gauge vars
${Q} = {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{ K}_i}_l}}}$
${{ Q}^i} = {{{{\frac{-1}{\alpha}}} {{{ \beta}^k}} {{{{ b}^i}_k}}} - {{{\alpha}} \cdot {{{{ \gamma}^k}^i}} {{\left({{{{{{{{ \gamma}_j}_k}_{,l}}} {{{{ \gamma}^j}^l}}} - {{{{ \Gamma}^j}_k}_j}} - {{ a}_k}}\right)}}}}$
primitive $\partial_t$ defs
${{ \alpha}_{,t}} = {{ {-{\alpha}} {{Q}}} + {{{{ \alpha}_{,i}}} {{{ \beta}^i}}}}$
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \alpha}_{,i}}} {{{ \beta}^i}}}}$
${{{ \beta}^k}_{,t}} = {{{ B}^k} + {{{{ \beta}^i}} {{{{ \beta}^k}_{,i}}}}}$
${{{ B}^i}_{,t}} = {{{{{{\alpha}^{2}}} {{\frac{3}{4}}} {{\left({{{{{{{{{{ \Gamma}^i}_j}_k}_{,t}}} {{{{ \gamma}^j}^k}}} - {{{{{{ \Gamma}^i}^j}^k}} {{{{{ \gamma}_j}_k}_{,t}}}}} - {{{{ \beta}^l}} {{{{{{ \Gamma}^i}_j}_k}_{,l}}} {{{{ \gamma}^j}^k}}}} + {{{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{\left({{{{{ \Gamma}_j}_k}_l} + {{{{ \Gamma}_k}_j}_l}}\right)}}}}\right)}}} - {{{\frac{3}{4}}} {{{ B}^i}}}} + {{{{ \beta}^k}} {{{{ B}^i}_{,k}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{{{ \gamma}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{ K}_i}_j}_{,t}} = {{-{{{ \alpha}_{,i}}_{,j}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{ \alpha}_{,k}}}} + {{{\alpha}} \cdot {{\left({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma}_i}_j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S}_i}_j}}}}\right)}}}}$
lapse vars
${{ f}_{,k}} = {{{f'}} \cdot {{\alpha}} \cdot {{{ a}_k}}}$
hyperbolic state variables
${{ a}_k} = {{\left( \log\left( \alpha\right)\right)}_{,k}}$
${{ a}_k} = {{\frac{1}{\alpha}} {{ \alpha}_{,k}}}$
${{ \alpha}_{,k}} = {{{\alpha}} \cdot {{{ a}_k}}}$
${{{ b}^i}_j} = {{{ \beta}^i}_{,j}}$
${{{ \beta}^i}_{,j}} = {{{ b}^i}_j}$
${{{{ d}_k}_i}_j} = {{{\frac{1}{2}}} {{{{{ \gamma}_i}_j}_{,k}}}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{2}} {{{{{ d}_k}_i}_j}}}$
connections wrt aux vars
${{{{ \Gamma}^k}_i}_j} = {{{\frac{1}{2}}} {{{{ \gamma}^k}^l}} {{\left({{{{{{ \gamma}_l}_i}_{,j}} + {{{{ \gamma}_l}_j}_{,i}}} - {{{{ \gamma}_i}_j}_{,l}}}\right)}}}$
${{{{ \Gamma}_i}_j}_k} = {{{\frac{1}{2}}} {{\left({{{{{{ \gamma}_i}_j}_{,k}} + {{{{ \gamma}_i}_k}_{,j}}} - {{{{ \gamma}_j}_k}_{,i}}}\right)}}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{{{ \Gamma}_i}_j}_k} + {{{{ \Gamma}_j}_i}_k}}$
${{{{ \gamma}_i}_j}_{,k}} = {{{{{{ \gamma}_i}_a}} {{{{{ \Gamma}^a}_j}_k}}} + {{{{{ \gamma}_j}_a}} {{{{{ \Gamma}^a}_i}_k}}}}$
${{{{ \Gamma}_i}_j}_k} = {-{\left({{{{{{ d}_i}_j}_k} - {{{{ d}_j}_i}_k}} - {{{{ d}_k}_i}_j}}\right)}}$
${{{{ \Gamma}^i}_j}_k} = {{{{{ \gamma}^i}^l}} {{\left({{{{{ d}_j}_l}_k} + {{{{{ d}_k}_l}_j} - {{{{ d}_l}_j}_k}}}\right)}}}$
${\gamma^{ij}}_{,k}$ wrt aux vars
${{{{ \gamma}^i}^j}_{,k}} = { {-{{{ \gamma}^i}^l}} {{{{{ \gamma}_l}_m}_{,k}}} {{{{ \gamma}^m}^j}}}$
${{{{ \gamma}^i}^j}_{,k}} = {-{{{2}} {{{{ \gamma}^i}^l}} {{{{ \gamma}^m}^j}} {{{{{ d}_k}_l}_m}}}}$
Ricci wrt aux vars
${{{ R}_i}_j} = {{{{{{{{ \Gamma}^k}_i}_j}_{,k}} - {{{{{ \Gamma}^k}_i}_k}_{,j}}} + {{{{{{ \Gamma}^k}_l}_k}} {{{{{ \Gamma}^l}_i}_j}}}} - {{{{{{ \Gamma}^k}_l}_j}} {{{{{ \Gamma}^l}_i}_k}}}}$
${{{ R}_i}_j} = {{{{{\left( {{{{ \gamma}^k}^a}} {{\left({{{{{ d}_i}_a}_j} + {{{{{ d}_j}_a}_i} - {{{{ d}_a}_i}_j}}}\right)}}\right)}_{,k}} - {{\left( {{{{ \gamma}^k}^a}} {{\left({{{{{ d}_i}_a}_k} + {{{{{ d}_k}_a}_i} - {{{{ d}_a}_i}_k}}}\right)}}\right)}_{,j}}} + {{{{{{{ \gamma}^k}^a}} {{\left({{{{{ d}_l}_a}_k} + {{{{{ d}_k}_a}_l} - {{{{ d}_a}_l}_k}}}\right)}}}} {{{{{{ \gamma}^l}^a}} {{\left({{{{{ d}_i}_a}_j} + {{{{{ d}_j}_a}_i} - {{{{ d}_a}_i}_j}}}\right)}}}}}} - {{{{{{{ \gamma}^k}^a}} {{\left({{{{{ d}_l}_a}_j} + {{{{{ d}_j}_a}_l} - {{{{ d}_a}_l}_j}}}\right)}}}} {{{{{{ \gamma}^l}^a}} {{\left({{{{{ d}_i}_a}_k} + {{{{{ d}_k}_a}_i} - {{{{ d}_a}_i}_k}}}\right)}}}}}}$
