}_a}_j}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_l}_a}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_a}_l}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_l}} {{{{{ d}_k}_a}_i}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_a}_l}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_i}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_k}_a}_i}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_a}_i}_k}}}}}$
symmetrizing
${{{ R}_i}_j} = {{{-{{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_b}_c}_k}} {{{{{ d}_i}_a}_j}}}} - {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_b}_c}_k}} {{{{{ d}_j}_a}_i}}}} + {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_a}_i}_j}} {{{{{ d}_b}_c}_k}}} + {{{{{ \gamma}^a}^k}} {{{{{{ d}_i}_a}_j}_{,k}}}} + {{{{{{ \gamma}^a}^k}} {{{{{{ d}_j}_a}_i}_{,k}}}} - {{{{{ \gamma}^a}^k}} {{{{{{ d}_a}_i}_j}_{,k}}}}} + {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_b}_c}}} + {{{{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_j}_b}_c}} {{{{{ d}_k}_a}_i}}} - {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_b}_c}}}} - {{{{{ \gamma}^a}^k}} {{{{{{ d}_i}_a}_k}_{,j}}}}} + {{{{{ \gamma}^a}^l}} {{{{ \gamma}^a}^k}} {{{{{ d}_a}_k}_l}} {{{{{ d}_i}_a}_j}}} + {{{{{{{ \gamma}^a}^k}} {{{{ \gamma}^a}^l}} {{{{{ d}_a}_k}_l}} {{{{{ d}_j}_a}_i}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^a}^l}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_k}_l}}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^a}^l}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_a}_l}}}}}$
${{{ R}_i}_j} = {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}$
${{{ R}_i}_j} = {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}$
${{{ R}_i}_j} = {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}$
${{{ R}_i}_j} = {{-{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} - {{{2}} {{{{{ d}^a}_b}_i}} {{{{{ d}_j}_a}^b}}}} + {{{{{{{{ d}^a}_i}_j}} {{{{{ d}^b}_a}_b}}} - {{{{{{ d}^b}_a}_b}} {{{{{ d}_i}^a}_j}}}} - {{{{{{ d}^b}_a}_b}} {{{{{ d}_j}^a}_i}}}} + {{{2}} {{{{{ d}_a}^b}_i}} {{{{{ d}_j}^a}_b}}} + {{{{{{ d}_i}^a}_b}} {{{{{ d}_j}_a}^b}}}}$
${{{ R}_i}_j} = {{-{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} - {{{2}} {{{{{ d}^a}_b}_i}} {{{{{ d}_j}_a}^b}}}} + {{{{{{ e}_a}} {{{{{ d}^a}_i}_j}}} - {{{{ e}_a}} {{{{{ d}_i}^a}_j}}}} - {{{{ e}_a}} {{{{{ d}_j}^a}_i}}}} + {{{2}} {{{{{ d}_a}^b}_i}} {{{{{ d}_j}^a}_b}}} + {{{{{{ d}_i}^a}_b}} {{{{{ d}_j}_a}^b}}}}$
${{{ R}_i}_j} = {{-{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {{{{{{ e}_a}} {{{{{ d}^a}_i}_j}}} - {{{{ e}_a}} {{{{{ d}_i}^a}_j}}}} - {{{{ e}_a}} {{{{{ d}_j}^a}_i}}}} + {{{{{{ d}_i}^a}_b}} {{{{{ d}_j}_a}^b}}}}$
${{{ R}_i}_j} = {{{{{ \gamma}^a}^b}} {{\left({{{{{{{{ \gamma}^c}^d}} {{{{{ d}_a}_b}_d}} {{{{{ d}_c}_i}_j}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_b}_d}} {{{{{ d}_i}_c}_j}}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_b}_d}} {{{{{ d}_j}_c}_i}}}} + {{{{{ d}_a}_b}_i}_{,j}} + {{{{{{{ d}_a}_b}_j}_{,i}} - {{{{{ d}_a}_i}_j}_{,b}}} - {{{{{ d}_i}_a}_b}_{,j}}} + {{{{{{ d}_i}_a}_c}} {{{{{ d}_j}_b}^c}}}}\right)}}}$
time derivative of $\alpha_{,t}$
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}}$
time derivative of $\gamma_{ij,t}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{{{ \gamma}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{ \beta}^k}} {{{{2}} {{{{{ d}_k}_i}_j}}}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
time derivative of $a_{k,t}$
${{{ a}_k}_{,t}} = {-{\frac{{{{{ \alpha}_{,k}}} {{{ \alpha}_{,t}}}} - {{{\alpha}} \cdot {{{{ \alpha}_{,k}}_{,t}}}}}{{\alpha}^{2}}}}$
${{{ a}_k}_{,t}} = {-{\frac{{{{{ \alpha}_{,k}}} {{\left({{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}}\right)}}} - {{{\alpha}} \cdot {{{\left( {-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}\right)}_{,k}}}}}{{\alpha}^{2}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{ f}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{ \alpha}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{ f}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{ \alpha}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{{f'}} \cdot {{\alpha}} \cdot {{{ a}_k}}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{{\alpha}} \cdot {{{ a}_k}}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
time derivative of $d_{kij,t}$
${{{{{ d}_k}_i}_j}_{,t}} = {\frac{{{{{ \gamma}_i}_j}_{,k}}_{,t}}{2}}$
${{{{{ d}_k}_i}_j}_{,t}} = {\frac{{\left( {{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{ \beta}^l}} {{{{2}} {{{{{ d}_l}_i}_j}}}}} + {{{{{ \gamma}_l}_j}} {{{{ \beta}^l}_{,i}}}} + {{{{{ \gamma}_i}_l}} {{{{ \beta}^l}_{,j}}}}\right)}_{,k}}{2}}$
