gauge vars
${Q} = {{{\alpha}} \cdot {{f}} {{\left({{{{{{ \gamma} ^i} ^l}} {{{{ K} _i} _l}}} - {{{2}} {{\Theta}}}}\right)}}}$
${{ Q} ^i} = {{{{\frac{-1}{\alpha}}} {{{ \beta} ^k}} {{{{ b} ^i} _k}}} - {{{\alpha}} \cdot {{{{ \gamma} ^k} ^i}} {{\left({{{{{{{{ \gamma} _j} _k} _{,l}}} {{{{ \gamma} ^j} ^l}}} - {{{{ \Gamma} ^j} _k} _j}} - {{ a} _k}}\right)}}}}$
primitive $\partial_t$ defs
${{ \alpha} _{,t}} = {{ {-{\alpha}} {{Q}}} + {{{{ \alpha} _{,i}}} {{{ \beta} ^i}}}}$
${{ \alpha} _{,t}} = {{-{{{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f}} {{{\alpha}^{2}}}} + {{{{ \alpha} _{,i}}} {{{ \beta} ^i}}}}$
${{{ \beta} ^k} _{,t}} = {{{ B} ^k} + {{{{ \beta} ^i}} {{{{ \beta} ^k} _{,i}}}}}$
${{{ B} ^i} _{,t}} = {{{{{{\alpha}^{2}}} {{\frac{3}{4}}} {{\left({{{{{{{{{{ \Gamma} ^i} _j} _k} _{,t}}} {{{{ \gamma} ^j} ^k}}} - {{{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \gamma} _j} _k} _{,t}}}}} - {{{{ \beta} ^l}} {{{{{{ \Gamma} ^i} _j} _k} _{,l}}} {{{{ \gamma} ^j} ^k}}}} + {{{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{\left({{{{{ \Gamma} _j} _k} _l} + {{{{ \Gamma} _k} _j} _l}}\right)}}}}\right)}}} - {{{\frac{3}{4}}} {{{ B} ^i}}}} + {{{{ \beta} ^k}} {{{{ B} ^i} _{,k}}}}}$
${{{{ \gamma} _i} _j} _{,t}} = {{{{-2}} {{\alpha}} \cdot {{{{ K} _i} _j}}} + {{{{{{ \gamma} _i} _j} _{,k}}} {{{ \beta} ^k}}} + {{{{{ \gamma} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{{{ \gamma} _i} _k}} {{{{ \beta} ^k} _{,j}}}}}$
${{{{ K} _i} _j} _{,t}} = {{-{{{ \alpha} _{,i}} _{,j}}} + {{{{{{ \Gamma} ^k} _i} _j}} {{{ \alpha} _{,k}}}} + {{{\alpha}} \cdot {{\left({{{{{{{{{ R} _i} _j} + {{{ Z} _j} _{,i}}} - {{{{{{ \Gamma} ^k} _j} _i}} {{{ Z} _k}}}} + {{{ Z} _i} _{,j}}} - {{{{{{ \Gamma} ^k} _i} _j}} {{{ Z} _k}}}} + {{{\left({{{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}}} - {{{2}} {{\Theta}}}}\right)}} {{{{ K} _i} _j}}}} - {{{2}} {{{{ K} _i} _k}} {{{{ \gamma} ^k} ^l}} {{{{ K} _j} _l}}}}\right)}}} + {{{{{{ K} _i} _j} _{,k}}} {{{ \beta} ^k}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma} _i} _j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S} _i} _j}}}}\right)}}}}$
${{ \Theta} _{,t}} = {{{{{{{ \beta} ^k}} {{{ \Theta} _{,k}}}} - {{\left( {{\alpha}} \cdot {{\left({{{{{{ d} ^k} ^j} _j} - {{{{ d} _j} ^j} ^k}} - {{ Z} ^k}}\right)}}\right)} _{,k}}} + {{{{\frac{1}{2}} {\alpha}}} {{\left({{{{{{{2}} {{{ a} _k}} {{\left({{{{{{ d} ^k} ^j} _j} - {{{{ d} _j} ^j} ^k}} - {{{2}} {{{ Z} ^k}}}}\right)}}} + {{{{{{ d} _k} ^r} ^s}} {{{{{ \Gamma} ^k} _r} _s}}}} - {{{{{{ d} ^k} ^j} _j}} {{\left({{{{{ d} _k} _l} ^l} - {{{2}} {{{ Z} _k}}}}\right)}}}} - {{{{{ K} _k} _l}} {{{{ \gamma} ^k} ^m}} {{{{ \gamma} ^l} ^n}} {{{{ K} _m} _n}}}} + {{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}} {{\left({{{{{{ \gamma} ^m} ^n}} {{{{ K} _m} _n}}} - {{{2}} {{\Theta}}}}\right)}}}}\right)}}}} - {{{8}} {{\pi}} \cdot {{\alpha}} \cdot {{\rho}}}}$
${{{ Z} _i} _{,t}} = {{{{{{{\left( {{{ \beta} ^k}} {{{ Z} _i}}\right)} _{,k}} + {{\left( {{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{ K} _l} _i}}\right)} _{,k}}} - {{\left( {{\alpha}} \cdot {{\left({{{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}}} - {\Theta}}\right)}}\right)} _{,i}}} - {{{{ Z} _i}} {{{{ b} ^k} _k}}}} + {{{{ Z} _k}} {{{{ b} ^k} _i}}} + {{{\alpha}} \cdot {{\left({{{{{{{ a} _i}} {{\left({{{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}}} - {{{2}} {{\Theta}}}}\right)}}} - {{{{ a} _k}} {{{{ \gamma} ^k} ^l}} {{{{ K} _l} _i}}}} - {{{{{ \gamma} ^k} ^l}} {{{{ K} _l} _r}} {{{{{ \Gamma} ^r} _k} _i}}}} + {{{{{ \gamma} ^k} ^l}} {{{{ K} _l} _i}} {{\left({{{{{ d} _k} _m} ^m} - {{{2}} {{{ Z} _k}}}}\right)}}}}\right)}}}} - {{{8}} {{\pi}} \cdot {{\alpha}} \cdot {{{ S} _i}}}}$
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
${{{{ \Gamma} ^k} _i} _j} = {{{\frac{1}{2}}} {{{{ \gamma} ^k} ^l}} {{\left({{{{{{ \gamma} _l} _i} _{,j}} + {{{{ \gamma} _l} _j} _{,i}}} - {{{{ \gamma} _i} _j} _{,l}}}\right)}}}$
${{{{ \Gamma} _i} _j} _k} = {{{\frac{1}{2}}} {{\left({{{{{{ \gamma} _i} _j} _{,k}} + {{{{ \gamma} _i} _k} _{,j}}} - {{{{ \gamma} _j} _k} _{,i}}}\right)}}}$
${{{{ \gamma} _i} _j} _{,k}} = {{{{{ \Gamma} _i} _j} _k} + {{{{ \Gamma} _j} _i} _k}}$
${{{{ \gamma} _i} _j} _{,k}} = {{{{{{ \gamma} _i} _a}} {{{{{ \Gamma} ^a} _j} _k}}} + {{{{{ \gamma} _j} _a}} {{{{{ \Gamma} ^a} _i} _k}}}}$
${{{{ \Gamma} _i} _j} _k} = {{-{{{{ d} _i} _j} _k}} + {{{{ d} _j} _i} _k} + {{{{ d} _k} _i} _j}}$
${{{{ \Gamma} ^i} _j} _k} = {{{{{ \gamma} ^i} ^l}} {{\left({{{{{ d} _j} _l} _k} + {{{{{ d} _k} _l} _j} - {{{{ d} _l} _j} _k}}}\right)}}}$
${\gamma^{ij}}_{,k}$ 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
${{{ R} _i} _j} = {{{-{{{2}} {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^b} ^k}} {{{{{ d} _b} _c} _k}} {{{{{ d} _i} _a} _j}}}} - {{{2}} {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^b} ^k}} {{{{{ d} _b} _c} _k}} {{{{{ d} _j} _a} _i}}}} + {{{2}} {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^b} ^k}} {{{{{ d} _a} _i} _j}} {{{{{ d} _b} _c} _k}}} + {{{{{ \gamma} ^a} ^k}} {{{{{{ d} _i} _a} _j} _{,k}}}} + {{{{{{ \gamma} ^a} ^k}} {{{{{{ d} _j} _a} _i} _{,k}}}} - {{{{{ \gamma} ^a} ^k}} {{{{{{ d} _a} _i} _j} _{,k}}}}} + {{{2}} {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^b} ^k}} {{{{{ d} _i} _a} _k}} {{{{{ d} _j} _b} _c}}} + {{{{{2}} {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^b} ^k}} {{{{{ d} _j} _b} _c}} {{{{{ d} _k} _a} _i}}} - {{{2}} {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^b} ^k}} {{{{{ d} _a} _i} _k}} {{{{{ d} _j} _b} _c}}}} - {{{{{ \gamma} ^a} ^k}} {{{{{{ d} _i} _a} _k} _{,j}}}}} + {{{{{ \gamma} ^a} ^l}} {{{{ \gamma} ^a} ^k}} {{{{{ d} _a} _k} _l}} {{{{{ d} _i} _a} _j}}} + {{{{{{{ \gamma} ^a} ^k}} {{{{ \gamma} ^a} ^l}} {{{{{ d} _a} _k} _l}} {{{{{ d} _j} _a} _i}}} - {{{{{ \gamma} ^a} ^k}} {{{{ \gamma} ^a} ^l}} {{{{{ d} _a} _i} _j}} {{{{{ d} _a} _k} _l}}}} - {{{{{ \gamma} ^a} ^k}} {{{{ \gamma} ^a} ^l}} {{{{{ d} _i} _a} _k}} {{{{{ d} _j} _a} _l}}}}}$