${{{ R}_i}_j} = {{{{{{{ \gamma}^k}^a}_{,k}}} {{{{{ d}_i}_a}_j}}} + {{{{{{{ \gamma}^k}^a}_{,k}}} {{{{{ d}_j}_a}_i}}} - {{{{{{ \gamma}^k}^a}_{,k}}} {{{{{ d}_a}_i}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_j}_{,k}}}} + {{{{{{{{ \gamma}^k}^a}} {{{{{{ d}_j}_a}_i}_{,k}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_j}_{,k}}}}} - {{{{{{ \gamma}^k}^a}_{,j}}} {{{{{ d}_i}_a}_k}}}} - {{{{{{ \gamma}^k}^a}_{,j}}} {{{{{ d}_k}_a}_i}}}} + {{{{{{{{ \gamma}^k}^a}_{,j}}} {{{{{ d}_a}_i}_k}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_k}_{,j}}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_k}_a}_i}_{,j}}}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_k}_{,j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_l}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_i}_a}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_l}_a}_k}}} + {{{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_j}_a}_i}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_l}_a}_k}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_k}_a}_l}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_l}_k}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_l}_a}_j}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_k}_a}_i}} {{{{{ d}_l}_a}_j}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_l}_a}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_a}_l}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_l}} {{{{{ d}_k}_a}_i}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_a}_l}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_i}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_k}_a}_i}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_a}_i}_k}}}}}$
${{{ R}_i}_j} = {{{-{{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_i}_a}_j}} {{{{{ d}_k}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_j}_a}_i}} {{{{{ d}_k}_b}_c}}}} + {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_a}_i}_j}} {{{{{ d}_k}_b}_c}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_j}_{,k}}}} + {{{{{{ \gamma}^k}^a}} {{{{{{ d}_j}_a}_i}_{,k}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_j}_{,k}}}}} + {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_b}_c}}} + {{{{{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_j}_b}_c}} {{{{{ d}_k}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^a}} {{{{ \gamma}^k}^b}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_b}_c}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_i}_a}_k}_{,j}}}}} - {{{{{ \gamma}^k}^a}} {{{{{{ d}_k}_a}_i}_{,j}}}}} + {{{{{ \gamma}^k}^a}} {{{{{{ d}_a}_i}_k}_{,j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_l}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_j}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_i}_a}_j}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_l}_a}_k}}} + {{{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_i}} {{{{{ d}_k}_a}_l}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_k}} {{{{{ d}_j}_a}_i}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_l}_a}_k}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_k}_a}_l}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_l}_k}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_l}_a}_j}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_k}_a}_i}} {{{{{ d}_l}_a}_j}}}} + {{{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_l}_a}_j}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_a}_l}}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_j}_a}_l}} {{{{{ d}_k}_a}_i}}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_a}_l}}} + {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_i}_a}_k}}} + {{{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_k}_a}_i}}} - {{{{{ \gamma}^k}^a}} {{{{ \gamma}^l}^a}} {{{{{ d}_a}_l}_j}} {{{{{ d}_a}_i}_k}}}}}$
