Symbolic Lua Output

gauge vars
${Q} = {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{ K}_i}_l}}}$
primitive $\partial_t$ defs
${{ \alpha}_{,t}} = {{ {-{\alpha}} {{Q}}} + {{{{ \alpha}_{,i}}} {{{ \beta}^i}}}}$
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \alpha}_{,i}}} {{{ \beta}^i}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{{{ \gamma}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{ K}_i}_j}_{,t}} = {{-{{{ \alpha}_{,i}}_{,j}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{ \alpha}_{,k}}}} + {{{\alpha}} \cdot {{({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}})}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{({{{{{{ \gamma}_i}_j}} {{({{S} - {\rho}})}}} - {{{2}} {{{{ S}_i}_j}}}})}}}}$
lapse vars
${{ f}_{,k}} = {{{f'}} \cdot {{\alpha}} \cdot {{{ a}_k}}}$
hyperbolic state variables
${{ a}_k} = {{\left( log\left( \alpha\right)\right)}_{,k}}$
${{ a}_k} = {{\frac{1}{\alpha}}{({{ \alpha}_{,k}})}}$
${{ \alpha}_{,k}} = {{{\alpha}} \cdot {{{ a}_k}}}$
${{{{ d}_k}_i}_j} = {{{\frac{1}{2}}} {{{{{ \gamma}_i}_j}_{,k}}}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{2}} {{{{{ d}_k}_i}_j}}}$
connections wrt aux vars
${{{{ \Gamma}^k}_i}_j} = {{{\frac{1}{2}}} {{{{ \gamma}^k}^l}} {{({{{{{{ \gamma}_l}_i}_{,j}} + {{{{ \gamma}_l}_j}_{,i}}} - {{{{ \gamma}_i}_j}_{,l}}})}}}$
${{{{ \Gamma}_i}_j}_k} = {{{\frac{1}{2}}} {{({{{{{{ \gamma}_i}_j}_{,k}} + {{{{ \gamma}_i}_k}_{,j}}} - {{{{ \gamma}_j}_k}_{,i}}})}}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{{{ \Gamma}_i}_j}_k} + {{{{ \Gamma}_j}_i}_k}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{{{{ \gamma}_i}_a}} {{{{{ \Gamma}^a}_j}_k}}} + {{{{{ \gamma}_j}_a}} {{{{{ \Gamma}^a}_i}_k}}}}$
${{{{ \Gamma}_i}_j}_k} = {-{({{{{{{ d}_i}_j}_k} - {{{{ d}_j}_i}_k}} - {{{{ d}_k}_i}_j}})}}$
${{{{ \Gamma}^i}_j}_k} = {{{{{ \gamma}^i}^l}} {{({{{{{ d}_j}_l}_k} + {{{{{ d}_k}_l}_j} - {{{{ d}_l}_j}_k}}})}}}$
${\gamma^{ij}}_{,k}$ 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
${{{ R}_i}_j} = {{{{{{{{ \Gamma}^k}_i}_j}_{,k}} - {{{{{ \Gamma}^k}_i}_k}_{,j}}} + {{{{{{ \Gamma}^k}_l}_k}} {{{{{ \Gamma}^l}_i}_j}}}} - {{{{{{ \Gamma}^k}_l}_j}} {{{{{ \Gamma}^l}_i}_k}}}}$
${{{ R}_i}_j} = {{{{{\left( {{{{ \gamma}^k}^a}} {{({{{{{ d}_i}_a}_j} + {{{{{ d}_j}_a}_i} - {{{{ d}_a}_i}_j}}})}}\right)}_{,k}} - {{\left( {{{{ \gamma}^k}^a}} {{({{{{{ d}_i}_a}_k} + {{{{{ d}_k}_a}_i} - {{{{ d}_a}_i}_k}}})}}\right)}_{,j}}} + {{{{{{{ \gamma}^k}^a}} {{({{{{{ d}_l}_a}_k} + {{{{{ d}_k}_a}_l} - {{{{ d}_a}_l}_k}}})}}}} {{{{{{ \gamma}^l}^b}} {{({{{{{ d}_i}_b}_j} + {{{{{ d}_j}_b}_i} - {{{{ d}_b}_i}_j}}})}}}}}} - {{{{{{{ \gamma}^k}^a}} {{({{{{{ d}_l}_a}_j} + {{{{{ d}_j}_a}_l} - {{{{ d}_a}_l}_j}}})}}}} {{{{{{ \gamma}^l}^b}} {{({{{{{ d}_i}_b}_k} + {{{{{ d}_k}_b}_i} - {{{{ d}_b}_i}_k}}})}}}}}}$
${{{ 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}^b}} {{{{{ d}_i}_b}_j}} {{{{{ d}_l}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_i}_b}_j}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_k}} {{{{{ d}_i}_b}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_j}_b}_i}} {{{{{ d}_l}_a}_k}}} + {{{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_j}_b}_i}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_k}} {{{{{ d}_j}_b}_i}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_j}} {{{{{ d}_l}_a}_k}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_j}} {{{{{ d}_k}_a}_l}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_k}} {{{{{ d}_b}_i}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_i}_b}_k}} {{{{{ d}_l}_a}_j}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_k}_b}_i}} {{{{{ d}_l}_a}_j}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_k}} {{{{{ d}_l}_a}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_i}_b}_k}} {{{{{ d}_j}_a}_l}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_j}_a}_l}} {{{{{ d}_k}_b}_i}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_k}} {{{{{ d}_j}_a}_l}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_j}} {{{{{ d}_i}_b}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_j}} {{{{{ d}_k}_b}_i}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_j}} {{{{{ d}_b}_i}_k}}}}}$
${{{ R}_i}_j} = {{{-{{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^d}} {{{{{ d}_i}_a}_j}} {{{{{ d}_k}_d}_c}}}} - {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^d}} {{{{{ d}_j}_a}_i}} {{{{{ d}_k}_d}_c}}}} + {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_k}_d}_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}^d}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_d}_c}}} + {{{{{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^d}} {{{{{ d}_j}_d}_c}} {{{{{ d}_k}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^d}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_d}_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}^b}} {{{{{ d}_i}_b}_j}} {{{{{ d}_l}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_i}_b}_j}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_k}} {{{{{ d}_i}_b}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_j}_b}_i}} {{{{{ d}_l}_a}_k}}} + {{{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_j}_b}_i}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_k}} {{{{{ d}_j}_b}_i}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_j}} {{{{{ d}_l}_a}_k}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_j}} {{{{{ d}_k}_a}_l}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_k}} {{{{{ d}_b}_i}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_i}_b}_k}} {{{{{ d}_l}_a}_j}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_k}_b}_i}} {{{{{ d}_l}_a}_j}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_k}} {{{{{ d}_l}_a}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_i}_b}_k}} {{{{{ d}_j}_a}_l}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_j}_a}_l}} {{{{{ d}_k}_b}_i}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_b}_i}_k}} {{{{{ d}_j}_a}_l}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_j}} {{{{{ d}_i}_b}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_j}} {{{{{ d}_k}_b}_i}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^b}} {{{{{ d}_a}_l}_j}} {{{{{ d}_b}_i}_k}}}}}$
symmetrizing
${{{ R}_i}_j} = {{{-{{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^d}^k}} {{{{{ d}_d}_c}_k}} {{{{{ d}_i}_a}_j}}}} - {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^d}^k}} {{{{{ d}_d}_c}_k}} {{{{{ d}_j}_a}_i}}}} + {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^d}^k}} {{{{{ d}_a}_i}_j}} {{{{{ d}_d}_c}_k}}} + {{{{{ \gamma}^a}^k}} {{{{{{ d}_i}_a}_j}_{,k}}}} + {{{{{{ \gamma}^a}^k}} {{{{{{ d}_j}_a}_i}_{,k}}}} - {{{{{ \gamma}^a}^k}} {{{{{{ d}_a}_i}_j}_{,k}}}}} + {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^d}^k}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_c}_d}}} + {{{{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^d}^k}} {{{{{ d}_j}_c}_d}} {{{{{ d}_k}_a}_i}}} - {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^d}^k}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_c}_d}}}} - {{{{{ \gamma}^a}^k}} {{{{{{ d}_i}_a}_k}_{,j}}}}} + {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_i}_b}_j}} {{{{{ d}_l}_a}_k}}} + {{{{{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_j}_b}_i}} {{{{{ d}_l}_a}_k}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_b}_i}_j}} {{{{{ d}_l}_a}_k}}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_i}_b}_k}} {{{{{ d}_l}_a}_j}}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_k}_b}_i}} {{{{{ d}_l}_a}_j}}}} + {{{{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_b}_i}_k}} {{{{{ d}_l}_a}_j}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_i}_b}_k}} {{{{{ d}_j}_a}_l}}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_j}_a}_l}} {{{{{ d}_k}_b}_i}}}} + {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_b}_i}_k}} {{{{{ d}_j}_a}_l}}} + {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_a}_j}_l}} {{{{{ d}_i}_b}_k}}} + {{{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_a}_j}_l}} {{{{{ d}_k}_b}_i}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^b}^l}} {{{{{ d}_a}_j}_l}} {{{{{ d}_b}_i}_k}}}}}$
${{{ R}_i}_j} = {-{{{{{ \gamma}^a}^b}} {{({{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_b}_c}_i}}}} + {{{{{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_c}_b}_i}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_i}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_d}_a}_j}}} + {{{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_j}_a}_d}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_d}_a}_j}}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_j}_a}_d}}}} + {{{{{{{{ \gamma}^c}^d}} {{{{{ d}_c}_i}_j}} {{{{{ d}_d}_a}_b}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_i}_c}_j}}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_j}_c}_i}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_j}} {{{{{ d}_i}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{ \gamma}^c}^d}} {{{{{ d}_i}_b}_c}} {{{{{ d}_j}_a}_d}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{ d}_i}_a}_b}_{,j}}})}}}}$
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 $a_{k,t}$
${{{ a}_k}_{,t}} = {\frac{-{({{{{{ \alpha}_{,k}}} {{{ \alpha}_{,t}}}} - {{{\alpha}} \cdot {{{{ \alpha}_{,k}}_{,t}}}}})}}{{\alpha}^{2}}}$
${{{ a}_k}_{,t}} = {\frac{-{({{{{{ \alpha}_{,k}}} {{({{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}})}}} - {{{\alpha}} \cdot {{{\left( {-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}\right)}_{,k}}}}})}}{{\alpha}^{2}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{ f}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{ \alpha}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{ f}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{ \alpha}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{{f'}} \cdot {{\alpha}} \cdot {{{ a}_k}}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{{\alpha}} \cdot {{{ a}_k}}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^a}^l}} {{{{ \gamma}^i}^b}} {{{{{ d}_k}_b}_a}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^a}^l}} {{{{ \gamma}^i}^b}} {{{{{ d}_k}_b}_a}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
time derivative of $d_{kij,t}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{{ \gamma}_i}_j}_{,k}}_{,t}})}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}}{({{\left( {{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{ \beta}^l}} {{{{2}} {{{{{ d}_l}_i}_j}}}}} + {{{{{ \gamma}_l}_j}} {{{{ \beta}^l}_{,i}}}} + {{{{{ \gamma}_i}_l}} {{{{ \beta}^l}_{,j}}}}\right)}_{,k}})}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}}{({-{({{{{{{{{{{2}} {{{ \alpha}_{,k}}} {{{{ K}_i}_j}}} - {{{{{ \beta}^l}_{,i}}} {{{{{ \gamma}_l}_j}_{,k}}}}} - {{{{{ \beta}^l}_{,j}}} {{{{{ \gamma}_i}_l}_{,k}}}}} - {{{2}} {{{{ \beta}^l}_{,k}}} {{{{{ d}_l}_i}_j}}}} - {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}}} - {{{{{ \gamma}_i}_l}} {{{{{ \beta}^l}_{,j}}_{,k}}}}} - {{{{{ \gamma}_l}_j}} {{{{{ \beta}^l}_{,i}}_{,k}}}}} + {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}})}})}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}}{({{-{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}} + {{{2}} {{{{ \beta}^l}_{,i}}} {{{{{ d}_k}_l}_j}}} + {{{2}} {{{{ \beta}^l}_{,j}}} {{{{{ d}_k}_i}_l}}} + {{{2}} {{{{ \beta}^l}_{,k}}} {{{{{ d}_l}_i}_j}}} + {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}} + {{{{{ \gamma}_i}_l}} {{{{{ \beta}^l}_{,j}}_{,k}}}} + {{{{{{ \gamma}_l}_j}} {{{{{ \beta}^l}_{,i}}_{,k}}}} - {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}})}}$
$K_{ij,t}$ with hyperbolic terms
${{{{ K}_i}_j}_{,t}} = {{-{{{ \alpha}_{,i}}_{,j}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{ \alpha}_{,k}}}} + {{{\alpha}} \cdot {{({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}})}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{({{{{{{ \gamma}_i}_j}} {{({{S} - {\rho}})}}} - {{{2}} {{{{ S}_i}_j}}}})}}}}$