${{{{{ d}_k}_i}_j}_{,t}} = {-{\frac{{{{{{{{{{2}} {{{ \alpha}_{,k}}} {{{{ K}_i}_j}}} - {{{{{ \beta}^l}_{,i}}} {{{{{ \gamma}_l}_j}_{,k}}}}} - {{{{{ \beta}^l}_{,j}}} {{{{{ \gamma}_i}_l}_{,k}}}}} - {{{2}} {{{{ \beta}^l}_{,k}}} {{{{{ d}_l}_i}_j}}}} - {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}}} - {{{{{ \gamma}_i}_l}} {{{{{ \beta}^l}_{,j}}_{,k}}}}} - {{{{{ \gamma}_l}_j}} {{{{{ \beta}^l}_{,i}}_{,k}}}}} + {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}{2}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {\frac{{-{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}} + {{{2}} {{{{ \beta}^l}_{,i}}} {{{{{ d}_k}_l}_j}}} + {{{2}} {{{{ \beta}^l}_{,j}}} {{{{{ d}_k}_i}_l}}} + {{{2}} {{{{ \beta}^l}_{,k}}} {{{{{ d}_l}_i}_j}}} + {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}} + {{{{{ \gamma}_i}_l}} {{{{{ \beta}^l}_{,j}}_{,k}}}} + {{{{{{ \gamma}_l}_j}} {{{{{ \beta}^l}_{,i}}_{,k}}}} - {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}}{2}}$
$K_{ij,t}$ with hyperbolic terms
${{{{ K}_i}_j}_{,t}} = {{-{{{ \alpha}_{,i}}_{,j}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{ \alpha}_{,k}}}} + {{{\alpha}} \cdot {{\left({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma}_i}_j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S}_i}_j}}}}\right)}}}}$
${{{{ K}_i}_j}_{,t}} = {{-{{{\frac{1}{2}}} {{\left({{{\left( {{\alpha}} \cdot {{{ a}_i}}\right)}_{,j}} + {{\left( {{\alpha}} \cdot {{{ a}_j}}\right)}_{,i}}}\right)}}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{{\alpha}} \cdot {{{ a}_k}}}}} + {{{\alpha}} \cdot {{\left({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma}_i}_j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S}_i}_j}}}}\right)}}}}$
${{{{ K}_i}_j}_{,t}} = {\frac{{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{ \Gamma}^k}_i}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{{ \alpha}_{,j}}} {{{ a}_i}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{{ \alpha}_{,i}}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}{2}}$
${{{{ K}_i}_j}_{,t}} = {\frac{{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{ \Gamma}^k}_i}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}{2}}$
${{{{ K}_i}_j}_{,t}} = {\frac{{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{{ \gamma}^k}^l}} {{\left({{{{{ d}_i}_l}_j} + {{{{{ d}_j}_l}_i} - {{{{ d}_l}_i}_j}}}\right)}}}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}{2}}$
${{{{ K}_i}_j}_{,t}} = {\frac{{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{{ \gamma}^k}^l}} {{\left({{{{{ d}_i}_l}_j} + {{{{{ d}_j}_l}_i} - {{{{ d}_l}_i}_j}}}\right)}}}}} + {{{2}} {{\alpha}} \cdot {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}{2}}$
${{{{ K}_i}_j}_{,t}} = {\frac{{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ \gamma}^k}^l}} {{{{{ d}_i}_l}_j}}} + {{{{{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ \gamma}^k}^l}} {{{{{ d}_j}_l}_i}}} - {{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ \gamma}^k}^l}} {{{{{ d}_l}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_b}_c}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}}} + {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} + {{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_b}_a}_i}_{,j}}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_b}_a}_j}_{,i}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}{2}}$
${{{{ K}_i}_j}_{,t}} = {\frac{{{{2}} {{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_i}_b}_j}}} + {{{{{{{2}} {{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_j}_b}_i}}} - {{{2}} {{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_b}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_b}_c}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}}} + {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} + {{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_a}_b}} {{{{ K}_i}_j}} {{{{ \gamma}^a}^b}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_a}} {{{{ K}_j}_b}} {{{{ \gamma}^a}^b}}}} + {{{2}} {{{ \beta}^a}} {{{{{ K}_i}_j}_{,a}}}} + {{{2}} {{{{ K}_a}_i}} {{{{ \beta}^a}_{,j}}}} + {{{2}} {{{{ K}_a}_j}} {{{{ \beta}^a}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}{2}}$
partial derivatives
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{ \beta}^k}} {{{{2}} {{{{{ d}_k}_i}_j}}}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {\frac{{-{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}} + {{{2}} {{{{ \beta}^l}_{,i}}} {{{{{ d}_k}_l}_j}}} + {{{2}} {{{{ \beta}^l}_{,j}}} {{{{{ d}_k}_i}_l}}} + {{{2}} {{{{ \beta}^l}_{,k}}} {{{{{ d}_l}_i}_j}}} + {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}} + {{{{{ \gamma}_i}_l}} {{{{{ \beta}^l}_{,j}}_{,k}}}} + {{{{{{ \gamma}_l}_j}} {{{{{ \beta}^l}_{,i}}_{,k}}}} - {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}}{2}}$