${{{ R} _i} _j} = {{{{{ \gamma} ^a} ^b}} {{\left({{{{-{{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _i} _j}} {{{{{ d} _a} _b} _c}}}} - {{{{{ \gamma} ^a} ^c}} {{{{{ d} _i} _a} _c}} {{{{{ d} _j} _a} _b}}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _d} _i}} {{{{{ d} _j} _b} _c}}}} + {{{{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _i} _j}} {{{{{ d} _c} _b} _d}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _b} _d}} {{{{{ d} _i} _a} _j}}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _b} _d}} {{{{{ d} _j} _a} _i}}}} + {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _d} _a} _i}} {{{{{ d} _j} _b} _c}}} + {{{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _i} _a} _d}} {{{{{ d} _j} _b} _c}}} - {{{{{ d} _a} _i} _j} _{,b}}} + {{{{{ d} _b} _a} _i} _{,j}} + {{{{{{ d} _b} _a} _j} _{,i}} - {{{{{ d} _i} _a} _b} _{,j}}} + {{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _i} _a} _j}}} + {{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _j} _a} _i}}}}\right)}}}$
${{{ R} _i} _j} = {{{{{ \gamma} ^a} ^b}} {{\left({{{{-{{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _i} _j}} {{{{{ d} _a} _b} _c}}}} - {{{{{ \gamma} ^a} ^c}} {{{{{ d} _i} _a} _c}} {{{{{ d} _j} _a} _b}}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _d} _i}} {{{{{ d} _j} _b} _c}}}} + {{{{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _i} _j}} {{{{{ d} _c} _b} _d}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _b} _d}} {{{{{ d} _i} _a} _j}}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _b} _d}} {{{{{ d} _j} _a} _i}}}} + {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _d} _a} _i}} {{{{{ d} _j} _b} _c}}} + {{{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _i} _a} _d}} {{{{{ d} _j} _b} _c}}} - {{{{{ d} _a} _i} _j} _{,b}}} + {{{{{ d} _b} _a} _i} _{,j}} + {{{{{{ d} _b} _a} _j} _{,i}} - {{{{{ d} _i} _a} _b} _{,j}}} + {{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _i} _a} _j}}} + {{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _j} _a} _i}}}}\right)}}}$
${{{ R} _i} _j} = {{{{{ \gamma} ^a} ^b}} {{\left({{{{-{{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _i} _j}} {{{{{ d} _a} _b} _c}}}} - {{{{{ \gamma} ^a} ^c}} {{{{{ d} _i} _a} _c}} {{{{{ d} _j} _a} _b}}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _d} _i}} {{{{{ d} _j} _b} _c}}}} + {{{{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _i} _j}} {{{{{ d} _c} _b} _d}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _b} _d}} {{{{{ d} _i} _a} _j}}}} - {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _b} _d}} {{{{{ d} _j} _a} _i}}}} + {{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _d} _a} _i}} {{{{{ d} _j} _b} _c}}} + {{{{2}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _i} _a} _d}} {{{{{ d} _j} _b} _c}}} - {{{{{ d} _a} _i} _j} _{,b}}} + {{{{{ d} _b} _a} _i} _{,j}} + {{{{{{ d} _b} _a} _j} _{,i}} - {{{{{ d} _i} _a} _b} _{,j}}} + {{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _i} _a} _j}}} + {{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _j} _a} _i}}}}\right)}}}$
${{{ R} _i} _j} = {{{{{{{ d} ^a} _i} _j}} {{{{{ d} ^b} _a} _b}}} + {{{{{{{{{ d} _i} ^a} _b}} {{{{{ d} _j} _a} ^b}}} - {{{2}} {{{{{ d} ^a} _b} _i}} {{{{{ d} _j} _a} ^b}}}} - {{{{{{ d} ^b} _a} _b}} {{{{{ d} _i} ^a} _j}}}} - {{{{{{ d} ^b} _a} _b}} {{{{{ d} _j} ^a} _i}}}} + {{{{2}} {{{{{ d} _a} ^b} _i}} {{{{{ d} _j} ^a} _b}}} - {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _i} _j} _{,b}}}}} + {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _i} _{,j}}}} + {{{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _j} _{,i}}}} - {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _i} _a} _b} _{,j}}}}}}$
${{{ R} _i} _j} = {{{{{ e} _a}} {{{{{ d} ^a} _i} _j}}} + {{{{{{{{{ d} _i} ^a} _b}} {{{{{ d} _j} _a} ^b}}} - {{{2}} {{{{{ d} ^a} _b} _i}} {{{{{ d} _j} _a} ^b}}}} - {{{{ e} _a}} {{{{{ d} _i} ^a} _j}}}} - {{{{ e} _a}} {{{{{ d} _j} ^a} _i}}}} + {{{{2}} {{{{{ d} _a} ^b} _i}} {{{{{ d} _j} ^a} _b}}} - {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _i} _j} _{,b}}}}} + {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _i} _{,j}}}} + {{{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _j} _{,i}}}} - {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _i} _a} _b} _{,j}}}}}}$
${{{ R} _i} _j} = {{{{{ e} _a}} {{{{{ d} ^a} _i} _j}}} + {{{{{{{{{ d} _i} ^a} _b}} {{{{{ d} _j} _a} ^b}}} - {{{{ e} _a}} {{{{{ d} _i} ^a} _j}}}} - {{{{ e} _a}} {{{{{ d} _j} ^a} _i}}}} - {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _i} _j} _{,b}}}}} + {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _i} _{,j}}}} + {{{{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _j} _{,i}}}} - {{{{{ \gamma} ^a} ^b}} {{{{{{ d} _i} _a} _b} _{,j}}}}}}$
${{{ R} _i} _j} = {{{{{ \gamma} ^a} ^b}} {{\left({{{{{{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _b} _d}} {{{{{ d} _c} _i} _j}}} - {{{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _b} _d}} {{{{{ d} _i} _c} _j}}}} - {{{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _b} _d}} {{{{{ d} _j} _c} _i}}}} + {{{{{ d} _a} _b} _i} _{,j}} + {{{{{{{ d} _a} _b} _j} _{,i}} - {{{{{ d} _a} _i} _j} _{,b}}} - {{{{{ d} _i} _a} _b} _{,j}}} + {{{{{{ d} _i} _a} _c}} {{{{{ d} _j} _b} ^c}}}}\right)}}}$