${{{{ K}_i}_j}_{,t}} = {{-{{{\frac{1}{2}}} {{({{{\left( {{\alpha}} \cdot {{{ a}_i}}\right)}_{,j}} + {{\left( {{\alpha}} \cdot {{{ a}_j}}\right)}_{,i}}})}}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{{\alpha}} \cdot {{{ a}_k}}}}} + {{{\alpha}} \cdot {{({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}})}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{({{{{{{ \gamma}_i}_j}} {{({{S} - {\rho}})}}} - {{{2}} {{{{ S}_i}_j}}}})}}}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{ \Gamma}^k}_i}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{{ \alpha}_{,j}}} {{{ a}_i}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{{ \alpha}_{,i}}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}})}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{ \Gamma}^k}_i}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}})}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{{ \gamma}^k}^l}} {{({{{{{ d}_i}_l}_j} + {{{{{ d}_j}_l}_i} - {{{{ d}_l}_i}_j}}})}}}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}})}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{{ \gamma}^k}^l}} {{({{{{{ d}_i}_l}_j} + {{{{{ d}_j}_l}_i} - {{{{ d}_l}_i}_j}}})}}}}} + {{{2}} {{\alpha}} \cdot {-{{{{{ \gamma}^a}^b}} {{({{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_b}_c}_i}}}} + {{{{{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_c}_b}_i}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_i}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_d}_a}_j}}} + {{{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_j}_a}_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}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_d}_a}_j}}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_j}_a}_d}}}} + {{{{{{{{ \gamma}^c}^d}} {{{{{ d}_c}_i}_j}} {{{{{ d}_d}_a}_b}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_i}_c}_j}}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_j}_c}_i}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_j}} {{{{{ d}_i}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{ \gamma}^c}^d}} {{{{{ d}_i}_b}_c}} {{{{{ d}_j}_a}_d}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{ d}_i}_a}_b}_{,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}}}}}})}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{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}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_b}_c}_i}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_c}_b}_i}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_i}_b}_c}}} + {{{{{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_d}_a}_j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_j}_a}_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}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_d}_a}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_j}_a}_d}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_i}_j}} {{{{{ d}_d}_a}_b}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_i}_c}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_j}_c}_i}}} + {{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_j}} {{{{{ d}_i}_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}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_b}_c}} {{{{{ d}_j}_a}_d}}}} - {{{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 {{{{ 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}}}}}})}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{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}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_b}_c}_i}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_c}_b}_i}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_i}_b}_c}}} + {{{{{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_d}_a}_j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_j}_a}_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}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_d}_a}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_j}_a}_d}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_i}_j}} {{{{{ d}_d}_a}_b}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_i}_c}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_j}_c}_i}}} + {{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_j}} {{{{{ d}_i}_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}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_b}_c}} {{{{{ d}_j}_a}_d}}}} - {{{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 {{{{ 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}}}}}})}}$
partial derivatives
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^a}^l}} {{{{ \gamma}^i}^b}} {{{{{ d}_k}_b}_a}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{ \beta}^k}} {{{{2}} {{{{{ d}_k}_i}_j}}}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}}{({{-{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}} + {{{2}} {{{{ \beta}^l}_{,i}}} {{{{{ d}_k}_l}_j}}} + {{{2}} {{{{ \beta}^l}_{,j}}} {{{{{ d}_k}_i}_l}}} + {{{2}} {{{{ \beta}^l}_{,k}}} {{{{{ d}_l}_i}_j}}} + {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}} + {{{{{ \gamma}_i}_l}} {{{{{ \beta}^l}_{,j}}_{,k}}}} + {{{{{{ \gamma}_l}_j}} {{{{{ \beta}^l}_{,i}}_{,k}}}} - {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}})}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{({{{{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}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_b}_c}_i}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_c}_b}_i}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_i}_b}_c}}} + {{{{{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_d}_a}_j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_j}_a}_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}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_d}_a}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_j}_a}_d}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_i}_j}} {{{{{ d}_d}_a}_b}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_i}_c}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_j}_c}_i}}} + {{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_j}} {{{{{ d}_i}_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}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_b}_c}} {{{{{ d}_j}_a}_d}}}} - {{{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 {{{{ 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}}}}}})}}$
neglecting shift
${{ \alpha}_{,t}} = {{{-1}} {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}$
${{{ a}_k}_{,t}} = {{-{{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^a}^l}} {{{{ \gamma}^i}^b}} {{{{{ d}_k}_b}_a}}} + {-{{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}} + {-{{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{-{{{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}} + {-{{{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}}}$
${{{{ K}_i}_j}_{,t}} = {{{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} + {{{\alpha}} \cdot {{{{ K}_a}_b}} {{{{ K}_i}_j}} {{{{ \gamma}^a}^b}}} + {-{{{2}} {{\alpha}} \cdot {{{{ K}_i}_a}} {{{{ K}_j}_b}} {{{{ \gamma}^a}^b}}}} + {-{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_b}_c}_i}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_c}_b}_i}}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_i}_b}_c}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_d}_a}_j}}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_j}_a}_d}}}} + {-{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}}} + {-{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_d}_a}_j}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_j}_a}_d}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_i}_j}} {{{{{ d}_d}_a}_b}}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_i}_c}_j}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_j}_c}_i}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_j}} {{{{{ d}_i}_b}_c}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_b}_c}} {{{{{ d}_j}_a}_d}}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {-{{{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_b}_i}_j}}}} + {{{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_i}_b}_j}}} + {{{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_j}_b}_i}}} + {{\frac{1}{2}}{({-{{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}})}} + {-{{{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} + {{\frac{1}{2}}{({-{{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}})}} + {-{{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} + {-{{{4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}}}$
neglecting source terms
${{ \alpha}_{,t}} = {0}$
${{{ a}_k}_{,t}} = {{{-1}} {{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {0}$
${{{{{ d}_k}_i}_j}_{,t}} = {{{-1}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}$
${{{{ K}_i}_j}_{,t}} = {{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {-{{{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {{\frac{1}{2}}{({-{{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}})}} + {{\frac{1}{2}}{({-{{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}})}}}$
...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}} {{({{{{{2}} {{f}} {{{{ \gamma}^a}^l}} {{{{ \gamma}^i}^b}} {{{{{ d}_k}_b}_a}}} - {{{f}} {{{ a}_k}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f'}} \cdot {{{ a}_k}} {{{{ \gamma}^i}^l}}}})}}$
${{{ \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 {{({{{{4}} {{S}} {{\pi}} \cdot {{{{ \gamma}_i}_j}}} + {{{{{{{ K}_a}_b}} {{{{ K}_i}_j}} {{{{ \gamma}^a}^b}}} - {{{2}} {{{{ K}_i}_a}} {{{{ K}_j}_b}} {{{{ \gamma}^a}^b}}}} - {{{2}} {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_b}_c}_i}}} - {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_c}_b}_i}}}} + {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_j}} {{{{{ d}_i}_b}_c}}} + {{{{{{{2}} {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_d}_a}_j}}}} - {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_b}_c}_i}} {{{{{ d}_j}_a}_d}}}} - {{{2}} {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}}} - {{{2}} {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}}} + {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_d}_a}_j}}} + {{{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_i}} {{{{{ d}_j}_a}_d}}} - {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_i}_j}} {{{{{ d}_d}_a}_b}}}} + {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_i}_c}_j}}} + {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_b}} {{{{{ d}_j}_c}_i}}} + {{{{2}} {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} - {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_j}} {{{{{ d}_i}_b}_c}}}} + {{{{{2}} {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}} - {{{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_b}_c}} {{{{{ d}_j}_a}_d}}}} - {{{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_b}_i}_j}}}} + {{{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_i}_b}_j}}} + {{{{{{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_j}_b}_i}}} - {{{{ a}_i}} {{{ a}_j}}}} - {{{8}} {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{4}} {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}}})}}$
spelled out
mismatch
${\partial_ {{t}}( {K_{xy}})} = {{{\frac{1}{2}}{({-{{{\alpha}} \cdot {{\partial_ {{x}}( {a_y})}}}})}} + {-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\partial_ {{x}}( {d_{yxx}})}}}} + {{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{xyy}})}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{yxy}})}}}} + {{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{xyz}})}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{yxz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{yyz}})}}}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{zyy}})}}} + {-{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{yzz}})}}}} + {{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{zyz}})}}}}$
difference
${-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\partial_ {{x}}( {d_{yxx}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{yxy}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{yxz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{yyy}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{yyz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{yzz}})}}}}$