${{{{ K}_i}_j}_{,t}} = {{{{{{ {-{\frac{1}{2}}} {{\alpha}} \cdot {{{{ a}_i}_{,j}}}} - {{{\frac{1}{2}}} {{\alpha}} \cdot {{{{ a}_j}_{,i}}}}} + {{{\alpha}} \cdot {{\left({{{{\frac{1}{2}}} {{{{ \gamma}^p}^q}} {{\left({{{{{{ d}_i}_p}_q}_{,j}} + {{{{{ d}_j}_p}_q}_{,i}}}\right)}}} - {{{\frac{1}{2}}} {{{{ \gamma}^m}^r}} {{\left({{{{{{ d}_m}_i}_j}_{,r}} + {{{{{ d}_m}_j}_i}_{,r}}}\right)}}}}\right)}}}} - {{{\alpha}} \cdot {{{{ V}_j}_{,i}}}}} - {{{\alpha}} \cdot {{{{ V}_i}_{,j}}}}} + {{{\alpha}} \cdot {{\left({{{{{{ {-{{ a}_i}} {{{ a}_j}}} + {{{\left({{{{{{ d}_j}_i}^k} + {{{{ d}_i}_j}^k}} - {{{{ d}^k}_i}_j}}\right)}} {{\left({{{{ a}_k} + {{ V}_k}} - {{{{ d}^l}_l}_k}}\right)}}} + {{{2}} {{\left({{{{{ d}^k}^l}_j} - {{{{ d}^l}^k}_j}}\right)}} {{{{{ d}_k}_l}_i}}} + {{{2}} {{{{{ d}_i}^k}^l}} {{{{{ d}_k}_l}_j}}} + {{{2}} {{{{{ d}_j}^k}^l}} {{{{{ d}_k}_l}_i}}}} - {{{3}} {{{{{ d}_i}^k}^l}} {{{{{ d}_j}_k}_l}}}} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{{{ \gamma}^k}^l}} {{{{ K}_i}_l}} {{{{ K}_k}_j}}}} - {{{{{ \gamma}^k}^l}} {{{{ K}_j}_l}} {{{{ K}_k}_i}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ K}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{ V}_k}_{,t}} = {0}$
neglecting shift
${{ \alpha}_{,t}} = {{{-1}} {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}$
${{{ a}_k}_{,t}} = {{-{{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} + {-{{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}} + {-{{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{-{{{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}} + {-{{{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}}}$
${{{{ K}_i}_j}_{,t}} = {{{{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} + {-{{{\alpha}} \cdot {{{{ K}_i}_l}} {{{{ K}_k}_j}} {{{{ \gamma}^k}^l}}}} + {-{{{\alpha}} \cdot {{{{ K}_j}_l}} {{{{ K}_k}_i}} {{{{ \gamma}^k}^l}}}} + {-{{{\alpha}} \cdot {{{{ V}_i}_{,j}}}}} + {-{{{\alpha}} \cdot {{{{ V}_j}_{,i}}}}} + {-{{{\alpha}} \cdot {{{ V}_k}} {{{{{ d}^k}_i}_j}}}} + {{{\alpha}} \cdot {{{ V}_k}} {{{{{ d}_i}_j}^k}}} + {{{\alpha}} \cdot {{{ V}_k}} {{{{{ d}_j}_i}^k}}} + {-{\frac{{{\alpha}} \cdot {{{{ \gamma}^m}^r}} {{{{{{ d}_m}_i}_j}_{,r}}}}{2}}} + {-{\frac{{{\alpha}} \cdot {{{{ \gamma}^m}^r}} {{{{{{ d}_m}_j}_i}_{,r}}}}{2}}} + {\frac{{{\alpha}} \cdot {{{{ \gamma}^p}^q}} {{{{{{ d}_i}_p}_q}_{,j}}}}{2}} + {\frac{{{\alpha}} \cdot {{{{ \gamma}^p}^q}} {{{{{{ d}_j}_p}_q}_{,i}}}}{2}} + {-{\frac{{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}{2}}} + {-{{{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} + {-{\frac{{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}{2}}} + {-{{{\alpha}} \cdot {{{ a}_k}} {{{{{ d}^k}_i}_j}}}} + {{{\alpha}} \cdot {{{ a}_k}} {{{{{ d}_i}_j}^k}}} + {{{\alpha}} \cdot {{{ a}_k}} {{{{{ d}_j}_i}^k}}} + {{{\alpha}} \cdot {{{{{ d}^k}_i}_j}} {{{{{ d}^l}_l}_k}}} + {-{{{3}} {{\alpha}} \cdot {{{{{ d}_i}^k}^l}} {{{{{ d}_j}_k}_l}}}} + {{{2}} {{\alpha}} \cdot {{{{{ d}_i}^k}^l}} {{{{{ d}_k}_l}_j}}} + {-{{{\alpha}} \cdot {{{{{ d}_i}_j}^k}} {{{{{ d}^l}_l}_k}}}} + {{{2}} {{\alpha}} \cdot {{{{{ d}_j}^k}^l}} {{{{{ d}_k}_l}_i}}} + {-{{{\alpha}} \cdot {{{{{ d}_j}_i}^k}} {{{{{ d}^l}_l}_k}}}} + {{{2}} {{\alpha}} \cdot {{{{{ d}^k}^l}_j}} {{{{{ d}_k}_l}_i}}} + {-{{{2}} {{\alpha}} \cdot {{{{{ d}_k}_l}_i}} {{{{{ d}^l}^k}_j}}}}}$
${{{ V}_k}_{,t}} = {0}$
neglecting source terms
${{ \alpha}_{,t}} = {0}$
${{{ a}_k}_{,t}} = {{{-1}} {{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {0}$
${{{{{ d}_k}_i}_j}_{,t}} = {{{-1}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}$
${{{{ K}_i}_j}_{,t}} = {{-{{{\alpha}} \cdot {{{{ V}_i}_{,j}}}}} + {-{{{\alpha}} \cdot {{{{ V}_j}_{,i}}}}} + {-{\frac{{{\alpha}} \cdot {{{{ \gamma}^m}^r}} {{{{{{ d}_m}_i}_j}_{,r}}}}{2}}} + {-{\frac{{{\alpha}} \cdot {{{{ \gamma}^m}^r}} {{{{{{ d}_m}_j}_i}_{,r}}}}{2}}} + {\frac{{{\alpha}} \cdot {{{{ \gamma}^p}^q}} {{{{{{ d}_i}_p}_q}_{,j}}}}{2}} + {\frac{{{\alpha}} \cdot {{{{ \gamma}^p}^q}} {{{{{{ d}_j}_p}_q}_{,i}}}}{2}} + {-{\frac{{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}{2}}} + {-{\frac{{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}{2}}}}$
${{{ V}_k}_{,t}} = {0}$
...and those source terms are...
${ \alpha}_{,t}$$ + \dots = $$-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}$
${{ a}_k}_{,t}$$ + \dots = $${{\alpha}} \cdot {{{{ K}_i}_l}} {{\left({{{{{2}} {{f}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{f}} {{{ a}_k}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f'}} \cdot {{{ a}_k}} {{{{ \gamma}^i}^l}}}}\right)}}$
${{{ \gamma}_i}_j}_{,t}$$ + \dots = $$-{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}}}$
${{{{ d}_k}_i}_j}_{,t}$$ + \dots = $$-{{{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}$
${{{ K}_i}_j}_{,t}$$ + \dots = $${{\alpha}} \cdot {{\left({{{{{{{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{{{ K}_i}_l}} {{{{ K}_k}_j}} {{{{ \gamma}^k}^l}}}} - {{{{{ K}_j}_l}} {{{{ K}_k}_i}} {{{{ \gamma}^k}^l}}}} - {{{{ V}_k}} {{{{{ d}^k}_i}_j}}}} + {{{{ V}_k}} {{{{{ d}_i}_j}^k}}} + {{{{{{ V}_k}} {{{{{ d}_j}_i}^k}}} - {{{{ a}_i}} {{{ a}_j}}}} - {{{{ a}_k}} {{{{{ d}^k}_i}_j}}}} + {{{{ a}_k}} {{{{{ d}_i}_j}^k}}} + {{{{ a}_k}} {{{{{ d}_j}_i}^k}}} + {{{{{{{ d}^k}_i}_j}} {{{{{ d}^l}_l}_k}}} - {{{3}} {{{{{ d}_i}^k}^l}} {{{{{ d}_j}_k}_l}}}} + {{{{2}} {{{{{ d}_i}^k}^l}} {{{{{ d}_k}_l}_j}}} - {{{{{{ d}_i}_j}^k}} {{{{{ d}^l}_l}_k}}}} + {{{{2}} {{{{{ d}_j}^k}^l}} {{{{{ d}_k}_l}_i}}} - {{{{{{ d}_j}_i}^k}} {{{{{ d}^l}_l}_k}}}} + {{{{2}} {{{{{ d}^k}^l}_j}} {{{{{ d}_k}_l}_i}}} - {{{2}} {{{{{ d}_k}_l}_i}} {{{{{ d}^l}^k}_j}}}}}\right)}}$