symmetrizing
${{{ R}_i}_j} = {{{-{{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_b}_c}_k}} {{{{{ d}_i}_a}_j}}}} - {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_b}_c}_k}} {{{{{ d}_j}_a}_i}}}} + {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_a}_i}_j}} {{{{{ d}_b}_c}_k}}} + {{{{{ \gamma}^a}^k}} {{{{{{ d}_i}_a}_j}_{,k}}}} + {{{{{{ \gamma}^a}^k}} {{{{{{ d}_j}_a}_i}_{,k}}}} - {{{{{ \gamma}^a}^k}} {{{{{{ d}_a}_i}_j}_{,k}}}}} + {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_b}_c}}} + {{{{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_j}_b}_c}} {{{{{ d}_k}_a}_i}}} - {{{2}} {{{{ \gamma}^a}^c}} {{{{ \gamma}^b}^k}} {{{{{ d}_a}_i}_k}} {{{{{ d}_j}_b}_c}}}} - {{{{{ \gamma}^a}^k}} {{{{{{ d}_i}_a}_k}_{,j}}}}} + {{{{{ \gamma}^a}^l}} {{{{ \gamma}^a}^k}} {{{{{ d}_a}_k}_l}} {{{{{ d}_i}_a}_j}}} + {{{{{{{ \gamma}^a}^k}} {{{{ \gamma}^a}^l}} {{{{{ d}_a}_k}_l}} {{{{{ d}_j}_a}_i}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^a}^l}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_k}_l}}}} - {{{{{ \gamma}^a}^k}} {{{{ \gamma}^a}^l}} {{{{{ d}_i}_a}_k}} {{{{{ d}_j}_a}_l}}}}}$
${{{ R}_i}_j} = {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}$
${{{ R}_i}_j} = {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}$
${{{ R}_i}_j} = {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}$
${{{ R}_i}_j} = {{-{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} - {{{2}} {{{{{ d}^a}_b}_i}} {{{{{ d}_j}_a}^b}}}} + {{{{{{{{ d}^a}_i}_j}} {{{{{ d}^b}_a}_b}}} - {{{{{{ d}^b}_a}_b}} {{{{{ d}_i}^a}_j}}}} - {{{{{{ d}^b}_a}_b}} {{{{{ d}_j}^a}_i}}}} + {{{2}} {{{{{ d}_a}^b}_i}} {{{{{ d}_j}^a}_b}}} + {{{{{{ d}_i}^a}_b}} {{{{{ d}_j}_a}^b}}}}$
${{{ R}_i}_j} = {{-{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} - {{{2}} {{{{{ d}^a}_b}_i}} {{{{{ d}_j}_a}^b}}}} + {{{{{{ e}_a}} {{{{{ d}^a}_i}_j}}} - {{{{ e}_a}} {{{{{ d}_i}^a}_j}}}} - {{{{ e}_a}} {{{{{ d}_j}^a}_i}}}} + {{{2}} {{{{{ d}_a}^b}_i}} {{{{{ d}_j}^a}_b}}} + {{{{{{ d}_i}^a}_b}} {{{{{ d}_j}_a}^b}}}}$
${{{ R}_i}_j} = {{-{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {{{{{{ e}_a}} {{{{{ d}^a}_i}_j}}} - {{{{ e}_a}} {{{{{ d}_i}^a}_j}}}} - {{{{ e}_a}} {{{{{ d}_j}^a}_i}}}} + {{{{{{ d}_i}^a}_b}} {{{{{ d}_j}_a}^b}}}}$
${{{ R}_i}_j} = {{{{{ \gamma}^a}^b}} {{\left({{{{{{{{ \gamma}^c}^d}} {{{{{ d}_a}_b}_d}} {{{{{ d}_c}_i}_j}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_b}_d}} {{{{{ d}_i}_c}_j}}}} - {{{{{ \gamma}^c}^d}} {{{{{ d}_a}_b}_d}} {{{{{ d}_j}_c}_i}}}} + {{{{{ d}_a}_b}_i}_{,j}} + {{{{{{{ d}_a}_b}_j}_{,i}} - {{{{{ d}_a}_i}_j}_{,b}}} - {{{{{ d}_i}_a}_b}_{,j}}} + {{{{{{ d}_i}_a}_c}} {{{{{ d}_j}_b}^c}}}}\right)}}}$
time derivative of $\alpha_{,t}$
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}}$
time derivative of $\gamma_{ij,t}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{{{ \gamma}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K}_i}_j}}} + {{{{ \beta}^k}} {{{{2}} {{{{{ d}_k}_i}_j}}}}} + {{{{{ \gamma}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{ \gamma}_i}_k}} {{{{ \beta}^k}_{,j}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{-{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}}}} + {{{2}} {{{ \beta}^k}} {{{{{ d}_k}_i}_j}}} + {{{{{ \gamma}_k}_j}} {{{{ b}^k}_i}}} + {{{{{ \gamma}_i}_k}} {{{{ b}^k}_j}}}}$
time derivative of $a_{k,t}$
${{{ a}_k}_{,t}} = {-{\frac{{{{{ \alpha}_{,k}}} {{{ \alpha}_{,t}}}} - {{{\alpha}} \cdot {{{{ \alpha}_{,k}}_{,t}}}}}{{\alpha}^{2}}}}$