mismatch
${\partial_ {{t}}( {K_{xz}})} = {{{\frac{1}{2}}{({-{{{\alpha}} \cdot {{\partial_ {{x}}( {a_z})}}}})}} + {-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\partial_ {{x}}( {d_{zxx}})}}}} + {{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{xyz}})}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{zxy}})}}}} + {{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{xzz}})}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{zxz}})}}}} + {{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{yyz}})}}} + {-{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{zyy}})}}}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{yzz}})}}} + {-{{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{zyz}})}}}}}$
difference
${-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\partial_ {{x}}( {d_{zxx}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{zxy}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{zxz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{zyy}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{zyz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{zzz}})}}}}$

${\partial_ {{t}}( \alpha)} = {0}$
${\partial_ {{t}}( {a_x})} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{f}} {{\partial_ {{x}}( {K_{xx}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{f}} {{\partial_ {{x}}( {K_{xy}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{f}} {{\partial_ {{x}}( {K_{xz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{f}} {{\partial_ {{x}}( {K_{yy}})}}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{f}} {{\partial_ {{x}}( {K_{yz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{f}} {{\partial_ {{x}}( {K_{zz}})}}}}}$
${\partial_ {{t}}( {a_y})} = {0}$
${\partial_ {{t}}( {a_z})} = {0}$
${\partial_ {{t}}( {\gamma_{xx}})} = {0}$
${\partial_ {{t}}( {\gamma_{xy}})} = {0}$
${\partial_ {{t}}( {\gamma_{xz}})} = {0}$
${\partial_ {{t}}( {\gamma_{yy}})} = {0}$
${\partial_ {{t}}( {\gamma_{yz}})} = {0}$
${\partial_ {{t}}( {\gamma_{zz}})} = {0}$
${\partial_ {{t}}( {d_{xxx}})} = {{{-1}} {{\alpha}} \cdot {{\partial_ {{x}}( {K_{xx}})}}}$
${\partial_ {{t}}( {d_{xxy}})} = {{{-1}} {{\alpha}} \cdot {{\partial_ {{x}}( {K_{xy}})}}}$
${\partial_ {{t}}( {d_{xxz}})} = {{{-1}} {{\alpha}} \cdot {{\partial_ {{x}}( {K_{xz}})}}}$
${\partial_ {{t}}( {d_{xyy}})} = {{{-1}} {{\alpha}} \cdot {{\partial_ {{x}}( {K_{yy}})}}}$
${\partial_ {{t}}( {d_{xyz}})} = {{{-1}} {{\alpha}} \cdot {{\partial_ {{x}}( {K_{yz}})}}}$
${\partial_ {{t}}( {d_{xzz}})} = {{{-1}} {{\alpha}} \cdot {{\partial_ {{x}}( {K_{zz}})}}}$
${\partial_ {{t}}( {d_{yxx}})} = {0}$
${\partial_ {{t}}( {d_{yxy}})} = {0}$
${\partial_ {{t}}( {d_{yxz}})} = {0}$
${\partial_ {{t}}( {d_{yyy}})} = {0}$
${\partial_ {{t}}( {d_{yyz}})} = {0}$
${\partial_ {{t}}( {d_{yzz}})} = {0}$
${\partial_ {{t}}( {d_{zxx}})} = {0}$
${\partial_ {{t}}( {d_{zxy}})} = {0}$
${\partial_ {{t}}( {d_{zxz}})} = {0}$
${\partial_ {{t}}( {d_{zyy}})} = {0}$
${\partial_ {{t}}( {d_{zyz}})} = {0}$
${\partial_ {{t}}( {d_{zzz}})} = {0}$
${\partial_ {{t}}( {K_{xx}})} = {{-{{{\alpha}} \cdot {{\partial_ {{x}}( {a_x})}}}} + {{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{yxx}})}}} + {{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{zxx}})}}} + {-{{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{xyy}})}}}} + {{{2}} {{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{yxy}})}}} + {-{{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{xyz}})}}}} + {{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{yxz}})}}} + {{{2}} {{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{zxy}})}}} + {-{{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{xzz}})}}}} + {{{2}} {{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{zxz}})}}}}$
${\partial_ {{t}}( {K_{xy}})} = {{{\frac{1}{2}}{({-{{{\alpha}} \cdot {{\partial_ {{x}}( {a_y})}}}})}} + {{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{xyy}})}}} + {{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{xyz}})}}} + {{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{yyy}})}}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{yyz}})}}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{zyy}})}}} + {{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{zyz}})}}}}$
${\partial_ {{t}}( {K_{xz}})} = {{{\frac{1}{2}}{({-{{{\alpha}} \cdot {{\partial_ {{x}}( {a_z})}}}})}} + {{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{xyz}})}}} + {{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{xzz}})}}} + {{{\alpha}} \cdot {{{\gamma^{yy}}}} \cdot {{\partial_ {{x}}( {d_{yyz}})}}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{yzz}})}}} + {{{\alpha}} \cdot {{{\gamma^{yz}}}} \cdot {{\partial_ {{x}}( {d_{zyz}})}}} + {{{\alpha}} \cdot {{{\gamma^{zz}}}} \cdot {{\partial_ {{x}}( {d_{zzz}})}}}}$
${\partial_ {{t}}( {K_{yy}})} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\partial_ {{x}}( {d_{xyy}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{yyy}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{zyy}})}}}}}$
${\partial_ {{t}}( {K_{yz}})} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\partial_ {{x}}( {d_{xyz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{yyz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{zyz}})}}}}}$
${\partial_ {{t}}( {K_{zz}})} = {{-{{{\alpha}} \cdot {{{\gamma^{xx}}}} \cdot {{\partial_ {{x}}( {d_{xzz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xy}}}} \cdot {{\partial_ {{x}}( {d_{yzz}})}}}} + {-{{{\alpha}} \cdot {{{\gamma^{xz}}}} \cdot {{\partial_ {{x}}( {d_{zzz}})}}}}}$

as C code:



F.alpha = 0.;

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

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.a_l.x + Dx.d_lll.x.zz * gamma_uu.zz - Dx.d_lll.y.xx * gamma_uu.xy - 2. * Dx.d_lll.y.xy * gamma_uu.yy - 2. * Dx.d_lll.y.xz * gamma_uu.yz - Dx.d_lll.z.xx * gamma_uu.xz - 2. * Dx.d_lll.z.xy * gamma_uu.yz - 2. * Dx.d_lll.z.xz * gamma_uu.zz + 2. * Dx.d_lll.x.yz * gamma_uu.yz + Dx.d_lll.x.yy * gamma_uu.yy);

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

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

source terms:

as C code:



real tmp1 = K_ll.xy * gamma_uu.xy;

real tmp2 = K_ll.xz * gamma_uu.xz;

real tmp3 = K_ll.yz * gamma_uu.yz;

real tmp5 = K_ll.zz * gamma_uu.zz;

real tmp6 = K_ll.yy * gamma_uu.yy;

real tmp18 = f * gamma_uu.xx;

real tmp20 = dalpha_f * gamma_uu.xx;

real tmp21 = alpha * tmp20;

real tmp23 = gamma_uu.xx * gamma_uu.xx;

real tmp24 = f * tmp23;

real tmp29 = gamma_uu.xx * gamma_uu.xy;

real tmp30 = f * tmp29;

real tmp35 = gamma_uu.xx * gamma_uu.xz;

real tmp36 = f * tmp35;

real tmp41 = gamma_uu.xy * gamma_uu.xy;

real tmp42 = f * tmp41;

real tmp47 = gamma_uu.xy * gamma_uu.xz;

real tmp48 = f * tmp47;

real tmp53 = gamma_uu.xz * gamma_uu.xz;

real tmp54 = f * tmp53;

real tmp59 = f * gamma_uu.xy;

real tmp62 = dalpha_f * gamma_uu.xy;

real tmp63 = alpha * tmp62;

real tmp72 = gamma_uu.xx * gamma_uu.yy;

real tmp73 = f * tmp72;

real tmp84 = gamma_uu.xx * gamma_uu.yz;

real tmp85 = f * tmp84;

real tmp96 = gamma_uu.xy * gamma_uu.yy;

real tmp97 = f * tmp96;

real tmp102 = gamma_uu.xy * gamma_uu.yz;

real tmp103 = f * tmp102;

real tmp108 = gamma_uu.xz * gamma_uu.yy;

real tmp109 = f * tmp108;

real tmp114 = gamma_uu.xz * gamma_uu.yz;

real tmp115 = f * tmp114;

real tmp120 = f * gamma_uu.xz;

real tmp123 = dalpha_f * gamma_uu.xz;

real tmp124 = alpha * tmp123;

real tmp145 = gamma_uu.xx * gamma_uu.zz;

real tmp146 = f * tmp145;

real tmp163 = gamma_uu.xy * gamma_uu.zz;

real tmp164 = f * tmp163;

real tmp175 = gamma_uu.xz * gamma_uu.zz;

real tmp176 = f * tmp175;

real tmp181 = f * gamma_uu.yy;

real tmp183 = dalpha_f * gamma_uu.yy;

real tmp184 = alpha * tmp183;

real tmp204 = gamma_uu.yy * gamma_uu.yy;

real tmp205 = f * tmp204;

real tmp210 = gamma_uu.yy * gamma_uu.yz;

real tmp211 = f * tmp210;

real tmp216 = gamma_uu.yz * gamma_uu.yz;

real tmp217 = f * tmp216;

real tmp222 = f * gamma_uu.yz;

real tmp225 = dalpha_f * gamma_uu.yz;

real tmp226 = alpha * tmp225;

real tmp265 = gamma_uu.yy * gamma_uu.zz;

real tmp266 = f * tmp265;

real tmp277 = gamma_uu.yz * gamma_uu.zz;

real tmp278 = f * tmp277;

real tmp283 = f * gamma_uu.zz;

real tmp285 = dalpha_f * gamma_uu.zz;

real tmp286 = alpha * tmp285;

real tmp318 = gamma_uu.zz * gamma_uu.zz;

real tmp319 = f * tmp318;

real tmp1020 = a_l.x * alpha;

real tmp1032 = a_l.y * alpha;

real tmp1044 = a_l.z * alpha;

real tmp1090 = d_lll.y.xx * gamma_uu.xy;

real tmp1091 = d_lll.z.xx * gamma_uu.xz;

real tmp1123 = d_lll.x.yz * tmp84;

real tmp1128 = d_lll.x.zz * tmp145;

real tmp1132 = d_lll.y.xy * tmp72;

real tmp1135 = d_lll.y.xz * tmp84;

real tmp1138 = d_lll.y.yy * tmp96;

real tmp1140 = d_lll.y.yz * tmp108;

real tmp1143 = d_lll.y.zz * tmp163;

real tmp1147 = d_lll.y.zz * tmp114;

real tmp1150 = d_lll.z.xy * tmp84;

real tmp1153 = d_lll.z.xz * tmp145;

real tmp1156 = d_lll.z.yy * tmp102;

real tmp1159 = d_lll.z.yy * tmp108;

real tmp1163 = d_lll.z.yz * tmp163;

real tmp1166 = d_lll.z.zz * tmp175;

real tmp1173 = d_lll.x.yz * tmp102;

real tmp1178 = d_lll.x.zz * tmp163;

real tmp1183 = d_lll.y.xy * tmp96;

real tmp1186 = d_lll.y.xz * tmp102;

real tmp1189 = d_lll.y.yy * tmp204;

real tmp1192 = d_lll.y.yz * tmp210;

real tmp1195 = d_lll.y.zz * tmp265;

real tmp1200 = d_lll.y.zz * tmp216;

real tmp1203 = d_lll.z.xy * tmp102;

real tmp1206 = d_lll.z.xz * tmp163;

real tmp1209 = d_lll.z.yy * tmp210;

real tmp1212 = d_lll.z.yz * tmp265;

real tmp1215 = d_lll.z.zz * tmp277;

real tmp1228 = d_lll.x.zz * tmp175;

real tmp1233 = d_lll.y.xy * tmp108;

real tmp1239 = d_lll.y.yy * tmp210;

real tmp1242 = d_lll.y.yz * tmp265;

real tmp1245 = d_lll.y.zz * tmp277;

real tmp1248 = d_lll.z.xy * tmp114;

real tmp1251 = d_lll.z.xz * tmp175;

real tmp1254 = d_lll.z.yy * tmp265;

real tmp1259 = d_lll.z.yy * tmp216;

real tmp1262 = d_lll.z.yz * tmp277;

real tmp1265 = d_lll.z.zz * tmp318;

real tmp1287 = d_lll.z.xx * tmp108;

real tmp1288 = d_lll.x.yy * d_lll.x.yy;

real tmp1303 = d_lll.z.xx * tmp163;

real tmp1307 = d_lll.x.yz * d_lll.x.yz;

real tmp1323 = d_lll.x.zz * d_lll.x.zz;

real tmp1333 = d_lll.y.xz * tmp108;

real tmp1354 = d_lll.z.xx * tmp84;

real tmp1364 = d_lll.z.xy * tmp108;

real tmp1393 = d_lll.z.xx * tmp114;

real tmp1398 = d_lll.z.xy * tmp265;

real tmp1401 = d_lll.z.xy * tmp216;

real tmp1405 = d_lll.y.xz * d_lll.y.xz;

real tmp1422 = d_lll.z.xy * tmp163;

real tmp1447 = d_lll.z.xy * d_lll.z.xy;

real tmp1451 = d_lll.y.xx * d_lll.y.xx;

real tmp1455 = d_lll.z.xx * d_lll.z.xx;

real tmp1459 = K_ll.xy * K_ll.xy;

real tmp1462 = gamma_uu.zz * tmp1307;

real tmp1467 = gamma_uu.zz * tmp1405;

real tmp1472 = gamma_uu.zz * tmp1447;

real tmp1476 = K_ll.xz * K_ll.xz;

real tmp1587 = K_ll.yy * gamma_uu.xy;

real tmp1589 = K_ll.yz * gamma_uu.xz;

real tmp1606 = d_lll.x.yy * gamma_uu.xy;

real tmp1609 = d_lll.x.yz * gamma_uu.xz;

real tmp1615 = d_lll.y.xz * gamma_uu.xz;

real tmp1618 = d_lll.z.xy * gamma_uu.xz;

real tmp1622 = d_lll.x.yz * gamma_uu.yz;

real tmp1628 = d_lll.y.xz * gamma_uu.yz;