time derivative of $\alpha_{,t}$
${{ \alpha} _{,t}} = {{-{{{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f}} {{{\alpha}^{2}}}} + {{{{ \beta} ^i}} {{{{\alpha}} \cdot {{{ a} _i}}}}}}$
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 $a_{k,t}$
${{{ a} _k} _{,t}} = {\frac{{{{\alpha}} \cdot {{{{ \alpha} _{,k}} _{,t}}}} - {{{{ \alpha} _{,k}}} {{{ \alpha} _{,t}}}}}{{\alpha}^{2}}}$
${{{ a} _k} _{,t}} = {\frac{{{{\alpha}} \cdot {{{\left( {-{{{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f}} {{{\alpha}^{2}}}} + {{{{ \beta} ^i}} {{{{\alpha}} \cdot {{{ a} _i}}}}}\right)} _{,k}}}} - {{{{ \alpha} _{,k}}} {{\left({{-{{{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f}} {{{\alpha}^{2}}}} + {{{{ \beta} ^i}} {{{{\alpha}} \cdot {{{ a} _i}}}}}}\right)}}}}{{\alpha}^{2}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ \beta} ^i} _{,k}}}} - {{{\alpha}} \cdot {{{ f} _{,k}}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{ f} _{,k}}}} + {{{{2}} {{\Theta}} \cdot {{f}} {{{ \alpha} _{,k}}}} - {{{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{{ \gamma} ^i} ^l} _{,k}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{f}} {{{ \alpha} _{,k}}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ \beta} ^i} _{,k}}}} - {{{\alpha}} \cdot {{{ f} _{,k}}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{ f} _{,k}}}} + {{{{2}} {{\Theta}} \cdot {{f}} {{{ \alpha} _{,k}}}} - {{{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{{ \gamma} ^i} ^l} _{,k}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{f}} {{{ \alpha} _{,k}}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ \beta} ^i} _{,k}}}} - {{{\alpha}} \cdot {{{{f'}} \cdot {{\alpha}} \cdot {{{ a} _k}}}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{f'}} \cdot {{\alpha}} \cdot {{{ a} _k}}}}} + {{{{2}} {{\Theta}} \cdot {{f}} {{{{\alpha}} \cdot {{{ a} _k}}}}} - {{{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{{ \gamma} ^i} ^l} _{,k}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{f}} {{{{\alpha}} \cdot {{{ a} _k}}}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ \beta} ^i} _{,k}}}} - {{{f'}} \cdot {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f'}} \cdot {{{ a} _k}} {{{\alpha}^{2}}}} + {{{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{f}} {{{ a} _k}}} - {{{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{{ \gamma} ^i} ^l} _{,k}}}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{\alpha}} \cdot {{f}} {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ \beta} ^i} _{,k}}}} - {{{f'}} \cdot {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f'}} \cdot {{{ a} _k}} {{{\alpha}^{2}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{f}} {{{ a} _k}}} + {{{2}} {{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^b} ^l}} {{{{ \gamma} ^i} ^a}} {{{{{ d} _k} _a} _b}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{\alpha}} \cdot {{f}} {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ \beta} ^i} _{,k}}}} - {{{f'}} \cdot {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f'}} \cdot {{{ a} _k}} {{{\alpha}^{2}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{f}} {{{ a} _k}}} + {{{2}} {{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^b} ^l}} {{{{ \gamma} ^i} ^a}} {{{{{ d} _k} _a} _b}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{\alpha}} \cdot {{f}} {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ b} ^i} _k}}} - {{{f'}} \cdot {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f'}} \cdot {{{ a} _k}} {{{\alpha}^{2}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{f}} {{{ a} _k}}} + {{{2}} {{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^b} ^l}} {{{{ \gamma} ^i} ^a}} {{{{{ d} _k} _a} _b}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{\alpha}} \cdot {{f}} {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
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 ${B^i}_{,t}$
${{{ B} ^i} _{,t}} = {{{{{{\alpha}^{2}}} {{\frac{3}{4}}} {{\left({{{{{{{{{{ \Gamma} ^i} _j} _k} _{,t}}} {{{{ \gamma} ^j} ^k}}} - {{{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \gamma} _j} _k} _{,t}}}}} - {{{{ \beta} ^l}} {{{{{{ \Gamma} ^i} _j} _k} _{,l}}} {{{{ \gamma} ^j} ^k}}}} + {{{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{\left({{{{{ \Gamma} _j} _k} _l} + {{{{ \Gamma} _k} _j} _l}}\right)}}}}\right)}}} - {{{\frac{3}{4}}} {{{ B} ^i}}}} + {{{{ \beta} ^k}} {{{{ B} ^i} _{,k}}}}}$
${{{ B} ^i} _{,t}} = {{\frac{1}{4}}{\left({{{{{4}} {{{ \beta} ^k}} {{{{ B} ^i} _{,k}}}} - {{{3}} {{{ B} ^i}}}} + {{{{{3}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,t}}} {{{\alpha}^{2}}}} - {{{3}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta} ^l}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,l}}} {{{\alpha}^{2}}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _j} _k} _l}} {{{\alpha}^{2}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _k} _j} _l}} {{{\alpha}^{2}}}}}\right)}}$
${{{ B} ^i} _{,t}} = {{\frac{1}{4}}{\left({{{{{4}} {{{ \beta} ^k}} {{{{ B} ^i} _{,k}}}} - {{{3}} {{{ B} ^i}}}} + {{{{{3}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,t}}} {{{\alpha}^{2}}}} - {{{3}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta} ^l}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,l}}} {{{\alpha}^{2}}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _j} _k} _l}} {{{\alpha}^{2}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _k} _j} _l}} {{{\alpha}^{2}}}}}\right)}}$