${{ V}_k}_{,t}$$ + \dots = $$0$
spelled out
${\frac{d \alpha}{d t}} = {0}$
${\frac{d {a_x}}{d t}} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{f}} {{\frac{d {K_{xx}}}{d x}}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{f}} {{\frac{d {K_{xy}}}{d x}}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{f}} {{\frac{d {K_{xz}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{f}} {{\frac{d {K_{yy}}}{d x}}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{f}} {{\frac{d {K_{yz}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{f}} {{\frac{d {K_{zz}}}{d x}}}}}}$
${\frac{d {a_y}}{d t}} = {0}$
${\frac{d {a_z}}{d t}} = {0}$
${\frac{d {\gamma_{xx}}}{d t}} = {0}$
${\frac{d {\gamma_{xy}}}{d t}} = {0}$
${\frac{d {\gamma_{xz}}}{d t}} = {0}$
${\frac{d {\gamma_{yy}}}{d t}} = {0}$
${\frac{d {\gamma_{yz}}}{d t}} = {0}$
${\frac{d {\gamma_{zz}}}{d t}} = {0}$
${\frac{d {d_{xxx}}}{d t}} = {{{-1}} {{\alpha}} \cdot {{\frac{d {K_{xx}}}{d x}}}}$
${\frac{d {d_{xxy}}}{d t}} = {{{-1}} {{\alpha}} \cdot {{\frac{d {K_{xy}}}{d x}}}}$
${\frac{d {d_{xxz}}}{d t}} = {{{-1}} {{\alpha}} \cdot {{\frac{d {K_{xz}}}{d x}}}}$
${\frac{d {d_{xyy}}}{d t}} = {{{-1}} {{\alpha}} \cdot {{\frac{d {K_{yy}}}{d x}}}}$
${\frac{d {d_{xyz}}}{d t}} = {{{-1}} {{\alpha}} \cdot {{\frac{d {K_{yz}}}{d x}}}}$
${\frac{d {d_{xzz}}}{d t}} = {{{-1}} {{\alpha}} \cdot {{\frac{d {K_{zz}}}{d x}}}}$
${\frac{d {d_{yxx}}}{d t}} = {0}$
${\frac{d {d_{yxy}}}{d t}} = {0}$
${\frac{d {d_{yxz}}}{d t}} = {0}$
${\frac{d {d_{yyy}}}{d t}} = {0}$
${\frac{d {d_{yyz}}}{d t}} = {0}$
${\frac{d {d_{yzz}}}{d t}} = {0}$
${\frac{d {d_{zxx}}}{d t}} = {0}$
${\frac{d {d_{zxy}}}{d t}} = {0}$
${\frac{d {d_{zxz}}}{d t}} = {0}$
${\frac{d {d_{zyy}}}{d t}} = {0}$
${\frac{d {d_{zyz}}}{d t}} = {0}$
${\frac{d {d_{zzz}}}{d t}} = {0}$
${\frac{d {K_{xx}}}{d t}} = {{-{{{2}} {{\alpha}} \cdot {{\frac{d {V_x}}{d x}}}}} + {-{{{\alpha}} \cdot {{\frac{d {a_x}}{d x}}}}} + {{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{xxy}}}{d x}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yxx}}}{d x}}}}} + {{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{xxz}}}{d x}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zxx}}}{d x}}}}} + {{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\frac{d {d_{xyy}}}{d x}}}} + {{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\frac{d {d_{xyz}}}{d x}}}} + {{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\frac{d {d_{xzz}}}{d x}}}}}$
${\frac{d {K_{xy}}}{d t}} = {{-{{{\alpha}} \cdot {{\frac{d {V_y}}{d x}}}}} + {-{\frac{{{\alpha}} \cdot {{\frac{d {a_y}}{d x}}}}{2}}} + {-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\frac{d {d_{xxy}}}{d x}}}}} + {\frac{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\frac{d {d_{yxx}}}{d x}}}}{2}} + {{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{yxz}}}{d x}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zxy}}}{d x}}}}} + {\frac{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\frac{d {d_{yyy}}}{d x}}}}{2}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\frac{d {d_{yyz}}}{d x}}}} + {\frac{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\frac{d {d_{yzz}}}{d x}}}}{2}}}$
${\frac{d {K_{xz}}}{d t}} = {{-{{{\alpha}} \cdot {{\frac{d {V_z}}{d x}}}}} + {-{\frac{{{\alpha}} \cdot {{\frac{d {a_z}}{d x}}}}{2}}} + {-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\frac{d {d_{xxz}}}{d x}}}}} + {\frac{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\frac{d {d_{zxx}}}{d x}}}}{2}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yxz}}}{d x}}}}} + {{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{zxy}}}{d x}}}} + {\frac{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\frac{d {d_{zyy}}}{d x}}}}{2}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\frac{d {d_{zyz}}}{d x}}}} + {\frac{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\frac{d {d_{zzz}}}{d x}}}}{2}}}$
${\frac{d {K_{yy}}}{d t}} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\frac{d {d_{xyy}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yyy}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zyy}}}{d x}}}}}}$
${\frac{d {K_{yz}}}{d t}} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\frac{d {d_{xyz}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yyz}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zyz}}}{d x}}}}}}$
${\frac{d {K_{zz}}}{d t}} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\frac{d {d_{xzz}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yzz}}}{d x}}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zzz}}}{d x}}}}}}$
${\frac{d {V_x}}{d t}} = {0}$
${\frac{d {V_y}}{d t}} = {0}$
${\frac{d {V_z}}{d t}} = {0}$