real tmp1631 = d_lll.z.xy * gamma_uu.yz;

real tmp1646 = d_lll.x.yy * tmp29;

real tmp1648 = d_lll.x.yz * tmp35;

real tmp1650 = d_lll.y.xy * tmp29;

real tmp1655 = d_lll.y.xz * tmp35;

real tmp1668 = d_lll.y.zz * tmp53;

real tmp1672 = d_lll.z.xy * tmp35;

real tmp1679 = d_lll.x.yz * tmp47;

real tmp1682 = d_lll.y.xy * tmp41;

real tmp1687 = d_lll.y.xz * tmp47;

real tmp1697 = d_lll.y.yz * tmp102;

real tmp1707 = d_lll.z.xy * tmp47;

real tmp1726 = d_lll.y.xy * tmp47;

real tmp1731 = d_lll.y.xz * tmp53;

real tmp1741 = d_lll.y.yz * tmp163;

real tmp1746 = d_lll.y.zz * tmp175;

real tmp1751 = d_lll.z.xy * tmp53;

real tmp1756 = d_lll.z.yy * tmp163;

real tmp1759 = d_lll.z.yy * tmp114;

real tmp1773 = d_lll.x.zz * tmp114;

real tmp1807 = d_lll.y.xx * tmp84;

real tmp1823 = d_lll.y.yz * tmp216;

real tmp1832 = d_lll.z.xx * tmp145;

real tmp1909 = d_lll.z.xx * tmp35;

real tmp1931 = d_lll.z.xx * tmp47;

real tmp1950 = d_lll.z.xx * tmp53;

real tmp1971 = d_lll.z.xy * tmp210;

real tmp1992 = d_lll.z.xy * tmp145;

real tmp2024 = gamma_uu.yy * tmp1288;

real tmp2032 = gamma_uu.yz * tmp1447;

real tmp2176 = K_ll.xz * gamma_uu.xx;

real tmp2177 = K_ll.yz * gamma_uu.xy;

real tmp2179 = K_ll.zz * gamma_uu.xz;

real tmp2181 = K_ll.yz * gamma_uu.yy;

real tmp2183 = K_ll.zz * gamma_uu.yz;

real tmp2196 = d_lll.x.yz * gamma_uu.xy;

real tmp2199 = d_lll.x.zz * gamma_uu.xz;

real tmp2202 = d_lll.y.xz * gamma_uu.xy;

real tmp2206 = d_lll.z.xy * gamma_uu.xy;

real tmp2236 = d_lll.x.yz * tmp29;

real tmp2238 = d_lll.x.zz * tmp35;

real tmp2240 = d_lll.y.xz * tmp29;

real tmp2244 = d_lll.z.xy * tmp29;

real tmp2248 = d_lll.z.xz * tmp35;

real tmp2253 = d_lll.z.yy * tmp41;

real tmp2257 = d_lll.z.yz * tmp47;

real tmp2262 = d_lll.z.zz * tmp53;

real tmp2269 = d_lll.x.zz * tmp47;

real tmp2272 = d_lll.y.xz * tmp41;

real tmp2277 = d_lll.y.zz * tmp102;

real tmp2282 = d_lll.y.zz * tmp108;

real tmp2285 = d_lll.z.xy * tmp41;

real tmp2290 = d_lll.z.xz * tmp47;

real tmp2295 = d_lll.z.yy * tmp96;

real tmp2300 = d_lll.z.yz * tmp108;

real tmp2305 = d_lll.z.zz * tmp114;

real tmp2313 = d_lll.x.zz * tmp53;

real tmp2334 = d_lll.z.xz * tmp53;

real tmp2344 = d_lll.z.yz * tmp114;

real tmp2363 = d_lll.y.xz * tmp96;

real tmp2367 = d_lll.z.xx * tmp72;

real tmp2371 = d_lll.z.xx * tmp41;

real tmp2373 = d_lll.z.xy * tmp96;

real tmp2377 = d_lll.z.xz * tmp102;

real tmp2386 = d_lll.z.yz * tmp210;

real tmp2391 = d_lll.z.zz * tmp216;

real tmp2432 = d_lll.z.xz * tmp114;

real tmp2441 = d_lll.z.yz * tmp216;

real tmp2479 = d_lll.y.xz * tmp72;

real tmp2491 = d_lll.z.xx * tmp29;

real tmp2493 = d_lll.z.xy * tmp72;

real tmp2500 = d_lll.z.xz * tmp84;

real tmp2508 = d_lll.z.yz * tmp102;

real tmp2614 = gamma_uu.yz * tmp1405;

real tmp2619 = gamma_uu.yy * tmp1307;

real tmp2622 = gamma_uu.yy * tmp1447;

real tmp2625 = gamma_uu.zz * tmp1323;

real tmp2810 = d_lll.z.yy * gamma_uu.yz;

real tmp2853 = d_lll.y.yy * tmp72;

real tmp2858 = d_lll.y.yz * tmp84;

real tmp2863 = d_lll.z.yy * tmp84;

real tmp2884 = d_lll.z.yy * tmp145;

real tmp2897 = d_lll.y.xx * tmp29;

real tmp2935 = d_lll.y.yy * tmp108;

real tmp2940 = d_lll.y.yz * tmp114;

real tmp2986 = d_lll.z.yy * tmp175;

real tmp3006 = d_lll.z.yy * tmp47;

real tmp3117 = d_lll.y.zz * d_lll.y.zz;

real tmp3160 = d_lll.z.yy * d_lll.z.yy;

real tmp3162 = K_ll.yz * K_ll.yz;

real tmp3340 = d_lll.y.zz * gamma_uu.yz;

real tmp3377 = d_lll.y.zz * tmp84;

real tmp3387 = d_lll.z.yy * tmp72;

real tmp3417 = d_lll.x.zz * tmp84;

real tmp3630 = d_lll.y.zz * tmp210;

real tmp3748 = gamma_uu.zz * tmp3117;

real tmp3966 = d_lll.y.zz * tmp72;

real tmp3969 = d_lll.z.xz * tmp29;

real tmp3974 = d_lll.z.yz * tmp72;

real tmp3979 = d_lll.z.zz * tmp84;

real tmp3995 = d_lll.z.yz * tmp84;

real tmp4000 = d_lll.z.zz * tmp145;

real tmp4012 = d_lll.y.zz * tmp96;

real tmp4014 = d_lll.z.xz * tmp72;

real tmp4022 = d_lll.z.yz * tmp96;

real tmp4032 = d_lll.z.zz * tmp108;

real tmp4063 = d_lll.z.zz * tmp163;

real tmp4203 = d_lll.z.zz * tmp265;

F.alpha = -f * (tmp6 + tmp5 + 2. * tmp3 + 2. * tmp2 + 2. * tmp1 + K_ll.xx * gamma_uu.xx) * alpha * alpha;

F.a_l.x = -alpha * (K_ll.zz * a_l.x * tmp286 - 2. * K_ll.zz * d_lll.x.xx * tmp54 - 4. * K_ll.zz * d_lll.x.xy * tmp115 - 4. * K_ll.zz * d_lll.x.xz * tmp176 - 2. * K_ll.zz * d_lll.x.yy * tmp217 - 4. * K_ll.zz * d_lll.x.yz * tmp278 - 2. * K_ll.zz * d_lll.x.zz * tmp319 + K_ll.zz * a_l.x * tmp283 + 2. * K_ll.yz * a_l.x * tmp226 - 4. * K_ll.yz * d_lll.x.xx * tmp48 - 4. * K_ll.yz * d_lll.x.xy * tmp103 - 4. * K_ll.yz * d_lll.x.xy * tmp109 - 4. * K_ll.yz * d_lll.x.xz * tmp164 - 4. * K_ll.yz * d_lll.x.xz * tmp115 - 4. * K_ll.yz * d_lll.x.yy * tmp211 - 4. * K_ll.yz * d_lll.x.yz * tmp266 - 4. * K_ll.yz * d_lll.x.yz * tmp217 - 4. * K_ll.yz * d_lll.x.zz * tmp278 + 2. * K_ll.yz * a_l.x * tmp222 + K_ll.yy * a_l.x * tmp184 - 2. * K_ll.yy * d_lll.x.xx * tmp42 - 4. * K_ll.yy * d_lll.x.xy * tmp97 - 4. * K_ll.yy * d_lll.x.xz * tmp103 - 2. * K_ll.yy * d_lll.x.yy * tmp205 - 4. * K_ll.yy * d_lll.x.yz * tmp211 - 2. * K_ll.yy * d_lll.x.zz * tmp217 + K_ll.yy * a_l.x * tmp181 + 2. * K_ll.xz * a_l.x * tmp124 - 4. * K_ll.xz * d_lll.x.xx * tmp36 - 4. * K_ll.xz * d_lll.x.xy * tmp85 - 4. * K_ll.xz * d_lll.x.xy * tmp48 - 4. * K_ll.xz * d_lll.x.xz * tmp146 - 4. * K_ll.xz * d_lll.x.xz * tmp54 - 4. * K_ll.xz * d_lll.x.yy * tmp103 - 4. * K_ll.xz * d_lll.x.yz * tmp164 - 4. * K_ll.xz * d_lll.x.yz * tmp115 - 4. * K_ll.xz * d_lll.x.zz * tmp176 + 2. * K_ll.xz * a_l.x * tmp120 + 2. * K_ll.xy * a_l.x * tmp63 - 4. * K_ll.xy * d_lll.x.xx * tmp30 - 4. * K_ll.xy * d_lll.x.xy * tmp73 - 4. * K_ll.xy * d_lll.x.xy * tmp42 - 4. * K_ll.xy * d_lll.x.xz * tmp85 - 4. * K_ll.xy * d_lll.x.xz * tmp48 - 4. * K_ll.xy * d_lll.x.yy * tmp97 - 4. * K_ll.xy * d_lll.x.yz * tmp103 - 4. * K_ll.xy * d_lll.x.yz * tmp109 - 4. * K_ll.xy * d_lll.x.zz * tmp115 + 2. * K_ll.xy * a_l.x * tmp59 + K_ll.xx * a_l.x * tmp21 - 2. * K_ll.xx * d_lll.x.xx * tmp24 - 4. * K_ll.xx * d_lll.x.xy * tmp30 - 4. * K_ll.xx * d_lll.x.xz * tmp36 - 2. * K_ll.xx * d_lll.x.yy * tmp42 - 4. * K_ll.xx * d_lll.x.yz * tmp48 - 2. * K_ll.xx * d_lll.x.zz * tmp54 + K_ll.xx * a_l.x * tmp18);