${{{ B} ^i} _{,t}} = {{\frac{1}{4}}{\left({{{{{4}} {{{ \beta} ^k}} {{{{ B} ^i} _{,k}}}} - {{{3}} {{{ B} ^i}}}} + {{{{{3}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,t}}} {{{\alpha}^{2}}}} - {{{3}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta} ^l}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,l}}} {{{\alpha}^{2}}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _j} _k} _l}} {{{\alpha}^{2}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _k} _j} _l}} {{{\alpha}^{2}}}}}\right)}}$
time derivative of $d_{kij,t}$
${{{{{ d} _k} _i} _j} _{,t}} = {{\frac{1}{2}} {{{{{ \gamma} _i} _j} _{,k}} _{,t}}}$
${{{{{ d} _k} _i} _j} _{,t}} = {{\frac{1}{2}} {{\left( {-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}} + {{{2}} {{{ \beta} ^l}} {{{{{ d} _l} _i} _j}}} + {{{{{ \gamma} _l} _j}} {{{{ b} ^l} _i}}} + {{{{{ \gamma} _i} _l}} {{{{ b} ^l} _j}}}\right)} _{,k}}}$
${{{{{ d} _k} _i} _j} _{,t}} = {{\frac{1}{2}}{\left({{{-{{{2}} {{{ \alpha} _{,k}}} {{{{ K} _i} _j}}}} - {{{2}} {{\alpha}} \cdot {{{{{ K} _i} _j} _{,k}}}}} + {{{2}} {{{{ \beta} ^l} _{,k}}} {{{{{ d} _l} _i} _j}}} + {{{2}} {{{ \beta} ^l}} {{{{{{ d} _l} _i} _j} _{,k}}}} + {{{{{ b} ^l} _i}} {{{{{ \gamma} _l} _j} _{,k}}}} + {{{{{ \gamma} _l} _j}} {{{{{ b} ^l} _i} _{,k}}}} + {{{{{ b} ^l} _j}} {{{{{ \gamma} _i} _l} _{,k}}}} + {{{{{ \gamma} _i} _l}} {{{{{ b} ^l} _j} _{,k}}}}}\right)}}$
${{{{{ d} _k} _i} _j} _{,t}} = {{\frac{1}{2}}{\left({{{-{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{ K} _i} _j}}}} - {{{2}} {{\alpha}} \cdot {{{{{ K} _i} _j} _{,k}}}}} + {{{2}} {{{{ b} ^l} _k}} {{{{{ d} _l} _i} _j}}} + {{{2}} {{{ \beta} ^l}} {{{{{{ d} _l} _i} _j} _{,k}}}} + {{{2}} {{{{ b} ^l} _i}} {{{{{ d} _k} _l} _j}}} + {{{{{ \gamma} _l} _j}} {{{{{ b} ^l} _i} _{,k}}}} + {{{2}} {{{{ b} ^l} _j}} {{{{{ d} _k} _i} _l}}} + {{{{{ \gamma} _i} _l}} {{{{{ b} ^l} _j} _{,k}}}}}\right)}}$
$K_{ij,t}$ with hyperbolic terms
${{{{ K} _i} _j} _{,t}} = {{-{{{ \alpha} _{,i}} _{,j}}} + {{{{{{ \Gamma} ^k} _i} _j}} {{{ \alpha} _{,k}}}} + {{{\alpha}} \cdot {{\left({{{{{{{{{ R} _i} _j} + {{{ Z} _j} _{,i}}} - {{{{{{ \Gamma} ^k} _j} _i}} {{{ Z} _k}}}} + {{{ Z} _i} _{,j}}} - {{{{{{ \Gamma} ^k} _i} _j}} {{{ Z} _k}}}} + {{{\left({{{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}}} - {{{2}} {{\Theta}}}}\right)}} {{{{ K} _i} _j}}}} - {{{2}} {{{{ K} _i} _k}} {{{{ \gamma} ^k} ^l}} {{{{ K} _j} _l}}}}\right)}}} + {{{{{{ K} _i} _j} _{,k}}} {{{ \beta} ^k}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma} _i} _j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S} _i} _j}}}}\right)}}}}$
${{{{ K} _i} _j} _{,t}} = {{-{{{\frac{1}{2}}} {{\left({{{\left( {{\alpha}} \cdot {{{ a} _i}}\right)} _{,j}} + {{\left( {{\alpha}} \cdot {{{ a} _j}}\right)} _{,i}}}\right)}}}} + {{{{{{ \Gamma} ^k} _i} _j}} {{{{\alpha}} \cdot {{{ a} _k}}}}} + {{{\alpha}} \cdot {{\left({{{{{{{{{ R} _i} _j} + {{{ Z} _j} _{,i}}} - {{{{{{ \Gamma} ^k} _j} _i}} {{{ Z} _k}}}} + {{{ Z} _i} _{,j}}} - {{{{{{ \Gamma} ^k} _i} _j}} {{{ Z} _k}}}} + {{{\left({{{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}}} - {{{2}} {{\Theta}}}}\right)}} {{{{ K} _i} _j}}}} - {{{2}} {{{{ K} _i} _k}} {{{{ \gamma} ^k} ^l}} {{{{ K} _j} _l}}}}\right)}}} + {{{{{{ K} _i} _j} _{,k}}} {{{ \beta} ^k}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma} _i} _j}} {{\left({{S} - {\rho}}\right)}}} - {{{2}} {{{{ S} _i} _j}}}}\right)}}}}$
${{{{ K} _i} _j} _{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{{ \Gamma} ^k} _i} _j}}} + {{{2}} {{\alpha}} \cdot {{{{ R} _i} _j}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _j} _{,i}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{ \Gamma} ^k} _j} _i}}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{ \Gamma} ^k} _i} _j}}}} + {{{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}} - {{{4}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _i} _j}}}} - {{{4}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _j} _l}} {{{{ \gamma} ^k} ^l}}}} + {{{2}} {{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}}} + {{{2}} {{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{2}} {{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}}} - {{{{ \alpha} _{,j}}} {{{ a} _i}}}} - {{{\alpha}} \cdot {{{{ a} _i} _{,j}}}}} - {{{{ \alpha} _{,i}}} {{{ a} _j}}}} - {{{\alpha}} \cdot {{{{ a} _j} _{,i}}}}}}\right)}}$
${{{{ K} _i} _j} _{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{{ \Gamma} ^k} _i} _j}}} + {{{2}} {{\alpha}} \cdot {{{{ R} _i} _j}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _j} _{,i}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{ \Gamma} ^k} _j} _i}}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{ \Gamma} ^k} _i} _j}}}} + {{{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}} - {{{4}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _i} _j}}}} - {{{4}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _j} _l}} {{{{ \gamma} ^k} ^l}}}} + {{{2}} {{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}}} + {{{2}} {{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{2}} {{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}}} - {{{2}} {{\alpha}} \cdot {{{ a} _i}} {{{ a} _j}}}} - {{{\alpha}} \cdot {{{{ a} _i} _{,j}}}}} - {{{\alpha}} \cdot {{{{ a} _j} _{,i}}}}}}\right)}}$