${{{ a}_k}_{,t}} = {-{\frac{{{{{ \alpha}_{,k}}} {{\left({{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}}\right)}}} - {{{\alpha}} \cdot {{{\left( {-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}\right)}_{,k}}}}}{{\alpha}^{2}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{ f}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{ \alpha}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{ f}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{ \alpha}_{,k}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{\alpha}} \cdot {{{{f'}} \cdot {{\alpha}} \cdot {{{ a}_k}}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{f}} {{{{\alpha}} \cdot {{{ a}_k}}}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{{ \gamma}^i}^l}_{,k}}}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ \beta}^i}_{,k}}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ b}^i}_k}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
time derivative of ${\beta^i}_{,t}$
${{{ \beta}^k}_{,t}} = {{{ B}^k} + {{{{ \beta}^i}} {{{{ \beta}^k}_{,i}}}}}$
${{{ \beta}^k}_{,t}} = {{{ B}^k} + {{{{ \beta}^i}} {{{{ b}^k}_i}}}}$
time derivative of ${b^i}_{j,t}$
${{{{ \beta}^k}_{,t}}_{,i}} = {{{{ B}^k}_{,i}} + {{{{{ \beta}^j}_{,i}}} {{{{ b}^k}_j}}} + {{{{ \beta}^j}} {{{{{ b}^k}_j}_{,i}}}}}$
${{{{ b}^k}_i}_{,t}} = {{{{ B}^k}_{,i}} + {{{{{ b}^j}_i}} {{{{ b}^k}_j}}} + {{{{ \beta}^j}} {{{{{ b}^k}_j}_{,i}}}}}$
aux var $A^{ij}$
${{{ A}^i}^j} = {{{{ K}^i}^j} - {{{\frac{1}{3}}} {{{{ \gamma}^i}^j}} {{{{ \gamma}^k}^l}} {{{{ K}_k}_l}}}}$
time derivative of ${B^i}_{,t}$
${{{ B}^i}_{,t}} = {{{{{{\alpha}^{2}}} {{\frac{3}{4}}} {{\left({{{{{{{{{{ \Gamma}^i}_j}_k}_{,t}}} {{{{ \gamma}^j}^k}}} - {{{{{{ \Gamma}^i}^j}^k}} {{{{{ \gamma}_j}_k}_{,t}}}}} - {{{{ \beta}^l}} {{{{{{ \Gamma}^i}_j}_k}_{,l}}} {{{{ \gamma}^j}^k}}}} + {{{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{\left({{{{{ \Gamma}_j}_k}_l} + {{{{ \Gamma}_k}_j}_l}}\right)}}}}\right)}}} - {{{\frac{3}{4}}} {{{ B}^i}}}} + {{{{ \beta}^k}} {{{{ B}^i}_{,k}}}}}$
${{{ B}^i}_{,t}} = {-{{\frac{1}{4}}{\left({{{{3}} {{{ B}^i}}} + {{{{{{3}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \gamma}_j}_k}_{,t}}} {{{\alpha}^{2}}}} - {{{4}} {{{ \beta}^k}} {{{{ B}^i}_{,k}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_j}_k}_l}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_k}_j}_l}} {{{\alpha}^{2}}}}} + {{{{3}} {{{ \beta}^l}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,l}}} {{{\alpha}^{2}}}} - {{{3}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,t}}} {{{\alpha}^{2}}}}}}\right)}}}$
${{{ B}^i}_{,t}} = {-{{\frac{1}{4}}{\left({{{{3}} {{{ B}^i}}} + {{{{{{3}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \gamma}_j}_k}_{,t}}} {{{\alpha}^{2}}}} - {{{4}} {{{ \beta}^k}} {{{{ B}^i}_{,k}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_j}_k}_l}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_k}_j}_l}} {{{\alpha}^{2}}}}} + {{{{3}} {{{ \beta}^l}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,l}}} {{{\alpha}^{2}}}} - {{{3}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,t}}} {{{\alpha}^{2}}}}}}\right)}}}$
${{{ B}^i}_{,t}} = {-{{\frac{1}{4}}{\left({{{{3}} {{{ B}^i}}} + {{{{{{3}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \gamma}_j}_k}_{,t}}} {{{\alpha}^{2}}}} - {{{4}} {{{ \beta}^k}} {{{{ B}^i}_{,k}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_j}_k}_l}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_k}_j}_l}} {{{\alpha}^{2}}}}} + {{{{3}} {{{ \beta}^l}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,l}}} {{{\alpha}^{2}}}} - {{{3}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,t}}} {{{\alpha}^{2}}}}}}\right)}}}$