F.a_l.y = -alpha * (K_ll.zz * a_l.y * tmp286 - 2. * K_ll.zz * d_lll.y.xx * tmp54 - 4. * K_ll.zz * d_lll.y.xy * tmp115 - 4. * K_ll.zz * d_lll.y.xz * tmp176 - 2. * K_ll.zz * d_lll.y.yy * tmp217 - 4. * K_ll.zz * d_lll.y.yz * tmp278 - 2. * K_ll.zz * d_lll.y.zz * tmp319 + K_ll.zz * a_l.y * tmp283 + 2. * K_ll.yz * a_l.y * tmp226 - 4. * K_ll.yz * d_lll.y.xx * tmp48 - 4. * K_ll.yz * d_lll.y.xy * tmp103 - 4. * K_ll.yz * d_lll.y.xy * tmp109 - 4. * K_ll.yz * d_lll.y.xz * tmp164 - 4. * K_ll.yz * d_lll.y.xz * tmp115 - 4. * K_ll.yz * d_lll.y.yy * tmp211 - 4. * K_ll.yz * d_lll.y.yz * tmp266 - 4. * K_ll.yz * d_lll.y.yz * tmp217 - 4. * K_ll.yz * d_lll.y.zz * tmp278 + 2. * K_ll.yz * a_l.y * tmp222 + K_ll.yy * a_l.y * tmp184 - 2. * K_ll.yy * d_lll.y.xx * tmp42 - 4. * K_ll.yy * d_lll.y.xy * tmp97 - 4. * K_ll.yy * d_lll.y.xz * tmp103 - 2. * K_ll.yy * d_lll.y.yy * tmp205 - 4. * K_ll.yy * d_lll.y.yz * tmp211 - 2. * K_ll.yy * d_lll.y.zz * tmp217 + K_ll.yy * a_l.y * tmp181 + 2. * K_ll.xz * a_l.y * tmp124 - 4. * K_ll.xz * d_lll.y.xx * tmp36 - 4. * K_ll.xz * d_lll.y.xy * tmp85 - 4. * K_ll.xz * d_lll.y.xy * tmp48 - 4. * K_ll.xz * d_lll.y.xz * tmp146 - 4. * K_ll.xz * d_lll.y.xz * tmp54 - 4. * K_ll.xz * d_lll.y.yy * tmp103 - 4. * K_ll.xz * d_lll.y.yz * tmp164 - 4. * K_ll.xz * d_lll.y.yz * tmp115 - 4. * K_ll.xz * d_lll.y.zz * tmp176 + 2. * K_ll.xz * a_l.y * tmp120 + 2. * K_ll.xy * a_l.y * tmp63 - 4. * K_ll.xy * d_lll.y.xx * tmp30 - 4. * K_ll.xy * d_lll.y.xy * tmp73 - 4. * K_ll.xy * d_lll.y.xy * tmp42 - 4. * K_ll.xy * d_lll.y.xz * tmp85 - 4. * K_ll.xy * d_lll.y.xz * tmp48 - 4. * K_ll.xy * d_lll.y.yy * tmp97 - 4. * K_ll.xy * d_lll.y.yz * tmp103 - 4. * K_ll.xy * d_lll.y.yz * tmp109 - 4. * K_ll.xy * d_lll.y.zz * tmp115 + 2. * K_ll.xy * a_l.y * tmp59 + K_ll.xx * a_l.y * tmp21 - 2. * K_ll.xx * d_lll.y.xx * tmp24 - 4. * K_ll.xx * d_lll.y.xy * tmp30 - 4. * K_ll.xx * d_lll.y.xz * tmp36 - 2. * K_ll.xx * d_lll.y.yy * tmp42 - 4. * K_ll.xx * d_lll.y.yz * tmp48 - 2. * K_ll.xx * d_lll.y.zz * tmp54 + K_ll.xx * a_l.y * tmp18);

F.a_l.z = -alpha * (K_ll.zz * a_l.z * tmp286 - 2. * K_ll.zz * d_lll.z.xx * tmp54 - 4. * K_ll.zz * d_lll.z.xy * tmp115 - 4. * K_ll.zz * d_lll.z.xz * tmp176 - 2. * K_ll.zz * d_lll.z.yy * tmp217 - 4. * K_ll.zz * d_lll.z.yz * tmp278 - 2. * K_ll.zz * d_lll.z.zz * tmp319 + K_ll.zz * a_l.z * tmp283 + 2. * K_ll.yz * a_l.z * tmp226 - 4. * K_ll.yz * d_lll.z.xx * tmp48 - 4. * K_ll.yz * d_lll.z.xy * tmp103 - 4. * K_ll.yz * d_lll.z.xy * tmp109 - 4. * K_ll.yz * d_lll.z.xz * tmp164 - 4. * K_ll.yz * d_lll.z.xz * tmp115 - 4. * K_ll.yz * d_lll.z.yy * tmp211 - 4. * K_ll.yz * d_lll.z.yz * tmp266 - 4. * K_ll.yz * d_lll.z.yz * tmp217 - 4. * K_ll.yz * d_lll.z.zz * tmp278 + 2. * K_ll.yz * a_l.z * tmp222 + K_ll.yy * a_l.z * tmp184 - 2. * K_ll.yy * d_lll.z.xx * tmp42 - 4. * K_ll.yy * d_lll.z.xy * tmp97 - 4. * K_ll.yy * d_lll.z.xz * tmp103 - 2. * K_ll.yy * d_lll.z.yy * tmp205 - 4. * K_ll.yy * d_lll.z.yz * tmp211 - 2. * K_ll.yy * d_lll.z.zz * tmp217 + K_ll.yy * a_l.z * tmp181 + 2. * K_ll.xz * a_l.z * tmp124 - 4. * K_ll.xz * d_lll.z.xx * tmp36 - 4. * K_ll.xz * d_lll.z.xy * tmp85 - 4. * K_ll.xz * d_lll.z.xy * tmp48 - 4. * K_ll.xz * d_lll.z.xz * tmp146 - 4. * K_ll.xz * d_lll.z.xz * tmp54 - 4. * K_ll.xz * d_lll.z.yy * tmp103 - 4. * K_ll.xz * d_lll.z.yz * tmp164 - 4. * K_ll.xz * d_lll.z.yz * tmp115 - 4. * K_ll.xz * d_lll.z.zz * tmp176 + 2. * K_ll.xz * a_l.z * tmp120 + 2. * K_ll.xy * a_l.z * tmp63 - 4. * K_ll.xy * d_lll.z.xx * tmp30 - 4. * K_ll.xy * d_lll.z.xy * tmp73 - 4. * K_ll.xy * d_lll.z.xy * tmp42 - 4. * K_ll.xy * d_lll.z.xz * tmp85 - 4. * K_ll.xy * d_lll.z.xz * tmp48 - 4. * K_ll.xy * d_lll.z.yy * tmp97 - 4. * K_ll.xy * d_lll.z.yz * tmp103 - 4. * K_ll.xy * d_lll.z.yz * tmp109 - 4. * K_ll.xy * d_lll.z.zz * tmp115 + 2. * K_ll.xy * a_l.z * tmp59 + K_ll.xx * a_l.z * tmp21 - 2. * K_ll.xx * d_lll.z.xx * tmp24 - 4. * K_ll.xx * d_lll.z.xy * tmp30 - 4. * K_ll.xx * d_lll.z.xz * tmp36 - 2. * K_ll.xx * d_lll.z.yy * tmp42 - 4. * K_ll.xx * d_lll.z.yz * tmp48 - 2. * K_ll.xx * d_lll.z.zz * tmp54 + K_ll.xx * a_l.z * tmp18);

F.gamma_ll.xx = -2. * K_ll.xx * alpha;

F.gamma_ll.xy = -2. * K_ll.xy * alpha;

F.gamma_ll.xz = -2. * K_ll.xz * alpha;

F.gamma_ll.yy = -2. * K_ll.yy * alpha;

F.gamma_ll.yz = -2. * K_ll.yz * alpha;

F.gamma_ll.zz = -2. * K_ll.zz * alpha;

F.d_lll.x.xx = -K_ll.xx * tmp1020;

F.d_lll.x.xy = -K_ll.xy * tmp1020;

F.d_lll.x.xz = -K_ll.xz * tmp1020;

F.d_lll.x.yy = -K_ll.yy * tmp1020;

F.d_lll.x.yz = -K_ll.yz * tmp1020;

F.d_lll.x.zz = -K_ll.zz * tmp1020;

F.d_lll.y.xx = -K_ll.xx * tmp1032;

F.d_lll.y.xy = -K_ll.xy * tmp1032;

F.d_lll.y.xz = -K_ll.xz * tmp1032;

F.d_lll.y.yy = -K_ll.yy * tmp1032;

F.d_lll.y.yz = -K_ll.yz * tmp1032;

F.d_lll.y.zz = -K_ll.zz * tmp1032;

F.d_lll.z.xx = -K_ll.xx * tmp1044;

F.d_lll.z.xy = -K_ll.xy * tmp1044;

F.d_lll.z.xz = -K_ll.xz * tmp1044;

F.d_lll.z.yy = -K_ll.yy * tmp1044;

F.d_lll.z.yz = -K_ll.yz * tmp1044;

F.d_lll.z.zz = -K_ll.zz * tmp1044;

F.K_ll.xx = -alpha * (2. * gamma_uu.zz * tmp1476 + 2. * gamma_uu.yy * tmp1459 - 2. * gamma_uu.yy * tmp1462 - 2. * gamma_uu.yy * tmp1467 - 2. * gamma_uu.yy * tmp1472 + gamma_uu.xx * K_ll.xx * K_ll.xx - gamma_uu.xx * gamma_uu.yy * tmp1451 - gamma_uu.xx * gamma_uu.zz * tmp1455 + 2. * tmp1447 * tmp216 + d_lll.z.xx * tmp1254 - 2. * d_lll.z.xx * tmp1259 - 2. * d_lll.z.xx * tmp1262 - d_lll.z.xx * tmp1265 + 2. * tmp1405 * tmp216 - d_lll.y.yy * d_lll.z.xx * tmp210 - 2. * d_lll.y.yz * d_lll.z.xx * tmp265 - d_lll.y.zz * d_lll.z.xx * tmp277 - 2. * d_lll.z.xx * tmp1422 - 2. * d_lll.z.xx * tmp1251 + 4. * d_lll.y.xz * tmp1398 - 4. * d_lll.y.xz * tmp1401 + 2. * d_lll.y.xz * tmp1303 - 4. * d_lll.y.xz * tmp1393 + 2. * d_lll.y.xx * tmp1364 - 2. * d_lll.y.xx * tmp1206 - d_lll.y.xx * tmp1209 - 2. * d_lll.y.xx * tmp1212 - d_lll.y.xx * tmp1215 - 2. * d_lll.y.xy * tmp1287 + d_lll.y.xx * tmp1195 - 2. * d_lll.y.xx * tmp1200 - 2. * d_lll.y.xx * tmp1354 - 4. * d_lll.y.xx * tmp1203 + d_lll.x.zz * d_lll.y.xx * tmp163 - 2. * d_lll.x.zz * d_lll.y.xx * tmp114 - d_lll.x.zz * d_lll.z.xx * tmp175 - tmp1323 * tmp318 - 2. * d_lll.y.xx * tmp1183 - 2. * d_lll.y.xx * tmp1333 - d_lll.y.xx * tmp1189 - 2. * d_lll.y.xx * tmp1192 + d_lll.x.yy * tmp1287 - tmp1288 * tmp204 - 4. * d_lll.x.yz * d_lll.x.zz * tmp277 - 2. * d_lll.x.yz * d_lll.y.xx * tmp108 - 2. * d_lll.x.yz * tmp1303 - 2. * tmp1307 * tmp216 + 2. * d_lll.x.xz * tmp1265 - 4. * d_lll.x.yy * d_lll.x.yz * tmp210 - 2. * d_lll.x.yy * d_lll.x.zz * tmp216 - d_lll.x.yy * d_lll.y.xx * tmp96 - 2. * d_lll.x.yy * d_lll.z.xx * tmp102 + 4. * d_lll.x.xz * tmp1262 + 4. * d_lll.x.xz * tmp1259 + 4. * d_lll.x.xz * tmp1251 - 2. * d_lll.x.xz * tmp1254 + 4. * d_lll.x.xz * tmp1248 + 2. * d_lll.x.xz * tmp1245 + 4. * d_lll.x.xz * tmp1242 + 2. * d_lll.x.xz * tmp1239 + 4. * d_lll.x.xz * d_lll.y.xz * tmp114 + 4. * d_lll.x.xz * tmp1233 + 2. * d_lll.x.xy * tmp1215 - 2. * d_lll.x.xz * d_lll.x.yy * tmp108 - 4. * d_lll.x.xz * d_lll.x.yz * tmp114 - 2. * d_lll.x.xz * tmp1228 + 4. * d_lll.x.xy * tmp1212 + 2. * d_lll.x.xy * tmp1209 + 4. * d_lll.x.xy * tmp1206 + 4. * d_lll.x.xy * tmp1203 + 4. * d_lll.x.xy * tmp1200 + 4. * d_lll.x.xy * tmp1192 - 2. * d_lll.x.xy * tmp1195 + 2. * d_lll.x.xy * tmp1189 + 4. * d_lll.x.xy * tmp1186 + 4. * d_lll.x.xy * tmp1183 + d_lll.x.xx * tmp1166 - 2. * d_lll.x.xy * d_lll.x.yy * tmp96 - 4. * d_lll.x.xy * tmp1173 - 2. * d_lll.x.xy * tmp1178 + 2. * d_lll.x.xx * tmp1163 + 2. * d_lll.x.xx * tmp1156 - d_lll.x.xx * tmp1159 + 2. * d_lll.x.xx * tmp1153 + 2. * d_lll.x.xx * tmp1150 + 2. * d_lll.x.xx * tmp1147 + 2. * d_lll.x.xx * tmp1140 - d_lll.x.xx * tmp1143 + d_lll.x.xx * tmp1138 + 2. * d_lll.x.xx * tmp1135 + 2. * d_lll.x.xx * tmp1132 + a_l.x * a_l.x - d_lll.x.xx * d_lll.x.yy * tmp72 - 2. * d_lll.x.xx * tmp1123 - d_lll.x.xx * tmp1128 + a_l.z * d_lll.z.xx * gamma_uu.zz + a_l.z * d_lll.y.xx * gamma_uu.yz + a_l.y * d_lll.z.xx * gamma_uu.yz - a_l.z * d_lll.x.xx * gamma_uu.xz - 2. * a_l.z * d_lll.x.xy * gamma_uu.yz - 2. * a_l.z * d_lll.x.xz * gamma_uu.zz + a_l.y * d_lll.y.xx * gamma_uu.yy + a_l.x * tmp1091 - a_l.y * d_lll.x.xx * gamma_uu.xy - 2. * a_l.y * d_lll.x.xy * gamma_uu.yy - 2. * a_l.y * d_lll.x.xz * gamma_uu.yz + a_l.x * tmp1090 + 4. * M_PI * gamma_ll.xx * rho - a_l.x * d_lll.x.xx * gamma_uu.xx - 2. * a_l.x * d_lll.x.xy * gamma_uu.xy - 2. * a_l.x * d_lll.x.xz * gamma_uu.xz + 8. * M_PI * S_ll.xx + 4. * K_ll.xy * K_ll.xz * gamma_uu.yz - 4. * M_PI * S * gamma_ll.xx + 2. * K_ll.xx * tmp2 - K_ll.xx * tmp6 - 2. * K_ll.xx * tmp3 - K_ll.xx * tmp5 + 2. * K_ll.xx * tmp1);