as C code:



F.alpha = 0.;

F.a_l.x = f_alpha * (Dx.K_ll.zz * gamma_uu.zz + 2. * Dx.K_ll.yz * gamma_uu.yz + Dx.K_ll.yy * gamma_uu.yy + 2. * Dx.K_ll.xz * gamma_uu.xz + 2. * Dx.K_ll.xy * gamma_uu.xy + Dx.K_ll.xx * gamma_uu.xx);

F.a_l.y = 0.;

F.a_l.z = 0.;

F.gamma_ll.xx = 0.;

F.gamma_ll.xy = 0.;

F.gamma_ll.xz = 0.;

F.gamma_ll.yy = 0.;

F.gamma_ll.yz = 0.;

F.gamma_ll.zz = 0.;

F.d_lll.x.xx = Dx.K_ll.xx * alpha;

F.d_lll.x.xy = Dx.K_ll.xy * alpha;

F.d_lll.x.xz = Dx.K_ll.xz * alpha;

F.d_lll.x.yy = Dx.K_ll.yy * alpha;

F.d_lll.x.yz = Dx.K_ll.yz * alpha;

F.d_lll.x.zz = Dx.K_ll.zz * alpha;

F.d_lll.y.xx = 0.;

F.d_lll.y.xy = 0.;

F.d_lll.y.xz = 0.;

F.d_lll.y.yy = 0.;

F.d_lll.y.yz = 0.;

F.d_lll.y.zz = 0.;

F.d_lll.z.xx = 0.;

F.d_lll.z.xy = 0.;

F.d_lll.z.xz = 0.;

F.d_lll.z.yy = 0.;

F.d_lll.z.yz = 0.;

F.d_lll.z.zz = 0.;

F.K_ll.xx = alpha * (Dx.d_lll.z.xx * gamma_uu.xz + Dx.d_lll.y.xx * gamma_uu.xy + Dx.a_l.x - 2. * Dx.d_lll.x.xy * gamma_uu.xy - 2. * Dx.d_lll.x.xz * gamma_uu.xz - Dx.d_lll.x.yy * gamma_uu.yy - 2. * Dx.d_lll.x.yz * gamma_uu.yz - Dx.d_lll.x.zz * gamma_uu.zz + 2. * Dx.V_x);

F.K_ll.xy = (alpha * (Dx.a_l.y + 2. * Dx.d_lll.z.xy * gamma_uu.xz + 2. * Dx.d_lll.x.xy * gamma_uu.xx - Dx.d_lll.y.xx * gamma_uu.xx - 2. * Dx.d_lll.y.xz * gamma_uu.xz - Dx.d_lll.y.yy * gamma_uu.yy - 2. * Dx.d_lll.y.yz * gamma_uu.yz - Dx.d_lll.y.zz * gamma_uu.zz + 2. * Dx.V_y)) / 2.;

F.K_ll.xz = (alpha * (Dx.a_l.z + 2. * Dx.d_lll.y.xz * gamma_uu.xy - Dx.d_lll.z.xx * gamma_uu.xx - 2. * Dx.d_lll.z.xy * gamma_uu.xy - Dx.d_lll.z.yy * gamma_uu.yy - 2. * Dx.d_lll.z.yz * gamma_uu.yz - Dx.d_lll.z.zz * gamma_uu.zz + 2. * Dx.d_lll.x.xz * gamma_uu.xx + 2. * Dx.V_z)) / 2.;

F.K_ll.yy = alpha * (Dx.d_lll.z.yy * gamma_uu.xz + Dx.d_lll.y.yy * gamma_uu.xy + Dx.d_lll.x.yy * gamma_uu.xx);

F.K_ll.yz = alpha * (Dx.d_lll.z.yz * gamma_uu.xz + Dx.d_lll.y.yz * gamma_uu.xy + Dx.d_lll.x.yz * gamma_uu.xx);