time derivative of $d_{kij,t}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}} {{{{{ \gamma}_i}_j}_{,k}}_{,t}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}} {{\left( {-{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}}}} + {{{2}} {{{ \beta}^l}} {{{{{ d}_l}_i}_j}}} + {{{{{ \gamma}_l}_j}} {{{{ b}^l}_i}}} + {{{{{ \gamma}_i}_l}} {{{{ b}^l}_j}}}\right)}_{,k}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {-{{\frac{1}{2}}{\left({{{{{{{{{{2}} {{{ \alpha}_{,k}}} {{{{ K}_i}_j}}} - {{{2}} {{{{ \beta}^l}_{,k}}} {{{{{ d}_l}_i}_j}}}} - {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}}} - {{{{{ \gamma}_i}_l}} {{{{{ b}^l}_j}_{,k}}}}} - {{{{{ \gamma}_l}_j}} {{{{{ b}^l}_i}_{,k}}}}} - {{{{{ b}^l}_i}} {{{{{ \gamma}_l}_j}_{,k}}}}} - {{{{{ b}^l}_j}} {{{{{ \gamma}_i}_l}_{,k}}}}} + {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}\right)}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{-{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}} + {{{2}} {{{{ b}^l}_k}} {{{{{ d}_l}_i}_j}}} + {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}} + {{{{{ \gamma}_i}_l}} {{{{{ b}^l}_j}_{,k}}}} + {{{{{ \gamma}_l}_j}} {{{{{ b}^l}_i}_{,k}}}} + {{{2}} {{{{ b}^l}_i}} {{{{{ d}_k}_l}_j}}} + {{{{2}} {{{{ b}^l}_j}} {{{{{ d}_k}_i}_l}}} - {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}}\right)}}$
$K_{ij,t}$ with hyperbolic terms
${{{{ K}_i}_j}_{,t}} = {{-{{{ \alpha}_{,i}}_{,j}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{ \alpha}_{,k}}}} + {{{\alpha}} \cdot {{\left({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma}_i}_j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S}_i}_j}}}}\right)}}}}$
${{{{ K}_i}_j}_{,t}} = {{-{{{\frac{1}{2}}} {{\left({{{\left( {{\alpha}} \cdot {{{ a}_i}}\right)}_{,j}} + {{\left( {{\alpha}} \cdot {{{ a}_j}}\right)}_{,i}}}\right)}}}} + {{{{{{ \Gamma}^k}_i}_j}} {{{{\alpha}} \cdot {{{ a}_k}}}}} + {{{\alpha}} \cdot {{\left({{{{{ R}_i}_j} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{2}} {{{{ K}_i}_k}} {{{{ \gamma}^k}^l}} {{{{ K}_j}_l}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma}_i}_j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S}_i}_j}}}}\right)}}}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{ \Gamma}^k}_i}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{{ \alpha}_{,j}}} {{{ a}_i}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{{ \alpha}_{,i}}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}\right)}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{ \Gamma}^k}_i}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}\right)}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{{ \gamma}^k}^l}} {{\left({{{{{ d}_i}_l}_j} + {{{{{ d}_j}_l}_i} - {{{{ d}_l}_i}_j}}}\right)}}}}} + {{{2}} {{\alpha}} \cdot {{{{ R}_i}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}\right)}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{{{ \gamma}^k}^l}} {{\left({{{{{ d}_i}_l}_j} + {{{{{ d}_j}_l}_i} - {{{{ d}_l}_i}_j}}}\right)}}}}} + {{{2}} {{\alpha}} \cdot {-{{{{{ \gamma}^a}^b}} {{\left({{{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_a}_i}_j}}} + {{{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}} + {{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}}} + {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}} + {{{{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}}} - {{{2}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}}} + {{{{{{{ d}_a}_i}_j}_{,b}} - {{{{{ d}_b}_a}_i}_{,j}}} - {{{{{ d}_b}_a}_j}_{,i}}} + {{{{{{{ d}_i}_a}_b}_{,j}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}}} - {{{{{ \gamma}^a}^c}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}}}}\right)}}}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}\right)}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ \gamma}^k}^l}} {{{{{ d}_i}_l}_j}}} + {{{{{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ \gamma}^k}^l}} {{{{{ d}_j}_l}_i}}} - {{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ \gamma}^k}^l}} {{{{{ d}_l}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_b}_c}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}}} + {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} + {{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_b}_a}_i}_{,j}}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_b}_a}_j}_{,i}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_k}} {{{{ K}_j}_l}} {{{{ \gamma}^k}^l}}}} + {{{2}} {{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {{{2}} {{{{ K}_k}_i}} {{{{ \beta}^k}_{,j}}}} + {{{2}} {{{{ K}_k}_j}} {{{{ \beta}^k}_{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma}_i}_j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma}_i}_j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} - {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}} - {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}}\right)}}$
${{{{ K}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_i}_b}_j}}} + {{{{{{{2}} {{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_j}_b}_i}}} - {{{2}} {{\alpha}} \cdot {{{ a}_a}} {{{{ \gamma}^a}^b}} {{{{{ d}_b}_i}_j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_i}_j}} {{{{{ d}_a}_b}_c}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^a}^c}} {{{{{ d}_i}_a}_c}} {{{{{ d}_j}_a}_b}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_d}_i}} {{{{{ d}_j}_b}_c}}}} + {{{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_a}_i}_j}} {{{{{ d}_c}_b}_d}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_i}_a}_j}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_c}_b}_d}} {{{{{ d}_j}_a}_i}}}} + {{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_d}_a}_i}} {{{{{ d}_j}_b}_c}}} + {{{{4}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{ \gamma}^c}^d}} {{{{{ d}_i}_a}_d}} {{{{{ d}_j}_b}_c}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_i}_j}_{,b}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_i}_{,j}}}} + {{{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_a}_b}_j}_{,i}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^b}} {{{{{{ d}_i}_a}_b}_{,j}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_i}_a}_j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma}^a}^c}} {{{{ \gamma}^a}^b}} {{{{{ d}_a}_b}_c}} {{{{{ d}_j}_a}_i}}} + {{{{2}} {{\alpha}} \cdot {{{{ K}_a}_b}} {{{{ K}_i}_j}} {{{{ \gamma}^a}^b}}} - {{{4}} {{\alpha}} \cdot {{{{ K}_i}_a}} {{{{ K}_j}_b}} {{{{ \gamma}^a}^b}}}} + {{{2}} {{{ \beta}^a}} {{{{{ K}_i}_j}_{,a}}}} + {{{2}} {{{{ K}_a}_i}} {{{{ b}^a}_j}}} + {{{2}} {{{{ K}_a}_j}} {{{{ b}^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}}}}}}\right)}}$
partial derivatives
${{{ \beta}^k}_{,t}} = {{{ B}^k} + {{{{ \beta}^i}} {{{{ b}^k}_i}}}}$
${{{{ b}^k}_i}_{,t}} = {{{{ B}^k}_{,i}} + {{{{{ b}^j}_i}} {{{{ b}^k}_j}}} + {{{{ \beta}^j}} {{{{{ b}^k}_j}_{,i}}}}}$
${{{ B}^i}_{,t}} = {-{{\frac{1}{4}}{\left({{{{3}} {{{ B}^i}}} + {{{{{{3}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \gamma}_j}_k}_{,t}}} {{{\alpha}^{2}}}} - {{{4}} {{{ \beta}^k}} {{{{ B}^i}_{,k}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_j}_k}_l}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_k}_j}_l}} {{{\alpha}^{2}}}}} + {{{{3}} {{{ \beta}^l}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,l}}} {{{\alpha}^{2}}}} - {{{3}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,t}}} {{{\alpha}^{2}}}}}}\right)}}}$