${{{{ K} _i} _j} _{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{{{ \gamma} ^k} ^a}} {{\left({{{{{ d} _i} _a} _j} + {{{{{ d} _j} _a} _i} - {{{{ d} _a} _i} _j}}}\right)}}}}} + {{{2}} {{\alpha}} \cdot {{{{ R} _i} _j}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _j} _{,i}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{{ \gamma} ^k} ^a}} {{\left({{{{{ d} _j} _a} _i} + {{{{{ d} _i} _a} _j} - {{{{ d} _a} _j} _i}}}\right)}}}}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{{ \gamma} ^k} ^a}} {{\left({{{{{ d} _i} _a} _j} + {{{{{ d} _j} _a} _i} - {{{{ d} _a} _i} _j}}}\right)}}}}}} + {{{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}} - {{{4}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _i} _j}}}} - {{{4}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _j} _l}} {{{{ \gamma} ^k} ^l}}}} + {{{2}} {{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}}} + {{{2}} {{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{2}} {{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{{{{{{8}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}} - {{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}}} - {{{2}} {{\alpha}} \cdot {{{ a} _i}} {{{ a} _j}}}} - {{{\alpha}} \cdot {{{{ a} _i} _{,j}}}}} - {{{\alpha}} \cdot {{{{ a} _j} _{,i}}}}}}\right)}}$
${{{{ K} _i} _j} _{,t}} = {{\frac{1}{2}}{\left({{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{{{ \gamma} ^k} ^a}} {{\left({{{{{ d} _i} _a} _j} + {{{{{ d} _j} _a} _i} - {{{{ d} _a} _i} _j}}}\right)}}}}} + {{{2}} {{\alpha}} \cdot {{{{{{ \gamma} ^a} ^b}} {{\left({{{{-{{{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _i} _j}} {{{{{ d} _a} _b} _c}}}} - {{{{{ \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 {{{{ Z} _j} _{,i}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{{ \gamma} ^k} ^a}} {{\left({{{{{ d} _j} _a} _i} + {{{{{ d} _i} _a} _j} - {{{{ d} _a} _j} _i}}}\right)}}}}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{{{ \gamma} ^k} ^a}} {{\left({{{{{ d} _i} _a} _j} + {{{{{ d} _j} _a} _i} - {{{{ d} _a} _i} _j}}}\right)}}}}}} + {{{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}} - {{{4}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _i} _j}}}} - {{{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} ^a}} {{{{{ d} _i} _a} _j}}} + {{{{{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{ \gamma} ^k} ^a}} {{{{{ d} _j} _a} _i}}} - {{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{ \gamma} ^k} ^a}} {{{{{ d} _a} _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 {{{{ Z} _j} _{,i}}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ \gamma} ^k} ^a}} {{{{{ d} _j} _a} _i}}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ \gamma} ^k} ^a}} {{{{{ d} _i} _a} _j}}}} + {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ \gamma} ^k} ^a}} {{{{{ d} _a} _j} _i}}} + {{{2}} {{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} + {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ \gamma} ^k} ^a}} {{{{{ d} _a} _i} _j}}} + {{{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}} - {{{4}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _i} _j}}}} - {{{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} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _a} _i} _j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _i} _a} _b}} {{{{{ d} _j} _a} _c}}}} - {{{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} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _i} _a} _j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _j} _a} _i}}} + {{{{{2}} {{\alpha}} \cdot {{{{ Z} _j} _{,i}}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _j} _b} _i}}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _i} _b} _j}}}} + {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _b} _i} _j}}} + {{{2}} {{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} + {{{{{2}} {{\alpha}} \cdot {{{{ K} _a} _b}} {{{{ K} _i} _j}} {{{{ \gamma} ^a} ^b}}} - {{{4}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _i} _j}}}} - {{{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)}}$
Z4 terms
${{ Q} ^i} = {{{{\frac{-1}{\alpha}}} {{{ \beta} ^k}} {{{{ b} ^i} _k}}} - {{{\alpha}} \cdot {{{{ \gamma} ^k} ^i}} {{\left({{{{{{{{ \gamma} _j} _k} _{,l}}} {{{{ \gamma} ^j} ^l}}} - {{{{ \Gamma} ^j} _k} _j}} - {{ a} _k}}\right)}}}}$
${{ Q} ^i} = {{{{\frac{-1}{\alpha}}} {{{ \beta} ^k}} {{{{ b} ^i} _k}}} - {{{\alpha}} \cdot {{{{ \gamma} ^k} ^i}} {{\left({{{{{{{2}} {{{{{ d} _l} _j} _k}}}} {{{{ \gamma} ^j} ^l}}} - {{{{{ \gamma} ^j} ^a}} {{\left({{{{{ d} _k} _a} _j} + {{{{{ d} _j} _a} _k} - {{{{ d} _a} _k} _j}}}\right)}}}} - {{ a} _k}}\right)}}}}$
${{{{ b} ^i} _k} _{,t}} = {{{-{\left({{-{{\left( {{{ \beta} ^j}} {{{{ b} ^i} _k}}\right)} _{,j}}} + {{\left( {{{\alpha}} \cdot {{{ Q} ^i}}} + {{{{ \beta} ^j}} {{{{ b} ^i} _j}}}\right)} _{,k}}}\right)}} + {{{{{ b} ^i} _j}} {{{{ b} ^j} _k}}}} - {{{{{ b} ^j} _j}} {{{{ b} ^i} _k}}}}$
${{{{ b} ^i} _k} _{,t}} = {{{{{{ \beta} ^j} _{,j}}} {{{{ b} ^i} _k}}} + {{{{{{{{ \beta} ^j}} {{{{{ b} ^i} _k} _{,j}}}} - {{{{ Q} ^i}} {{{ \alpha} _{,k}}}}} - {{{\alpha}} \cdot {{{{ Q} ^i} _{,k}}}}} - {{{{{ \beta} ^j} _{,k}}} {{{{ b} ^i} _j}}}} - {{{{ \beta} ^j}} {{{{{ b} ^i} _j} _{,k}}}}} + {{{{{{ b} ^i} _j}} {{{{ b} ^j} _k}}} - {{{{{ b} ^i} _k}} {{{{ b} ^j} _j}}}}}$
${{ {{ b} ^i} _k} _{,t}} = {{{{{ { \beta} ^j} _{,j}}} {{{{ b} ^i} _k}}} + {{{{{{{{ \beta} ^j}} {{{ {{ b} ^i} _k} _{,j}}}} - {{{\left({{{{\frac{-1}{\alpha}}} {{{ \beta} ^a}} {{{{ b} ^i} _a}}} - {{{\alpha}} \cdot {{{{ \gamma} ^a} ^i}} {{\left({{{{{{{2}} {{{{{ d} _b} _c} _a}}}} {{{{ \gamma} ^c} ^b}}} - {{{{{ \gamma} ^c} ^d}} {{\left({{{{{ d} _a} _d} _c} + {{{{{ d} _c} _d} _a} - {{{{ d} _d} _a} _c}}}\right)}}}} - {{ a} _a}}\right)}}}}\right)}} {{{ \alpha} _{,k}}}}} - {{{\alpha}} \cdot {{{\left( {{{\frac{-1}{\alpha}}} {{{ \beta} ^a}} {{{{ b} ^i} _a}}} - {{{\alpha}} \cdot {{{{ \gamma} ^a} ^i}} {{\left({{{{{{{2}} {{{{{ d} _b} _c} _a}}}} {{{{ \gamma} ^c} ^b}}} - {{{{{ \gamma} ^c} ^d}} {{\left({{{{{ d} _a} _d} _c} + {{{{{ d} _c} _d} _a} - {{{{ d} _d} _a} _c}}}\right)}}}} - {{ a} _a}}\right)}}}\right)} _{,k}}}}} - {{{{ { \beta} ^j} _{,k}}} {{{{ b} ^i} _j}}}} - {{{{ \beta} ^j}} {{{ {{ b} ^i} _j} _{,k}}}}} + {{{{{{ b} ^i} _j}} {{{{ b} ^j} _k}}} - {{{{{ b} ^i} _k}} {{{{ b} ^j} _j}}}}}$