F.K_ll.zz = alpha * (Dx.d_lll.z.zz * gamma_uu.xz + Dx.d_lll.y.zz * gamma_uu.xy + Dx.d_lll.x.zz * gamma_uu.xx);

F.V_x = 0.;

F.V_y = 0.;

F.V_z = 0.;

removing zero row ${\frac{d \alpha}{d t}} = {0}$
removing zero row ${\frac{d {a_y}}{d t}} = {0}$
removing zero row ${\frac{d {a_z}}{d t}} = {0}$
removing zero row ${\frac{d {\gamma_{xx}}}{d t}} = {0}$
removing zero row ${\frac{d {\gamma_{xy}}}{d t}} = {0}$
removing zero row ${\frac{d {\gamma_{xz}}}{d t}} = {0}$
removing zero row ${\frac{d {\gamma_{yy}}}{d t}} = {0}$
removing zero row ${\frac{d {\gamma_{yz}}}{d t}} = {0}$
removing zero row ${\frac{d {\gamma_{zz}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{yxx}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{yxy}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{yxz}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{yyy}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{yyz}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{yzz}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{zxx}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{zxy}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{zxz}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{zyy}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{zyz}}}{d t}} = {0}$
removing zero row ${\frac{d {d_{zzz}}}{d t}} = {0}$
removing zero row ${\frac{d {V_x}}{d t}} = {0}$
removing zero row ${\frac{d {V_y}}{d t}} = {0}$
removing zero row ${\frac{d {V_z}}{d t}} = {0}$
${{\left[ \begin{matrix} \frac{d {a_x}}{d t} \\ \frac{d {d_{xxx}}}{d t} \\ \frac{d {d_{xxy}}}{d t} \\ \frac{d {d_{xxz}}}{d t} \\ \frac{d {d_{xyy}}}{d t} \\ \frac{d {d_{xyz}}}{d t} \\ \frac{d {d_{xzz}}}{d t} \\ \frac{d {K_{xx}}}{d t} \\ \frac{d {K_{xy}}}{d t} \\ \frac{d {K_{xz}}}{d t} \\ \frac{d {K_{yy}}}{d t} \\ \frac{d {K_{yz}}}{d t} \\ \frac{d {K_{zz}}}{d t}\end{matrix} \right]} + { {\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{f}} & {{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{f}} & {{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{f}} & {{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{f}} & {{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{f}} & {{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{f}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha \\ \alpha & 0 & -{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}}} & -{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}}} & -{{{\alpha}} \cdot {{{\gamma^{yy}}}}} & -{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}}} & -{{{\alpha}} \cdot {{{\gamma^{zz}}}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0\end{matrix} \right]} {\left[ \begin{matrix} \frac{d {a_x}}{d x} \\ \frac{d {d_{xxx}}}{d x} \\ \frac{d {d_{xxy}}}{d x} \\ \frac{d {d_{xxz}}}{d x} \\ \frac{d {d_{xyy}}}{d x} \\ \frac{d {d_{xyz}}}{d x} \\ \frac{d {d_{xzz}}}{d x} \\ \frac{d {K_{xx}}}{d x} \\ \frac{d {K_{xy}}}{d x} \\ \frac{d {K_{xz}}}{d x} \\ \frac{d {K_{yy}}}{d x} \\ \frac{d {K_{yz}}}{d x} \\ \frac{d {K_{zz}}}{d x}\end{matrix} \right]}}} = {\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ -{{{\alpha}} \cdot {{\left({{{{2}} {{\frac{d {V_x}}{d x}}}} + {{{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yxx}}}{d x}}}} + {{{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zxx}}}{d x}}}}}\right)}}} \\ -{\frac{{{\alpha}} \cdot {{\left({{{{2}} {{\frac{d {V_y}}{d x}}}} + {{{\frac{d {a_y}}{d x}} - {{{{\gamma^{xx}}}} \cdot {{\frac{d {d_{yxx}}}{d x}}}}} - {{{2}} {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{yxz}}}{d x}}}}} + {{{{{{2}} {{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zxy}}}{d x}}}} - {{{{\gamma^{yy}}}} \cdot {{\frac{d {d_{yyy}}}{d x}}}}} - {{{2}} {{{\gamma^{yz}}}} \cdot {{\frac{d {d_{yyz}}}{d x}}}}} - {{{{\gamma^{zz}}}} \cdot {{\frac{d {d_{yzz}}}{d x}}}}}}\right)}}}{2}} \\ -{\frac{{{\alpha}} \cdot {{\left({{{{2}} {{\frac{d {V_z}}{d x}}}} + {{\frac{d {a_z}}{d x}} - {{{{\gamma^{xx}}}} \cdot {{\frac{d {d_{zxx}}}{d x}}}}} + {{{{{{{2}} {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yxz}}}{d x}}}} - {{{2}} {{{\gamma^{xy}}}} \cdot {{\frac{d {d_{zxy}}}{d x}}}}} - {{{{\gamma^{yy}}}} \cdot {{\frac{d {d_{zyy}}}{d x}}}}} - {{{2}} {{{\gamma^{yz}}}} \cdot {{\frac{d {d_{zyz}}}{d x}}}}} - {{{{\gamma^{zz}}}} \cdot {{\frac{d {d_{zzz}}}{d x}}}}}}\right)}}}{2}} \\ -{{{\alpha}} \cdot {{\left({{{{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yyy}}}{d x}}}} + {{{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zyy}}}{d x}}}}}\right)}}} \\ -{{{\alpha}} \cdot {{\left({{{{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yyz}}}{d x}}}} + {{{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zyz}}}{d x}}}}}\right)}}} \\ -{{{\alpha}} \cdot {{\left({{{{{\gamma^{xy}}}} \cdot {{\frac{d {d_{yzz}}}{d x}}}} + {{{{\gamma^{xz}}}} \cdot {{\frac{d {d_{zzz}}}{d x}}}}}\right)}}}\end{matrix} \right]}$
characteristic polynomial:
$-{{{\lambda}} \cdot {{\left({{{{10}} {{{\alpha}^{4}}} {{{{\gamma^{xx}}}^{2}}} {{{\lambda}^{8}}}} + {{{{{{{5}} {{{\alpha}^{8}}} {{{{\gamma^{xx}}}^{4}}} {{{\lambda}^{4}}}} - {{{10}} {{{\alpha}^{6}}} {{{{\gamma^{xx}}}^{3}}} {{{\lambda}^{6}}}}} - {{{{\alpha}^{10}}} {{{{\gamma^{xx}}}^{5}}} {{{\lambda}^{2}}}}} - {{{5}} {{{\gamma^{xx}}}} \cdot {{{\alpha}^{2}}} {{{\lambda}^{10}}}}} - {{{{\gamma^{xx}}}} \cdot {{f}} {{{\alpha}^{2}}} {{{\lambda}^{10}}}}} + {{\lambda}^{12}} + {{{f}} {{{\alpha}^{12}}} {{{{\gamma^{xx}}}^{6}}}} + {{{5}} {{f}} {{{\alpha}^{4}}} {{{{\gamma^{xx}}}^{2}}} {{{\lambda}^{8}}}} + {{{{{10}} {{f}} {{{\alpha}^{8}}} {{{{\gamma^{xx}}}^{4}}} {{{\lambda}^{4}}}} - {{{5}} {{f}} {{{\alpha}^{10}}} {{{{\gamma^{xx}}}^{5}}} {{{\lambda}^{2}}}}} - {{{10}} {{f}} {{{\alpha}^{6}}} {{{{\gamma^{xx}}}^{3}}} {{{\lambda}^{6}}}}}}\right)}}}$