F.K_ll.xy = -alpha * (2. * gamma_uu.xy * tmp1467 - 2. * gamma_uu.xz * tmp2032 + 2. * gamma_uu.xy * tmp1462 + gamma_uu.xy * tmp2024 + gamma_uu.xx * gamma_uu.xy * tmp1451 + 2. * d_lll.z.xx * tmp1759 - 2. * d_lll.z.xy * tmp1251 - d_lll.z.xy * tmp1254 - 2. * d_lll.z.xy * tmp1262 - d_lll.z.xy * tmp1265 + 2. * d_lll.y.yz * tmp1303 - 2. * d_lll.y.yz * tmp1393 - 2. * d_lll.y.yz * tmp1401 - d_lll.y.zz * d_lll.z.xy * tmp277 - d_lll.z.xx * tmp1992 - 2. * d_lll.z.xx * tmp1756 + d_lll.y.xz * tmp1265 - d_lll.y.yy * tmp1971 + 2. * d_lll.y.xz * tmp1262 + d_lll.y.xz * tmp1254 + 2. * d_lll.y.xz * tmp1251 + 2. * d_lll.y.xz * tmp1248 + 2. * d_lll.y.xz * tmp1950 - 2. * d_lll.y.xz * tmp1422 + d_lll.y.xz * tmp1245 - d_lll.y.xz * tmp1832 + 2. * d_lll.y.xz * tmp1823 + d_lll.y.xz * tmp1239 + 2. * d_lll.y.xy * tmp1931 - 2. * d_lll.y.xy * tmp1203 + 2. * d_lll.y.xy * tmp1186 - 2. * d_lll.y.xy * tmp1354 + d_lll.y.xx * tmp1166 + 2. * d_lll.y.xx * tmp1163 + d_lll.y.xx * tmp1159 + 2. * d_lll.y.xx * tmp1153 + d_lll.y.xx * tmp1150 + d_lll.y.xx * tmp1909 + 2. * d_lll.y.xx * tmp1147 + 2. * d_lll.y.xx * tmp1697 - d_lll.y.xx * tmp1143 + d_lll.y.xx * tmp1138 + 2. * d_lll.y.xx * tmp1687 + d_lll.y.xx * tmp1135 + 2. * d_lll.y.xx * tmp1682 + d_lll.x.zz * d_lll.y.xx * tmp53 - 2. * d_lll.x.zz * d_lll.y.xy * tmp114 - d_lll.x.zz * d_lll.y.xz * tmp175 - d_lll.x.zz * d_lll.y.yy * tmp216 - 2. * d_lll.x.zz * d_lll.y.yz * tmp277 - d_lll.x.zz * d_lll.y.zz * tmp318 - d_lll.x.zz * d_lll.z.xy * tmp175 + d_lll.x.yz * tmp1265 - d_lll.x.zz * d_lll.y.xx * tmp145 + 2. * d_lll.x.yz * tmp1262 + 2. * d_lll.x.yz * tmp1259 + 2. * d_lll.x.yz * tmp1251 - d_lll.x.yz * tmp1254 + 2. * d_lll.x.yz * tmp1248 + d_lll.x.yz * tmp1832 - 2. * d_lll.x.yz * tmp1422 + 2. * d_lll.x.yz * d_lll.y.xx * tmp47 - 2. * d_lll.x.yz * d_lll.y.xy * tmp102 - d_lll.x.yz * tmp1239 - 2. * d_lll.x.yz * tmp1823 - d_lll.x.yz * tmp1245 + d_lll.x.yz * tmp1228 - d_lll.x.yz * tmp1807 + d_lll.x.yy * tmp1215 + 2. * d_lll.x.yy * tmp1212 + d_lll.x.yy * tmp1209 + 2. * d_lll.x.yy * tmp1206 + d_lll.x.yy * tmp1364 + d_lll.x.yy * tmp1354 + d_lll.x.yy * tmp1200 + 2. * d_lll.x.yy * tmp1186 - d_lll.x.yy * tmp1333 - d_lll.x.yy * tmp1195 + d_lll.x.yy * d_lll.y.xx * tmp41 + 2. * d_lll.x.yy * tmp1773 + d_lll.x.yy * d_lll.x.yz * tmp108 - d_lll.x.yy * tmp1178 + 2. * d_lll.x.yy * tmp1173 + 2. * d_lll.x.xz * tmp1756 - 2. * d_lll.x.xz * tmp1759 + 2. * d_lll.x.xz * d_lll.x.yz * tmp53 - 4. * d_lll.x.xz * tmp1726 - 2. * d_lll.x.xz * tmp1731 - 2. * d_lll.x.xz * d_lll.y.yy * tmp102 - 4. * d_lll.x.xz * tmp1741 - 2. * d_lll.x.xz * tmp1746 - 2. * d_lll.x.xz * tmp1751 + 2. * d_lll.x.xz * d_lll.x.yy * tmp47 + 2. * d_lll.x.xy * tmp1156 - 2. * d_lll.x.xy * tmp1159 + 2. * d_lll.x.xy * tmp1679 - 4. * d_lll.x.xy * tmp1682 - 2. * d_lll.x.xy * tmp1687 - 2. * d_lll.x.xy * tmp1138 - 4. * d_lll.x.xy * tmp1697 - 2. * d_lll.x.xy * tmp1147 - 2. * d_lll.x.xy * tmp1707 + 2. * d_lll.x.xy * d_lll.x.yy * tmp41 + d_lll.x.xx * tmp1648 - 2. * d_lll.x.xx * tmp1650 - d_lll.x.xx * tmp1655 - d_lll.x.xx * d_lll.y.yy * tmp41 - 2. * d_lll.x.xx * d_lll.y.yz * tmp47 - d_lll.x.xx * tmp1668 - d_lll.x.xx * tmp1672 + d_lll.x.xx * tmp1646 + a_l.z * d_lll.z.xy * gamma_uu.zz + a_l.y * tmp1631 - a_l.z * d_lll.x.yy * gamma_uu.yz - a_l.z * d_lll.x.yz * gamma_uu.zz - a_l.z * d_lll.y.xx * gamma_uu.xz - a_l.z * d_lll.y.xz * gamma_uu.zz + a_l.x * tmp1618 - a_l.y * d_lll.x.yy * gamma_uu.yy - a_l.y * tmp1622 - a_l.y * tmp1090 - a_l.y * tmp1628 + a_l.x * a_l.y - a_l.x * tmp1606 - a_l.x * tmp1609 - a_l.x * d_lll.y.xx * gamma_uu.xx - a_l.x * tmp1615 + 4. * M_PI * gamma_ll.xy * rho + 8. * M_PI * S_ll.xy + 2. * K_ll.xz * K_ll.yz * gamma_uu.zz - 4. * M_PI * S * gamma_ll.xy + 2. * K_ll.xz * K_ll.yy * gamma_uu.yz + K_ll.xy * tmp6 - K_ll.xy * tmp5 + 2. * K_ll.xx * tmp1589 + 2. * K_ll.xx * tmp1587 + K_ll.xx * K_ll.xy * gamma_uu.xx);