${{{{ b} ^i} _k} _{,t}} = {{{{{{4}} {{\alpha}} \cdot {{{ \alpha} _{,k}}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^b}} {{{{{ d} _b} _c} _a}}} - {{{2}} {{\alpha}} \cdot {{{ \alpha} _{,k}}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _d} _c}}}} - {{{2}} {{\alpha}} \cdot {{{ \alpha} _{,k}}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _d} _a}}}} + {{{{2}} {{\alpha}} \cdot {{{ \alpha} _{,k}}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _d} _a} _c}}} - {{{2}} {{\alpha}} \cdot {{{ \alpha} _{,k}}} {{{ a} _a}} {{{{ \gamma} ^a} ^i}}}} + {{{2}} {{{{ \gamma} ^a} ^i}} {{{{{ \gamma} ^c} ^b} _{,k}}} {{{{{ d} _b} _c} _a}} {{{\alpha}^{2}}}} + {{{{{2}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^b}} {{{{{{ d} _b} _c} _a} _{,k}}} {{{\alpha}^{2}}}} - {{{{{ \gamma} ^a} ^i}} {{{{{ \gamma} ^c} ^d} _{,k}}} {{{{{ d} _a} _d} _c}} {{{\alpha}^{2}}}}} - {{{{{ \gamma} ^a} ^i}} {{{{{ \gamma} ^c} ^d} _{,k}}} {{{{{ d} _c} _d} _a}} {{{\alpha}^{2}}}}} + {{{{{{{ \gamma} ^a} ^i}} {{{{{ \gamma} ^c} ^d} _{,k}}} {{{{{ d} _d} _a} _c}} {{{\alpha}^{2}}}} - {{{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{{ d} _a} _d} _c} _{,k}}} {{{\alpha}^{2}}}}} - {{{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{{ d} _c} _d} _a} _{,k}}} {{{\alpha}^{2}}}}} + {{{{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{{ d} _d} _a} _c} _{,k}}} {{{\alpha}^{2}}}} - {{{{{ \gamma} ^a} ^i}} {{{{ a} _a} _{,k}}} {{{\alpha}^{2}}}}} + {{{{{2}} {{{{ \gamma} ^c} ^b}} {{{{{ \gamma} ^a} ^i} _{,k}}} {{{{{ d} _b} _c} _a}} {{{\alpha}^{2}}}} - {{{{{ \gamma} ^c} ^d}} {{{{{ \gamma} ^a} ^i} _{,k}}} {{{{{ d} _a} _d} _c}} {{{\alpha}^{2}}}}} - {{{{{ \gamma} ^c} ^d}} {{{{{ \gamma} ^a} ^i} _{,k}}} {{{{{ d} _c} _d} _a}} {{{\alpha}^{2}}}}} + {{{{{{ \gamma} ^c} ^d}} {{{{{ \gamma} ^a} ^i} _{,k}}} {{{{{ d} _d} _a} _c}} {{{\alpha}^{2}}}} - {{{{ a} _a}} {{{{{ \gamma} ^a} ^i} _{,k}}} {{{\alpha}^{2}}}}} + {{{{{ \beta} ^a} _{,k}}} {{{{ b} ^i} _a}}} + {{{{ \beta} ^a}} {{{{{ b} ^i} _a} _{,k}}}} + {{{{{{{ \beta} ^j} _{,j}}} {{{{ b} ^i} _k}}} - {{{{{ \beta} ^j} _{,k}}} {{{{ b} ^i} _j}}}} - {{{{ \beta} ^j}} {{{{{ b} ^i} _j} _{,k}}}}} + {{{{ \beta} ^j}} {{{{{ b} ^i} _k} _{,j}}}} + {{{{{{ b} ^i} _j}} {{{{ b} ^j} _k}}} - {{{{{ b} ^i} _k}} {{{{ b} ^j} _j}}}}}$
${{{{ b} ^i} _k} _{,t}} = {{{{{{4}} {{{ a} _k}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^b} ^c}} {{{{{ d} _b} _c} _a}} {{{\alpha}^{2}}}} - {{{2}} {{{ a} _k}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _a} _c} _d}} {{{\alpha}^{2}}}}} - {{{2}} {{{ a} _k}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _d} _a}} {{{\alpha}^{2}}}}} + {{{{{2}} {{{ a} _k}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{ d} _c} _a} _d}} {{{\alpha}^{2}}}} - {{{2}} {{{ a} _a}} {{{ a} _k}} {{{{ \gamma} ^a} ^i}} {{{\alpha}^{2}}}}} - {{{4}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^b} ^f}} {{{{ \gamma} ^c} ^e}} {{{{{ d} _b} _c} _a}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}}} + {{{2}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^b} ^c}} {{{{{{ d} _b} _c} _a} _{,k}}} {{{\alpha}^{2}}}} + {{{2}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^e}} {{{{ \gamma} ^d} ^f}} {{{{{ d} _a} _d} _c}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}} + {{{{{{2}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^e}} {{{{ \gamma} ^d} ^f}} {{{{{ d} _c} _d} _a}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}} - {{{2}} {{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^e}} {{{{ \gamma} ^d} ^f}} {{{{{ d} _d} _a} _c}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}}} - {{{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{{ d} _a} _c} _d} _{,k}}} {{{\alpha}^{2}}}}} - {{{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{{ d} _c} _d} _a} _{,k}}} {{{\alpha}^{2}}}}} + {{{{{{{ \gamma} ^a} ^i}} {{{{ \gamma} ^c} ^d}} {{{{{{ d} _c} _a} _d} _{,k}}} {{{\alpha}^{2}}}} - {{{{{ \gamma} ^a} ^i}} {{{{ a} _a} _{,k}}} {{{\alpha}^{2}}}}} - {{{4}} {{{{ \gamma} ^a} ^e}} {{{{ \gamma} ^b} ^c}} {{{{ \gamma} ^f} ^i}} {{{{{ d} _b} _c} _a}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}}} + {{{2}} {{{{ \gamma} ^a} ^e}} {{{{ \gamma} ^c} ^d}} {{{{ \gamma} ^f} ^i}} {{{{{ d} _a} _c} _d}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}} + {{{{2}} {{{{ \gamma} ^a} ^e}} {{{{ \gamma} ^c} ^d}} {{{{ \gamma} ^f} ^i}} {{{{{ d} _c} _d} _a}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}} - {{{2}} {{{{ \gamma} ^a} ^e}} {{{{ \gamma} ^c} ^d}} {{{{ \gamma} ^f} ^i}} {{{{{ d} _c} _a} _d}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}}} + {{{2}} {{{ a} _a}} {{{{ \gamma} ^a} ^e}} {{{{ \gamma} ^f} ^i}} {{{{{ d} _k} _e} _f}} {{{\alpha}^{2}}}} + {{{{{ b} ^a} _k}} {{{{ b} ^i} _a}}} + {{{{{ \beta} ^a}} {{{{{ b} ^i} _a} _{,k}}}} - {{{{ \beta} ^j}} {{{{{ b} ^i} _j} _{,k}}}}} + {{{{ \beta} ^j}} {{{{{ b} ^i} _k} _{,j}}}}}$