simplified...
$-{{{\lambda}} \cdot {{\left({{{{{{10}} {{{\alpha}^{4}}} {{{{\gamma^{xx}}}^{2}}} {{{\lambda}^{8}}}} - {{{{\alpha}^{10}}} {{{{\gamma^{xx}}}^{5}}} {{{\lambda}^{2}}}}} - {{{10}} {{{\alpha}^{6}}} {{{{\gamma^{xx}}}^{3}}} {{{\lambda}^{6}}}}} + {{{{{5}} {{{\alpha}^{8}}} {{{{\gamma^{xx}}}^{4}}} {{{\lambda}^{4}}}} - {{{5}} {{{\gamma^{xx}}}} \cdot {{{\alpha}^{2}}} {{{\lambda}^{10}}}}} - {{{{\gamma^{xx}}}} \cdot {{f}} {{{\alpha}^{2}}} {{{\lambda}^{10}}}}} + {{\lambda}^{12}} + {{{f}} {{{\alpha}^{12}}} {{{{\gamma^{xx}}}^{6}}}} + {{{10}} {{f}} {{{\alpha}^{8}}} {{{{\gamma^{xx}}}^{4}}} {{{\lambda}^{4}}}} + {{{{{5}} {{f}} {{{\alpha}^{4}}} {{{{\gamma^{xx}}}^{2}}} {{{\lambda}^{8}}}} - {{{5}} {{f}} {{{\alpha}^{10}}} {{{{\gamma^{xx}}}^{5}}} {{{\lambda}^{2}}}}} - {{{10}} {{f}} {{{\alpha}^{6}}} {{{{\gamma^{xx}}}^{3}}} {{{\lambda}^{6}}}}}}\right)}}}$
simplified:
$\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{f}} & {{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{f}} & {{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{f}} & {{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{f}} & {{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{f}} & {{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{f}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha \\ \alpha & 0 & -{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}}} & -{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}}} & -{{{\alpha}} \cdot {{{\gamma^{yy}}}}} & -{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}}} & -{{{\alpha}} \cdot {{{\gamma^{zz}}}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & 0 & 0\end{matrix} \right]$
$\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\alpha} & \frac{{{2}} {{{\gamma^{xy}}}}}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & \frac{{{2}} {{{\gamma^{xz}}}}}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & \frac{{\gamma^{yy}}}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & \frac{{{2}} {{{\gamma^{yz}}}}}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & \frac{{\gamma^{zz}}}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{{{\alpha}} \cdot {{{\gamma^{xx}}}}} \\ 0 & \frac{1}{\alpha} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{\alpha} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{\alpha} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{\alpha} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{f}}} & -{\frac{{\gamma^{xx}}}{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}}}} & -{\frac{{\gamma^{xy}}}{{{\alpha}} \cdot {{{\gamma^{yz}}}}}} & -{\frac{{\gamma^{xz}}}{{{\alpha}} \cdot {{{\gamma^{yz}}}}}} & -{\frac{{\gamma^{yy}}}{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}}}} & 0 & -{\frac{{\gamma^{zz}}}{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\alpha} & 0 & 0 & 0 & 0 & 0 & 0 \\ -{\frac{1}{{{2}} {{{\gamma^{yz}}}} \cdot {{f}}}} & \frac{{\gamma^{xx}}}{{{2}} {{{\gamma^{yz}}}}} & \frac{{\gamma^{xy}}}{{\gamma^{yz}}} & \frac{{\gamma^{xz}}}{{\gamma^{yz}}} & \frac{{\gamma^{yy}}}{{{2}} {{{\gamma^{yz}}}}} & 1 & \frac{{\gamma^{zz}}}{{{2}} {{{\gamma^{yz}}}}} & 0 & 0 & 0 & 0 & 0 & 0\end{matrix} \right]$
$\left[ \begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{matrix} \right]$
eigenvalue: 0
eigenvector:
$\left[ \begin{matrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]$
eigenvalue: ${{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}}$
eigenvector:
$\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 \\ -{\frac{{{2}} {{{\gamma^{xy}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{{2}} {{{\gamma^{xz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{\gamma^{yy}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{{2}} {{{\gamma^{yz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{\gamma^{zz}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} \\ \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} \\ -{\frac{{{2}} {{{\gamma^{xy}}}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{xz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{yy}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{yz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{zz}}}{{\gamma^{xx}}}} \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1\end{matrix} \right]$
eigenvalue: ${-{\alpha}} {{\sqrt{{\gamma^{xx}}}}}$
eigenvector:
$\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 \\ \frac{{{2}} {{{\gamma^{xy}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{{2}} {{{\gamma^{xz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{\gamma^{yy}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{{2}} {{{\gamma^{yz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{\gamma^{zz}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} \\ -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 \\ 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 \\ 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 \\ 0 & 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 \\ 0 & 0 & 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} \\ -{\frac{{{2}} {{{\gamma^{xy}}}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{xz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{yy}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{yz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{zz}}}{{\gamma^{xx}}}} \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1\end{matrix} \right]$