F.K_ll.xz = -alpha * (gamma_uu.xz * tmp2625 + 2. * gamma_uu.xz * tmp2622 + 2. * gamma_uu.xz * tmp2619 + gamma_uu.xx * gamma_uu.xz * tmp1455 - 2. * gamma_uu.xy * tmp2614 + d_lll.z.xy * tmp1215 + 2. * d_lll.z.xy * tmp2441 + d_lll.z.xy * tmp1209 + 2. * d_lll.z.xy * tmp2432 + d_lll.z.xx * tmp1166 + 2. * d_lll.z.xx * tmp2344 + 2. * d_lll.z.xx * tmp1156 - d_lll.z.xx * tmp1159 + 2. * d_lll.z.xx * tmp2334 + 2. * d_lll.z.xx * tmp1707 + d_lll.z.xx * tmp1150 + d_lll.y.zz * tmp1398 + d_lll.y.zz * tmp1303 + 2. * d_lll.y.yz * tmp1971 + 2. * d_lll.y.yz * tmp1287 + d_lll.y.yy * d_lll.z.xy * tmp204 + d_lll.y.yy * d_lll.z.xx * tmp96 + 2. * d_lll.y.xz * tmp1203 - 2. * d_lll.y.xz * tmp1364 - 2. * d_lll.y.xz * tmp2432 - d_lll.y.xz * tmp1209 - 2. * d_lll.y.xz * tmp2441 - d_lll.y.xz * tmp1215 + d_lll.y.xz * tmp1354 + 2. * d_lll.y.xy * tmp2373 - d_lll.y.xz * tmp1189 - 2. * d_lll.y.xz * tmp1192 - d_lll.y.xz * tmp1195 + 2. * d_lll.y.xy * tmp2367 + 2. * d_lll.y.xx * tmp2300 - 2. * d_lll.y.xy * tmp2363 + 2. * d_lll.y.xx * tmp2290 - 2. * d_lll.y.xx * tmp2508 + 2. * d_lll.y.xx * tmp2285 - 2. * d_lll.y.xx * tmp2500 + d_lll.y.xx * tmp2491 - d_lll.y.xx * tmp2493 + 2. * d_lll.y.xx * tmp2277 - 2. * d_lll.y.xx * tmp2282 + d_lll.x.zz * tmp1259 - d_lll.y.xx * tmp2479 + 2. * d_lll.x.zz * tmp1248 - d_lll.x.zz * tmp1254 + d_lll.x.zz * tmp1950 - d_lll.x.zz * tmp1422 + d_lll.x.zz * tmp1245 + 2. * d_lll.x.zz * tmp1242 + d_lll.x.zz * tmp1239 + d_lll.x.zz * d_lll.y.xz * tmp163 + 2. * d_lll.x.zz * tmp1233 + d_lll.x.zz * tmp1807 + 2. * d_lll.x.yz * tmp1931 - 2. * d_lll.x.yz * tmp2432 - d_lll.x.yz * tmp1209 - 2. * d_lll.x.yz * tmp2441 - d_lll.x.yz * tmp1215 + 2. * d_lll.x.yz * tmp1200 - d_lll.x.yz * tmp1354 + 2. * d_lll.x.yz * tmp1192 - d_lll.x.yz * tmp1195 + d_lll.x.yz * tmp1189 + 2. * d_lll.x.yz * tmp1186 - 2. * d_lll.x.yz * tmp1333 + 2. * d_lll.x.yz * tmp1183 + d_lll.x.yz * d_lll.y.xx * tmp72 + 2. * d_lll.x.yz * tmp1773 + d_lll.x.yz * tmp1178 + d_lll.x.yy * tmp2371 - d_lll.x.yy * tmp2373 - 2. * d_lll.x.yy * tmp2377 - d_lll.x.yy * d_lll.z.yy * tmp204 - 2. * d_lll.x.yy * tmp2386 - d_lll.x.yy * tmp2391 + 2. * d_lll.x.yy * d_lll.x.zz * tmp102 - d_lll.x.yy * d_lll.x.zz * tmp108 - d_lll.x.yy * tmp2363 - d_lll.x.yy * tmp2367 + d_lll.x.yy * d_lll.x.yz * tmp96 + 2. * d_lll.x.xz * tmp1147 - 2. * d_lll.x.xz * tmp1707 - 4. * d_lll.x.xz * tmp2334 - 2. * d_lll.x.xz * tmp1156 - 4. * d_lll.x.xz * tmp2344 - 2. * d_lll.x.xz * tmp1166 + 2. * d_lll.x.xz * tmp2313 - 2. * d_lll.x.xz * tmp1687 - 2. * d_lll.x.xz * tmp1143 + 2. * d_lll.x.xz * tmp1679 + 2. * d_lll.x.xy * tmp2282 - 2. * d_lll.x.xy * tmp2285 - 4. * d_lll.x.xy * tmp2290 - 2. * d_lll.x.xy * tmp2295 - 4. * d_lll.x.xy * tmp2300 - 2. * d_lll.x.xy * tmp2305 + 2. * d_lll.x.xy * tmp2269 - 2. * d_lll.x.xy * tmp2272 - 2. * d_lll.x.xy * tmp2277 + 2. * d_lll.x.xy * d_lll.x.yz * tmp41 + d_lll.x.xx * tmp2238 - d_lll.x.xx * tmp2240 - d_lll.x.xx * tmp2244 - 2. * d_lll.x.xx * tmp2248 - d_lll.x.xx * tmp2253 - 2. * d_lll.x.xx * tmp2257 - d_lll.x.xx * tmp2262 + d_lll.x.xx * tmp2236 + a_l.z * tmp1628 - a_l.z * tmp1091 - a_l.z * tmp1631 + a_l.y * d_lll.y.xz * gamma_uu.yy - a_l.y * d_lll.z.xx * gamma_uu.xy - a_l.y * d_lll.z.xy * gamma_uu.yy - a_l.z * tmp1622 - a_l.z * d_lll.x.zz * gamma_uu.zz + a_l.x * tmp2202 - a_l.x * d_lll.z.xx * gamma_uu.xx - a_l.x * tmp2206 - a_l.y * d_lll.x.yz * gamma_uu.yy - a_l.y * d_lll.x.zz * gamma_uu.yz + a_l.x * a_l.z - a_l.x * tmp2196 - a_l.x * tmp2199 + 4. * M_PI * gamma_ll.xz * rho + 8. * M_PI * S_ll.xz + K_ll.xz * tmp5 - 4. * M_PI * S * gamma_ll.xz + 2. * K_ll.xy * tmp2183 - K_ll.xz * tmp6 + 2. * K_ll.xy * tmp2181 + 2. * K_ll.xx * tmp2179 + 2. * K_ll.xx * tmp2177 + K_ll.xx * tmp2176);

F.K_ll.yy = alpha * (gamma_uu.yy * gamma_uu.zz * tmp3160 - 2. * gamma_uu.zz * tmp3162 + 2. * gamma_uu.xx * tmp1472 - gamma_uu.yy * K_ll.yy * K_ll.yy + 2. * gamma_uu.xx * tmp1467 + 2. * gamma_uu.xx * tmp1462 + gamma_uu.xx * tmp2024 + d_lll.z.yy * tmp1265 - 2. * gamma_uu.xx * tmp1459 + 2. * d_lll.z.yy * tmp1262 + 2. * d_lll.z.xz * tmp2986 + 2. * d_lll.z.xy * tmp1756 - 2. * tmp1447 * tmp53 + 2. * d_lll.z.xx * d_lll.z.yy * tmp53 + tmp3117 * tmp318 - d_lll.z.xx * tmp2884 + d_lll.y.zz * d_lll.z.yy * tmp277 + 2. * d_lll.y.yz * tmp1832 - 4. * d_lll.y.yz * tmp1950 - 4. * d_lll.y.yz * tmp1248 - 4. * d_lll.y.yz * tmp1251 - 4. * d_lll.y.yz * tmp1262 - 2. * d_lll.y.yz * tmp1265 + 2. * d_lll.y.yz * tmp1245 + d_lll.y.yy * tmp1354 - 2. * d_lll.y.yy * tmp1931 - 2. * d_lll.y.yy * tmp1364 - 2. * d_lll.y.yy * tmp1206 - 2. * d_lll.y.yy * tmp1212 - d_lll.y.yy * tmp1215 + d_lll.y.yy * tmp1195 + 2. * tmp1405 * tmp53 + 2. * d_lll.y.xz * tmp1756 + 4. * d_lll.y.xz * tmp1746 + 4. * d_lll.y.xz * tmp2940 + 2. * d_lll.y.xz * tmp2935 + 2. * d_lll.y.xy * tmp1143 - 2. * d_lll.y.xy * tmp1909 - 4. * d_lll.y.xy * tmp1707 - 4. * d_lll.y.xy * tmp1153 - 4. * d_lll.y.xy * tmp1163 - 2. * d_lll.y.xy * tmp1166 + 4. * d_lll.y.xy * tmp1687 + tmp1451 * tmp23 + 2. * d_lll.y.xx * tmp3006 + 2. * d_lll.y.xx * tmp1668 - d_lll.y.xx * tmp2863 + 2. * d_lll.y.xx * tmp2858 + d_lll.y.xx * tmp2853 + 4. * d_lll.y.xx * tmp1655 + 2. * d_lll.y.xx * tmp1650 + d_lll.x.zz * tmp2986 + d_lll.x.zz * d_lll.y.yy * tmp163 - 2. * d_lll.x.zz * d_lll.y.yy * tmp114 - 2. * d_lll.x.zz * d_lll.y.yz * tmp175 + 2. * d_lll.x.zz * d_lll.y.xy * tmp145 - 4. * d_lll.x.zz * d_lll.y.xy * tmp53 + 4. * d_lll.x.yz * tmp1759 - 2. * tmp1307 * tmp53 + 4. * d_lll.x.yz * tmp1751 - 2. * d_lll.x.yz * tmp1756 + d_lll.x.yy * tmp1166 - 4. * d_lll.x.yz * tmp1726 - 2. * d_lll.x.yz * tmp2935 - 4. * d_lll.x.yz * tmp2940 - 4. * d_lll.x.yz * tmp1992 + 2. * d_lll.x.yy * tmp1163 + 2. * d_lll.x.yy * tmp1159 + 2. * d_lll.x.yy * tmp1153 + 4. * d_lll.x.yy * tmp1707 + d_lll.x.yy * tmp1909 - 2. * d_lll.x.yy * tmp1150 + 2. * d_lll.x.yy * tmp1147 + 2. * d_lll.x.yy * tmp1135 - d_lll.x.yy * tmp1143 + d_lll.x.yy * tmp2897 + 2. * d_lll.x.yy * tmp2313 + 2. * d_lll.x.yy * tmp1123 - d_lll.x.yy * tmp1128 + 2. * d_lll.x.xz * tmp2884 + 2. * d_lll.x.xz * d_lll.x.yy * tmp35 - 4. * d_lll.x.xz * d_lll.y.xy * tmp35 - 2. * d_lll.x.xz * d_lll.y.yy * tmp84 - 4. * d_lll.x.xz * d_lll.y.yz * tmp145 + 2. * d_lll.x.xy * tmp2863 + 2. * d_lll.x.xy * tmp1646 - 4. * d_lll.x.xy * tmp1650 - 2. * d_lll.x.xy * tmp2853 - 4. * d_lll.x.xy * tmp2858 + d_lll.x.xx * d_lll.z.yy * tmp35 + d_lll.x.xx * d_lll.x.yy * tmp23 - 2. * d_lll.x.xx * d_lll.y.xy * tmp23 - d_lll.x.xx * d_lll.y.yy * tmp29 - 2. * d_lll.x.xx * d_lll.y.yz * tmp35 + 2. * a_l.z * d_lll.y.yz * gamma_uu.zz - a_l.z * d_lll.z.yy * gamma_uu.zz - a_l.y * a_l.y + a_l.z * d_lll.y.yy * gamma_uu.yz + 2. * a_l.z * d_lll.y.xy * gamma_uu.xz + 2. * a_l.y * d_lll.y.yz * gamma_uu.yz - a_l.y * tmp2810 - a_l.z * d_lll.x.yy * gamma_uu.xz + a_l.y * d_lll.y.yy * gamma_uu.yy + 2. * a_l.y * d_lll.y.xy * gamma_uu.xy + 2. * a_l.x * d_lll.y.yz * gamma_uu.xz - a_l.x * d_lll.z.yy * gamma_uu.xz - a_l.y * tmp1606 + a_l.x * d_lll.y.yy * gamma_uu.xy + 2. * a_l.x * d_lll.y.xy * gamma_uu.xx + 4. * M_PI * S * gamma_ll.yy - 8. * M_PI * S_ll.yy - 4. * M_PI * gamma_ll.yy * rho - a_l.x * d_lll.x.yy * gamma_uu.xx + K_ll.yy * tmp5 + 2. * K_ll.xz * K_ll.yy * gamma_uu.xz - 2. * K_ll.yy * tmp3 + K_ll.xx * K_ll.yy * gamma_uu.xx - 2. * K_ll.xy * tmp1587 - 4. * K_ll.xy * tmp1589);