${{ \Theta} _{,t}} = {{\frac{1}{2}}{\left({{-{{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}} + {{{{2}} {{{ \Theta} _{,k}}} {{{ \beta} ^k}}} - {{{2}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}}}} + {{{2}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^k}} {{{{{ d} _j} _d} _e}}} + {{{{{{2}} {{{ Z} _a}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^a}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^k} ^b}} {{{{{ \gamma} ^j} ^c} _{,k}}} {{{{{ d} _b} _c} _j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^j} ^c}} {{{{{ \gamma} ^k} ^b} _{,k}}} {{{{{ d} _b} _c} _j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{{ d} _b} _c} _j} _{,k}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^e} ^k}} {{{{{ \gamma} ^d} ^j} _{,k}}} {{{{{ d} _j} _d} _e}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^j}} {{{{{ \gamma} ^e} ^k} _{,k}}} {{{{{ d} _j} _d} _e}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^k}} {{{{{{ d} _j} _d} _e} _{,k}}}} + {{{2}} {{\alpha}} \cdot {{{{ Z} _a} _{,k}}} {{{{ \gamma} ^k} ^a}}} + {{{2}} {{\alpha}} \cdot {{{ Z} _a}} {{{{{ \gamma} ^k} ^a} _{,k}}}} + {{{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}}} - {{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^k}} {{{{{ d} _j} _d} _e}}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{ a} _k}} {{{{ \gamma} ^k} ^a}}}} + {{{{\alpha}} \cdot {{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{{ \Gamma} ^k} _r} _s}} {{{{{ d} _k} _d} _e}}} - {{{\alpha}} \cdot {{{{ \gamma} ^f} ^l}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _l} _f}}}} + {{{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}}} - {{{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^m}} {{{{ \gamma} ^l} ^n}}}} + {{{{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^m} ^n}}} - {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}}}}\right)}}$
${{ \Theta} _{,t}} = {{\frac{1}{2}}{\left({{-{{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}} + {{{{2}} {{{ \Theta} _{,k}}} {{{ \beta} ^k}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _a}} {{{ a} _k}} {{{{ \gamma} ^k} ^a}}}} + {{{4}} {{\alpha}} \cdot {{{{ \gamma} ^h} ^c}} {{{{ \gamma} ^j} ^g}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _g} _h}}} + {{{{{{4}} {{\alpha}} \cdot {{{{ \gamma} ^h} ^b}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^g}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _g} _h}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{{ d} _b} _c} _j} _{,k}}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^g}} {{{{ \gamma} ^e} ^k}} {{{{ \gamma} ^h} ^j}} {{{{{ d} _j} _d} _e}} {{{{{ d} _k} _g} _h}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^g}} {{{{ \gamma} ^h} ^k}} {{{{{ d} _j} _d} _e}} {{{{{ d} _k} _g} _h}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^k}} {{{{{{ d} _j} _d} _e} _{,k}}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _a} _{,k}}} {{{{ \gamma} ^k} ^a}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^h} ^a}} {{{{ \gamma} ^k} ^g}} {{{{{ d} _k} _g} _h}}}} + {{{\alpha}} \cdot {{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _k} _d} _e}} {{{{{ d} _r} _i} _s}}} + {{{{{\alpha}} \cdot {{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _k} _d} _e}} {{{{{ d} _s} _i} _r}}} - {{{\alpha}} \cdot {{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _i} _r} _s}} {{{{{ d} _k} _d} _e}}}} - {{{\alpha}} \cdot {{{{ \gamma} ^f} ^l}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _l} _f}}}} + {{{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}}} - {{{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^m}} {{{{ \gamma} ^l} ^n}}}} + {{{{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^m} ^n}}} - {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}}}}\right)}}$
${{{ Z} _i} _{,t}} = {{{{{{{\left( {{{ \beta} ^k}} {{{ Z} _i}}\right)} _{,k}} + {{\left( {{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{ K} _l} _i}}\right)} _{,k}}} - {{\left( {{\alpha}} \cdot {{\left({{{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}}} - {\Theta}}\right)}}\right)} _{,i}}} - {{{{ Z} _i}} {{{{ b} ^k} _k}}}} + {{{{ Z} _k}} {{{{ b} ^k} _i}}} + {{{\alpha}} \cdot {{\left({{{{{{{ a} _i}} {{\left({{{{{{ \gamma} ^k} ^l}} {{{{ K} _k} _l}}} - {{{2}} {{\Theta}}}}\right)}}} - {{{{ a} _k}} {{{{ \gamma} ^k} ^l}} {{{{ K} _l} _i}}}} - {{{{{ \gamma} ^k} ^l}} {{{{ K} _l} _r}} {{{{{ \Gamma} ^r} _k} _i}}}} + {{{{{ \gamma} ^k} ^l}} {{{{ K} _l} _i}} {{\left({{{{{ d} _k} _m} ^m} - {{{2}} {{{ Z} _k}}}}\right)}}}}\right)}}}} - {{{8}} {{\pi}} \cdot {{\alpha}} \cdot {{{ S} _i}}}}$
${{{ Z} _i} _{,t}} = {{{{{ \beta} ^k}} {{{{ Z} _i} _{,k}}}} + {{{{ Z} _i}} {{{{ \beta} ^k} _{,k}}}} + {{{{ \alpha} _{,k}}} {{{{ K} _l} _i}} {{{{ \gamma} ^k} ^l}}} + {{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _l} _i} _{,k}}}} + {{{{\alpha}} \cdot {{{{ K} _l} _i}} {{{{{ \gamma} ^k} ^l} _{,k}}}} - {{{{ \alpha} _{,i}}} {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}}} + {{{{{\Theta}} \cdot {{{ \alpha} _{,i}}}} - {{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _k} _l} _{,i}}}}} - {{{\alpha}} \cdot {{{{ K} _k} _l}} {{{{{ \gamma} ^k} ^l} _{,i}}}}} + {{{{\alpha}} \cdot {{{ \Theta} _{,i}}}} - {{{{ Z} _i}} {{{{ b} ^k} _k}}}} + {{{{ Z} _k}} {{{{ b} ^k} _i}}} + {{{{{{\alpha}} \cdot {{{ a} _i}} {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}} - {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{ a} _i}}}} - {{{\alpha}} \cdot {{{ a} _k}} {{{{ K} _l} _i}} {{{{ \gamma} ^k} ^l}}}} - {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{{ \Gamma} ^r} _k} _i}}}} + {{{{{\alpha}} \cdot {{{{ K} _l} _i}} {{{{ \gamma} ^a} ^m}} {{{{ \gamma} ^k} ^l}} {{{{{ d} _k} _m} _a}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ K} _l} _i}} {{{{ \gamma} ^k} ^l}}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{{ S} _i}}}}}$