${{ \alpha}_{,t}} = {{-{{{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{ \beta}^i}} {{{{\alpha}} \cdot {{{ a}_i}}}}}}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {{{{{ a}_i}} {{{{ b}^i}_k}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {{-{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}}}} + {{{2}} {{{ \beta}^k}} {{{{{ d}_k}_i}_j}}} + {{{{{ \gamma}_k}_j}} {{{{ b}^k}_i}}} + {{{{{ \gamma}_i}_k}} {{{{ b}^k}_j}}}}$
${{{{{ d}_k}_i}_j}_{,t}} = {{\frac{1}{2}}{\left({{-{{{2}} {{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}} + {{{2}} {{{{ b}^l}_k}} {{{{{ d}_l}_i}_j}}} + {{{2}} {{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}} + {{{{{ \gamma}_i}_l}} {{{{{ b}^l}_j}_{,k}}}} + {{{{{ \gamma}_l}_j}} {{{{{ b}^l}_i}_{,k}}}} + {{{2}} {{{{ b}^l}_i}} {{{{{ d}_k}_l}_j}}} + {{{{2}} {{{{ b}^l}_j}} {{{{{ d}_k}_i}_l}}} - {{{2}} {{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}}\right)}}$
${{{{ K}_i}_j}_{,t}} = {{{{{{ {-{\frac{1}{2}}} {{\alpha}} \cdot {{{{ a}_i}_{,j}}}} - {{{\frac{1}{2}}} {{\alpha}} \cdot {{{{ a}_j}_{,i}}}}} + {{{\alpha}} \cdot {{\left({{{{\frac{1}{2}}} {{{{ \gamma}^p}^q}} {{\left({{{{{{ d}_i}_p}_q}_{,j}} + {{{{{ d}_j}_p}_q}_{,i}}}\right)}}} - {{{\frac{1}{2}}} {{{{ \gamma}^m}^r}} {{\left({{{{{{ d}_m}_i}_j}_{,r}} + {{{{{ d}_m}_j}_i}_{,r}}}\right)}}}}\right)}}}} - {{{\alpha}} \cdot {{{{ V}_j}_{,i}}}}} - {{{\alpha}} \cdot {{{{ V}_i}_{,j}}}}} + {{{\alpha}} \cdot {{\left({{{{{{ {-{{ a}_i}} {{{ a}_j}}} + {{{\left({{{{{{ d}_j}_i}^k} + {{{{ d}_i}_j}^k}} - {{{{ d}^k}_i}_j}}\right)}} {{\left({{{{ a}_k} + {{ V}_k}} - {{{{ d}^l}_l}_k}}\right)}}} + {{{2}} {{\left({{{{{ d}^k}^l}_j} - {{{{ d}^l}^k}_j}}\right)}} {{{{{ d}_k}_l}_i}}} + {{{2}} {{{{{ d}_i}^k}^l}} {{{{{ d}_k}_l}_j}}} + {{{2}} {{{{{ d}_j}^k}^l}} {{{{{ d}_k}_l}_i}}}} - {{{3}} {{{{{ d}_i}^k}^l}} {{{{{ d}_j}_k}_l}}}} + {{{{{ \gamma}^k}^l}} {{{{ K}_k}_l}} {{{{ K}_i}_j}}}} - {{{{{ \gamma}^k}^l}} {{{{ K}_i}_l}} {{{{ K}_k}_j}}}} - {{{{{ \gamma}^k}^l}} {{{{ K}_j}_l}} {{{{ K}_k}_i}}}}\right)}}} + {{{{{{ K}_i}_j}_{,k}}} {{{ \beta}^k}}} + {{{{{ K}_k}_j}} {{{{ b}^k}_i}}} + {{{{{ K}_i}_k}} {{{{ b}^k}_j}}}}$
${{{ V}_k}_{,t}} = {0}$
neglecting source terms
${{{ \beta}^k}_{,t}} = {0}$
${{{{ b}^k}_i}_{,t}} = {{{{ B}^k}_{,i}} + {{{{ \beta}^j}} {{{{{ b}^k}_j}_{,i}}}}}$
${{{ B}^i}_{,t}} = {{-{{\frac{1}{4}} {{{3}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \gamma}_j}_k}_{,t}}} {{{\alpha}^{2}}}}}} + {{{{ \beta}^k}} {{{{ B}^i}_{,k}}}} + {-{{\frac{1}{4}} {{{3}} {{{ \beta}^l}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,l}}} {{{\alpha}^{2}}}}}} + {{\frac{1}{4}} {{{3}} {{{{ \gamma}^j}^k}} {{{{{{ \Gamma}^i}_j}_k}_{,t}}} {{{\alpha}^{2}}}}}}$
${{ \alpha}_{,t}} = {0}$
${{{ a}_k}_{,t}} = {{{{{ \beta}^i}} {{{{ a}_i}_{,k}}}} + {-{{{\alpha}} \cdot {{f}} {{{{ \gamma}^i}^l}} {{{{{ K}_i}_l}_{,k}}}}}}$
${{{{ \gamma}_i}_j}_{,t}} = {0}$
${{{{{ d}_k}_i}_j}_{,t}} = {{{{{ \beta}^l}} {{{{{{ d}_l}_i}_j}_{,k}}}} + {{\frac{1}{2}} {{{{{ \gamma}_i}_l}} {{{{{ b}^l}_j}_{,k}}}}} + {{\frac{1}{2}} {{{{{ \gamma}_l}_j}} {{{{{ b}^l}_i}_{,k}}}}} + {-{{{\alpha}} \cdot {{{{{ K}_i}_j}_{,k}}}}}}$
${{{{ K}_i}_j}_{,t}} = {{{{{ \beta}^k}} {{{{{ K}_i}_j}_{,k}}}} + {-{{{\alpha}} \cdot {{{{ V}_i}_{,j}}}}} + {-{{{\alpha}} \cdot {{{{ V}_j}_{,i}}}}} + {-{{\frac{1}{2}} {{{\alpha}} \cdot {{{{ \gamma}^m}^r}} {{{{{{ d}_m}_i}_j}_{,r}}}}}} + {-{{\frac{1}{2}} {{{\alpha}} \cdot {{{{ \gamma}^m}^r}} {{{{{{ d}_m}_j}_i}_{,r}}}}}} + {{\frac{1}{2}} {{{\alpha}} \cdot {{{{ \gamma}^p}^q}} {{{{{{ d}_i}_p}_q}_{,j}}}}} + {{\frac{1}{2}} {{{\alpha}} \cdot {{{{ \gamma}^p}^q}} {{{{{{ d}_j}_p}_q}_{,i}}}}} + {-{{\frac{1}{2}} {{{\alpha}} \cdot {{{{ a}_j}_{,i}}}}}} + {-{{\frac{1}{2}} {{{\alpha}} \cdot {{{{ a}_i}_{,j}}}}}}}$
${{{ V}_k}_{,t}} = {0}$
...and those source terms are...