F.K_ll.yz = alpha * (2. * gamma_uu.xy * gamma_uu.xz * tmp1307 - gamma_uu.yy * gamma_uu.yz * tmp3160 - gamma_uu.yz * tmp3748 + d_lll.z.xx * tmp2863 - 2. * d_lll.z.xx * tmp3006 - 2. * d_lll.z.xy * tmp2334 - 2. * d_lll.z.xy * tmp1156 - d_lll.z.xy * tmp1159 - 2. * d_lll.z.xy * tmp2344 - d_lll.z.xy * tmp1166 - 2. * d_lll.z.xz * tmp1759 - 2. * d_lll.z.yy * tmp2441 - d_lll.z.yy * tmp1215 - 2. * gamma_uu.xx * tmp2614 - 2. * gamma_uu.xx * tmp2032 + d_lll.y.zz * tmp1422 - 2. * d_lll.y.zz * tmp1248 - d_lll.y.zz * tmp1259 - d_lll.z.xx * tmp1672 + d_lll.y.zz * tmp1832 - d_lll.y.zz * tmp1950 + 2. * d_lll.y.yz * tmp1215 + 4. * d_lll.y.yz * tmp2441 + 4. * d_lll.y.yz * tmp2432 + 2. * d_lll.y.yz * tmp1203 + 2. * d_lll.y.yz * tmp1931 + d_lll.y.yy * tmp2391 - 2. * d_lll.y.yz * tmp1200 + 2. * d_lll.y.yy * tmp2386 + 2. * d_lll.y.yy * tmp2377 + d_lll.y.yy * tmp2373 + d_lll.y.yy * tmp2371 + d_lll.y.xz * tmp1166 - d_lll.y.yy * tmp3630 + 2. * d_lll.y.xz * tmp2344 + d_lll.y.xz * tmp1159 + 2. * d_lll.y.xz * tmp2334 - 2. * d_lll.y.xz * tmp1156 + d_lll.y.xz * tmp1909 + 2. * d_lll.y.xy * tmp2305 - d_lll.y.xz * tmp1138 - 2. * d_lll.y.xz * tmp1697 - d_lll.y.xz * tmp1143 - 2. * d_lll.y.xz * tmp1147 + 4. * d_lll.y.xy * tmp2508 + 4. * d_lll.y.xy * tmp2500 + 2. * d_lll.y.xy * tmp2285 + 2. * d_lll.y.xy * tmp2491 + d_lll.y.xx * tmp2262 - 2. * d_lll.y.xy * tmp2272 - 2. * d_lll.y.xy * tmp2277 + 2. * d_lll.y.xx * tmp2257 + d_lll.y.xx * tmp3387 - d_lll.y.xx * tmp2253 + 2. * d_lll.y.xx * tmp2248 + d_lll.y.xx * tmp2244 + d_lll.y.xx * d_lll.z.xx * tmp23 + d_lll.y.xx * tmp3377 - 2. * d_lll.y.xx * d_lll.y.zz * tmp47 + 2. * d_lll.x.zz * tmp1741 - 2. * d_lll.x.zz * tmp2940 - d_lll.x.zz * tmp1746 - d_lll.x.zz * tmp1992 - d_lll.x.zz * tmp1756 - d_lll.y.xx * tmp2240 + d_lll.x.zz * d_lll.y.xz * tmp145 - 2. * d_lll.x.zz * tmp1731 + 2. * d_lll.x.zz * tmp1726 + d_lll.x.yz * tmp1166 - 2. * d_lll.x.zz * d_lll.y.xy * tmp84 + 2. * d_lll.x.yz * tmp2344 + 2. * d_lll.x.yz * tmp2334 - d_lll.x.yz * tmp1159 + 2. * d_lll.x.yz * tmp1150 - 2. * d_lll.x.yz * tmp1707 + d_lll.x.yz * tmp1909 + 2. * d_lll.x.yz * tmp1697 - d_lll.x.yz * tmp1143 + d_lll.x.yz * tmp1138 + 2. * d_lll.x.yz * tmp1135 - 2. * d_lll.x.yz * tmp1687 + 2. * d_lll.x.yz * tmp1682 + d_lll.x.yz * tmp2897 + d_lll.x.yz * tmp1128 + 2. * d_lll.x.yy * tmp2300 + 2. * d_lll.x.yy * tmp2290 - d_lll.x.yy * tmp2295 - 2. * d_lll.x.yy * tmp2508 + d_lll.x.yy * tmp2493 - 2. * d_lll.x.yy * tmp2285 - 2. * d_lll.x.yy * tmp2500 + 2. * d_lll.x.yy * tmp3417 - 2. * d_lll.x.yy * tmp2269 - d_lll.x.yy * tmp2479 - d_lll.x.yy * tmp2282 + d_lll.x.yy * d_lll.x.yz * tmp72 + 2. * d_lll.x.xz * tmp1648 - 2. * d_lll.x.xz * tmp1655 - 2. * d_lll.x.xz * d_lll.y.zz * tmp145 - 2. * d_lll.x.xz * tmp1672 - 2. * d_lll.x.xz * tmp2863 + 2. * d_lll.x.xy * tmp2236 - 2. * d_lll.x.xy * tmp2240 - 2. * d_lll.x.xy * tmp3377 - 2. * d_lll.x.xy * tmp2244 - 2. * d_lll.x.xy * tmp3387 + d_lll.x.xx * d_lll.x.yz * tmp23 - d_lll.x.xx * d_lll.y.xz * tmp23 - d_lll.x.xx * d_lll.y.zz * tmp35 - d_lll.x.xx * d_lll.z.xy * tmp23 - d_lll.x.xx * d_lll.z.yy * tmp29 + a_l.z * tmp2810 + a_l.z * tmp1618 + a_l.z * d_lll.y.zz * gamma_uu.zz + a_l.z * tmp1615 + a_l.y * d_lll.z.yy * gamma_uu.yy - a_l.z * tmp1609 + a_l.y * tmp2206 + a_l.y * tmp3340 + a_l.y * tmp2202 + a_l.x * d_lll.z.yy * gamma_uu.xy - a_l.y * a_l.z - a_l.y * tmp2196 + a_l.x * d_lll.z.xy * gamma_uu.xx + a_l.x * d_lll.y.zz * gamma_uu.xz + a_l.x * d_lll.y.xz * gamma_uu.xx + 4. * M_PI * S * gamma_ll.yz - 8. * M_PI * S_ll.yz - 4. * M_PI * gamma_ll.yz * rho - a_l.x * d_lll.x.yz * gamma_uu.xx + K_ll.xx * K_ll.yz * gamma_uu.xx - 2. * K_ll.xy * tmp2176 - 2. * K_ll.xy * tmp2179 - 2. * K_ll.xz * tmp1587 - K_ll.yy * tmp2181 - 2. * K_ll.yy * tmp2183 - K_ll.yz * tmp5);

F.K_ll.zz = alpha * (gamma_uu.yy * tmp3748 - gamma_uu.zz * K_ll.zz * K_ll.zz + gamma_uu.xx * tmp2625 - 2. * gamma_uu.yy * tmp3162 + 2. * gamma_uu.xx * tmp2622 + 2. * gamma_uu.xx * gamma_uu.yy * tmp1405 + 2. * gamma_uu.xx * tmp2619 + tmp3160 * tmp204 - 2. * gamma_uu.xx * tmp1476 + d_lll.z.yy * tmp4203 + 2. * d_lll.z.yy * tmp2386 + 2. * d_lll.z.xz * tmp1159 + 2. * tmp1447 * tmp41 + 2. * d_lll.z.xy * tmp4063 + 4. * d_lll.z.xy * tmp2508 + 4. * d_lll.z.xy * tmp2295 + 4. * d_lll.z.xy * tmp2290 + tmp1455 * tmp23 + d_lll.z.xx * tmp4000 + 2. * d_lll.z.xx * tmp3995 + 2. * d_lll.z.xx * tmp2253 + 2. * d_lll.z.xx * tmp2248 + 4. * d_lll.z.xx * tmp2244 + d_lll.y.zz * tmp1209 + 2. * d_lll.y.zz * tmp1364 + 2. * d_lll.y.zz * tmp1931 + 2. * d_lll.y.yz * tmp3630 - 4. * d_lll.y.yz * d_lll.z.xz * tmp108 - 4. * d_lll.y.yz * tmp2386 - 2. * d_lll.y.yz * tmp4203 - d_lll.y.zz * tmp1354 + d_lll.y.yy * d_lll.y.zz * tmp204 - 2. * d_lll.y.yy * d_lll.z.xz * tmp96 - 2. * d_lll.y.yy * d_lll.z.yz * tmp204 - d_lll.y.yy * d_lll.z.zz * tmp210 + 2. * d_lll.y.xz * tmp2282 - 4. * d_lll.y.xz * tmp2290 - 4. * d_lll.y.xz * tmp2508 - 2. * d_lll.y.xz * tmp4063 - 2. * tmp1405 * tmp41 + 2. * d_lll.y.xy * tmp4012 - 4. * d_lll.y.xy * tmp4014 - 4. * d_lll.y.xy * tmp4022 - 2. * d_lll.y.xy * tmp4032 + d_lll.y.xx * tmp3979 - 2. * d_lll.y.xx * d_lll.z.zz * tmp47 + 2. * d_lll.y.xx * tmp3974 - 4. * d_lll.y.xx * d_lll.z.yz * tmp41 + 2. * d_lll.y.xx * d_lll.y.zz * tmp41 - 2. * d_lll.y.xx * tmp3969 + 2. * d_lll.x.zz * tmp1156 - d_lll.x.zz * tmp1159 - d_lll.y.xx * tmp3966 + 2. * d_lll.x.zz * tmp1150 + d_lll.x.zz * tmp1909 + 2. * d_lll.x.zz * tmp1143 + 2. * d_lll.x.zz * tmp1140 + d_lll.x.zz * tmp1138 + 4. * d_lll.x.zz * tmp1687 + 2. * d_lll.x.zz * tmp1132 - 2. * d_lll.x.zz * tmp1135 + d_lll.x.zz * tmp2897 + 4. * d_lll.x.yz * tmp2277 - 2. * d_lll.x.yz * tmp2282 - 4. * d_lll.x.yz * tmp2290 - 4. * d_lll.x.yz * tmp2508 - 2. * d_lll.x.yz * tmp4063 - 2. * tmp1307 * tmp41 + 4. * d_lll.x.yz * tmp2272 + 2. * d_lll.x.yz * tmp3417 - 4. * d_lll.x.yz * tmp2479 + d_lll.x.yy * tmp4032 + 2. * d_lll.x.yy * tmp4014 - 4. * d_lll.x.yy * d_lll.z.xz * tmp41 - 2. * d_lll.x.yy * tmp4022 - 2. * d_lll.x.yy * d_lll.z.zz * tmp102 + d_lll.x.yy * tmp4012 + 2. * d_lll.x.yy * d_lll.x.zz * tmp41 + 2. * d_lll.x.xz * tmp3377 - 4. * d_lll.x.xz * tmp2248 - 4. * d_lll.x.xz * tmp3995 - 2. * d_lll.x.xz * tmp4000 - d_lll.x.yy * d_lll.x.zz * tmp72 + 2. * d_lll.x.xz * tmp2238 + 2. * d_lll.x.xy * tmp3966 - 4. * d_lll.x.xy * tmp3969 - 4. * d_lll.x.xy * tmp3974 - 2. * d_lll.x.xy * tmp3979 + 2. * d_lll.x.xy * d_lll.x.zz * tmp29 + d_lll.x.xx * d_lll.y.zz * tmp29 - 2. * d_lll.x.xx * d_lll.z.xz * tmp23 - 2. * d_lll.x.xx * d_lll.z.yz * tmp29 - d_lll.x.xx * d_lll.z.zz * tmp35 + d_lll.x.xx * d_lll.x.zz * tmp23 + a_l.z * d_lll.z.zz * gamma_uu.zz - a_l.z * a_l.z + 2. * a_l.z * d_lll.z.yz * gamma_uu.yz + 2. * a_l.z * d_lll.z.xz * gamma_uu.xz + a_l.y * d_lll.z.zz * gamma_uu.yz - a_l.z * tmp2199 - a_l.z * tmp3340 + 2. * a_l.y * d_lll.z.yz * gamma_uu.yy + 2. * a_l.y * d_lll.z.xz * gamma_uu.xy + a_l.x * d_lll.z.zz * gamma_uu.xz - a_l.y * d_lll.x.zz * gamma_uu.xy - a_l.y * d_lll.y.zz * gamma_uu.yy + 2. * a_l.x * d_lll.z.yz * gamma_uu.xy + 2. * a_l.x * d_lll.z.xz * gamma_uu.xx + 4. * M_PI * S * gamma_ll.zz - 8. * M_PI * S_ll.zz - 4. * M_PI * gamma_ll.zz * rho - a_l.x * d_lll.x.zz * gamma_uu.xx - a_l.x * d_lll.y.zz * gamma_uu.xy + K_ll.yy * K_ll.zz * gamma_uu.yy - 2. * K_ll.yz * tmp2183 + 2. * K_ll.xy * K_ll.zz * gamma_uu.xy - 4. * K_ll.xz * tmp2177 - 2. * K_ll.xz * tmp2179 + K_ll.xx * K_ll.zz * gamma_uu.xx);

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