eigenvalue: ${{\alpha}} \cdot {{\sqrt{{{f}} {{{\gamma^{xx}}}}}}}$
eigenvector:
$\left[ \begin{matrix} \sqrt{{{{\gamma^{xx}}}} \cdot {{f}}} \\ \frac{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}}{{{{\gamma^{xx}}}} \cdot {{f}}} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]$
eigenvalue: ${-{\alpha}} {{\sqrt{{{f}} {{{\gamma^{xx}}}}}}}$
eigenvector:
$\left[ \begin{matrix} -{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}} \\ -{\frac{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}}{{{{\gamma^{xx}}}} \cdot {{f}}}} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]$
$\Lambda$:
$\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {-{\alpha}} {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {-{\alpha}} {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {-{\alpha}} {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {-{\alpha}} {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {-{\alpha}} {{\sqrt{{\gamma^{xx}}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\alpha}} \cdot {{\sqrt{{{f}} {{{\gamma^{xx}}}}}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {-{\alpha}} {{\sqrt{{{f}} {{{\gamma^{xx}}}}}}}\end{matrix} \right]$
R:
$\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{{{{\gamma^{xx}}}} \cdot {{f}}} & -{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}} \\ 1 & -{\frac{{{2}} {{{\gamma^{xy}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{{2}} {{{\gamma^{xz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{\gamma^{yy}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{{2}} {{{\gamma^{yz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & -{\frac{{\gamma^{zz}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}}} & \frac{{{2}} {{{\gamma^{xy}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{{2}} {{{\gamma^{xz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{\gamma^{yy}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{{2}} {{{\gamma^{yz}}}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{{\gamma^{zz}}}{{{{\gamma^{xx}}}} \cdot {{\sqrt{{\gamma^{xx}}}}}} & \frac{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}}{{{{\gamma^{xx}}}} \cdot {{f}}} & -{\frac{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}}{{{{\gamma^{xx}}}} \cdot {{f}}}} \\ 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{{\gamma^{xx}}}} & 0 & 0 & 0 & 0 & -{\frac{1}{\sqrt{{\gamma^{xx}}}}} & 0 & 0 \\ 0 & -{\frac{{{2}} {{{\gamma^{xy}}}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{xz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{yy}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{yz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{zz}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{xy}}}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{xz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{yy}}}{{\gamma^{xx}}}} & -{\frac{{{2}} {{{\gamma^{yz}}}}}{{\gamma^{xx}}}} & -{\frac{{\gamma^{zz}}}{{\gamma^{xx}}}} & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\end{matrix} \right]$
L:
$\left[ \begin{matrix} -{\frac{1}{{{{\gamma^{xx}}}} \cdot {{f}}}} & 1 & \frac{{{2}} {{{\gamma^{xy}}}}}{{\gamma^{xx}}} & \frac{{{2}} {{{\gamma^{xz}}}}}{{\gamma^{xx}}} & \frac{{\gamma^{yy}}}{{\gamma^{xx}}} & \frac{{{2}} {{{\gamma^{yz}}}}}{{\gamma^{xx}}} & \frac{{\gamma^{zz}}}{{\gamma^{xx}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{\sqrt{{\gamma^{xx}}}}{2} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\sqrt{{\gamma^{xx}}}}{2} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\sqrt{{\gamma^{xx}}}}{2} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{\sqrt{{\gamma^{xx}}}}{2} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\sqrt{{\gamma^{xx}}}}{2} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & -{\frac{\sqrt{{\gamma^{xx}}}}{2}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -{\frac{\sqrt{{\gamma^{xx}}}}{2}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -{\frac{\sqrt{{\gamma^{xx}}}}{2}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -{\frac{\sqrt{{\gamma^{xx}}}}{2}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{\sqrt{{\gamma^{xx}}}}{2}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ \frac{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}}{{{2}} {{{\gamma^{xx}}}} \cdot {{f}}} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & \frac{{\gamma^{xy}}}{{\gamma^{xx}}} & \frac{{\gamma^{xz}}}{{\gamma^{xx}}} & \frac{{\gamma^{yy}}}{{{2}} {{{\gamma^{xx}}}}} & \frac{{\gamma^{yz}}}{{\gamma^{xx}}} & \frac{{\gamma^{zz}}}{{{2}} {{{\gamma^{xx}}}}} \\ -{\frac{\sqrt{{{{\gamma^{xx}}}} \cdot {{f}}}}{{{2}} {{{\gamma^{xx}}}} \cdot {{f}}}} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & \frac{{\gamma^{xy}}}{{\gamma^{xx}}} & \frac{{\gamma^{xz}}}{{\gamma^{xx}}} & \frac{{\gamma^{yy}}}{{{2}} {{{\gamma^{xx}}}}} & \frac{{\gamma^{yz}}}{{\gamma^{xx}}} & \frac{{\gamma^{zz}}}{{{2}} {{{\gamma^{xx}}}}}\end{matrix} \right]$