${{ \beta}^k}_{,t}$$ + \dots = $${{ B}^k} + {{{{ \beta}^i}} {{{{ b}^k}_i}}}$
${{{ b}^k}_i}_{,t}$$ + \dots = $${{{{ b}^j}_i}} {{{{ b}^k}_j}}$
${{ B}^i}_{,t}$$ + \dots = $$-{{\frac{1}{4}} {{{3}} {{\left({{{{ B}^i} - {{{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_j}_k}_l}} {{{\alpha}^{2}}}}} - {{{{ \beta}^l}} {{{{{ \Gamma}^i}^j}^k}} {{{{{ \Gamma}_k}_j}_l}} {{{\alpha}^{2}}}}}\right)}}}}$
${ \alpha}_{,t}$$ + \dots = $${{\alpha}} \cdot {{\left({{{{{ \beta}^i}} {{{ a}_i}}} - {{{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}}\right)}}$
${{ a}_k}_{,t}$$ + \dots = $${{{{{ a}_i}} {{{{ b}^i}_k}}} - {{{f'}} \cdot {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}} {{{\alpha}^{2}}}}} + {{{{2}} {{\alpha}} \cdot {{f}} {{{{ K}_i}_l}} {{{{ \gamma}^b}^l}} {{{{ \gamma}^i}^a}} {{{{{ d}_k}_a}_b}}} - {{{\alpha}} \cdot {{f}} {{{ a}_k}} {{{{ K}_i}_l}} {{{{ \gamma}^i}^l}}}}$
${{{ \gamma}_i}_j}_{,t}$$ + \dots = $${-{{{2}} {{\alpha}} \cdot {{{{ K}_i}_j}}}} + {{{2}} {{{ \beta}^k}} {{{{{ d}_k}_i}_j}}} + {{{{{ \gamma}_k}_j}} {{{{ b}^k}_i}}} + {{{{{ \gamma}_i}_k}} {{{{ b}^k}_j}}}$
${{{{ d}_k}_i}_j}_{,t}$$ + \dots = $${{{{{ b}^l}_i}} {{{{{ d}_k}_l}_j}}} + {{{{{ b}^l}_j}} {{{{{ d}_k}_i}_l}}} + {{{{{{ b}^l}_k}} {{{{{ d}_l}_i}_j}}} - {{{\alpha}} \cdot {{{ a}_k}} {{{{ K}_i}_j}}}}$
${{{ K}_i}_j}_{,t}$$ + \dots = $${-{{{\alpha}} \cdot {{{ a}_i}} {{{ a}_j}}}} + {{{\alpha}} \cdot {{{ a}_k}} {{{{{ d}_j}_i}^k}}} + {{{{\alpha}} \cdot {{{ a}_k}} {{{{{ d}_i}_j}^k}}} - {{{\alpha}} \cdot {{{ a}_k}} {{{{{ d}^k}_i}_j}}}} + {{{\alpha}} \cdot {{{ V}_k}} {{{{{ d}_j}_i}^k}}} + {{{{{{\alpha}} \cdot {{{ V}_k}} {{{{{ d}_i}_j}^k}}} - {{{\alpha}} \cdot {{{ V}_k}} {{{{{ d}^k}_i}_j}}}} - {{{\alpha}} \cdot {{{{{ d}_j}_i}^k}} {{{{{ d}^l}_l}_k}}}} - {{{\alpha}} \cdot {{{{{ d}_i}_j}^k}} {{{{{ d}^l}_l}_k}}}} + {{{\alpha}} \cdot {{{{{ d}^k}_i}_j}} {{{{{ d}^l}_l}_k}}} + {{{{2}} {{\alpha}} \cdot {{{{{ d}_k}_l}_i}} {{{{{ d}^k}^l}_j}}} - {{{2}} {{\alpha}} \cdot {{{{{ d}_k}_l}_i}} {{{{{ d}^l}^k}_j}}}} + {{{2}} {{\alpha}} \cdot {{{{{ d}_i}^k}^l}} {{{{{ d}_k}_l}_j}}} + {{{{2}} {{\alpha}} \cdot {{{{{ d}_j}^k}^l}} {{{{{ d}_k}_l}_i}}} - {{{3}} {{\alpha}} \cdot {{{{{ d}_i}^k}^l}} {{{{{ d}_j}_k}_l}}}} + {{{{{\alpha}} \cdot {{{{ K}_i}_j}} {{{{ K}_k}_l}} {{{{ \gamma}^k}^l}}} - {{{\alpha}} \cdot {{{{ K}_i}_l}} {{{{ K}_k}_j}} {{{{ \gamma}^k}^l}}}} - {{{\alpha}} \cdot {{{{ K}_j}_l}} {{{{ K}_k}_i}} {{{{ \gamma}^k}^l}}}} + {{{{{ K}_k}_j}} {{{{ b}^k}_i}}} + {{{{{ K}_i}_k}} {{{{ b}^k}_j}}}$
${{ V}_k}_{,t}$$ + \dots = $$0$
spelled out