${{{ Z} _i} _{,t}} = {{{{{ \beta} ^k}} {{{{ Z} _i} _{,k}}}} + {{{{{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _l} _i} _{,k}}}} - {{{2}} {{\alpha}} \cdot {{{{ K} _l} _i}} {{{{ \gamma} ^c} ^l}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _k} _b} _c}}}} - {{{\Theta}} \cdot {{\alpha}} \cdot {{{ a} _i}}}} - {{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _k} _l} _{,i}}}}} + {{{2}} {{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ \gamma} ^c} ^l}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _i} _b} _c}}} + {{{\alpha}} \cdot {{{ \Theta} _{,i}}}} + {{{{{{ Z} _k}} {{{{ b} ^k} _i}}} - {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _k} _d} _i}}}} - {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _i} _d} _k}}}} + {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _d} _k} _i}}} + {{{{{\alpha}} \cdot {{{{ K} _l} _i}} {{{{ \gamma} ^a} ^m}} {{{{ \gamma} ^k} ^l}} {{{{{ d} _k} _m} _a}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ K} _l} _i}} {{{{ \gamma} ^k} ^l}}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{{ S} _i}}}}}$
partial derivatives
${{{ \beta} ^k} _{,t}} = {{{ B} ^k} + {{{{ \beta} ^i}} {{{{ b} ^k} _i}}}}$
${{{{ b} ^k} _i} _{,t}} = {{{{ B} ^k} _{,i}} + {{{{{ b} ^j} _i}} {{{{ b} ^k} _j}}} + {{{{ \beta} ^j}} {{{{{ b} ^k} _j} _{,i}}}}}$
${{{ B} ^i} _{,t}} = {{\frac{1}{4}}{\left({{{{{4}} {{{ \beta} ^k}} {{{{ B} ^i} _{,k}}}} - {{{3}} {{{ B} ^i}}}} + {{{{{3}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,t}}} {{{\alpha}^{2}}}} - {{{3}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{{\alpha}^{2}}}}} - {{{3}} {{{ \beta} ^l}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,l}}} {{{\alpha}^{2}}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _j} _k} _l}} {{{\alpha}^{2}}}} + {{{3}} {{{ \beta} ^l}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \Gamma} _k} _j} _l}} {{{\alpha}^{2}}}}}\right)}}$
${{ \alpha} _{,t}} = {{-{{{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f}} {{{\alpha}^{2}}}} + {{{{ \beta} ^i}} {{{{\alpha}} \cdot {{{ a} _i}}}}}}$
${{{{ \gamma} _i} _j} _{,t}} = {{-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}} + {{{2}} {{{ \beta} ^k}} {{{{{ d} _k} _i} _j}}} + {{{{{ \gamma} _k} _j}} {{{{ b} ^k} _i}}} + {{{{{ \gamma} _i} _k}} {{{{ b} ^k} _j}}}}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{{{ a} _i}} {{{{ b} ^i} _k}}} - {{{f'}} \cdot {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f'}} \cdot {{{ a} _k}} {{{\alpha}^{2}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{f}} {{{ a} _k}}} + {{{2}} {{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^b} ^l}} {{{{ \gamma} ^i} ^a}} {{{{{ d} _k} _a} _b}}} + {{{{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} - {{{\alpha}} \cdot {{f}} {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}} - {{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{{{ d} _k} _i} _j} _{,t}} = {{\frac{1}{2}}{\left({{{-{{{2}} {{\alpha}} \cdot {{{ a} _k}} {{{{ K} _i} _j}}}} - {{{2}} {{\alpha}} \cdot {{{{{ K} _i} _j} _{,k}}}}} + {{{2}} {{{{ b} ^l} _k}} {{{{{ d} _l} _i} _j}}} + {{{2}} {{{ \beta} ^l}} {{{{{{ d} _l} _i} _j} _{,k}}}} + {{{2}} {{{{ b} ^l} _i}} {{{{{ d} _k} _l} _j}}} + {{{{{ \gamma} _l} _j}} {{{{{ b} ^l} _i} _{,k}}}} + {{{2}} {{{{ b} ^l} _j}} {{{{{ d} _k} _i} _l}}} + {{{{{ \gamma} _i} _l}} {{{{{ b} ^l} _j} _{,k}}}}}\right)}}$
${{{{ 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} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _a} _i} _j}}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _i} _a} _b}} {{{{{ d} _j} _a} _c}}}} - {{{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} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _i} _a} _j}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _j} _a} _i}}} + {{{{{2}} {{\alpha}} \cdot {{{{ Z} _j} _{,i}}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _j} _b} _i}}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _i} _b} _j}}}} + {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _b} _i} _j}}} + {{{2}} {{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} + {{{{{2}} {{\alpha}} \cdot {{{{ K} _a} _b}} {{{{ K} _i} _j}} {{{{ \gamma} ^a} ^b}}} - {{{4}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _i} _j}}}} - {{{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)}}$
${{ \Theta} _{,t}} = {{\frac{1}{2}}{\left({{-{{{16}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}} + {{{{2}} {{{ \Theta} _{,k}}} {{{ \beta} ^k}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _a}} {{{ a} _k}} {{{{ \gamma} ^k} ^a}}}} + {{{4}} {{\alpha}} \cdot {{{{ \gamma} ^h} ^c}} {{{{ \gamma} ^j} ^g}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _g} _h}}} + {{{{{{4}} {{\alpha}} \cdot {{{{ \gamma} ^h} ^b}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^g}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _g} _h}}} - {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{{ d} _b} _c} _j} _{,k}}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^g}} {{{{ \gamma} ^e} ^k}} {{{{ \gamma} ^h} ^j}} {{{{{ d} _j} _d} _e}} {{{{{ d} _k} _g} _h}}}} - {{{4}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^g}} {{{{ \gamma} ^h} ^k}} {{{{{ d} _j} _d} _e}} {{{{{ d} _k} _g} _h}}}} + {{{2}} {{\alpha}} \cdot {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^k}} {{{{{{ d} _j} _d} _e} _{,k}}}} + {{{{2}} {{\alpha}} \cdot {{{{ Z} _a} _{,k}}} {{{{ \gamma} ^k} ^a}}} - {{{4}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^h} ^a}} {{{{ \gamma} ^k} ^g}} {{{{{ d} _k} _g} _h}}}} + {{{\alpha}} \cdot {{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _k} _d} _e}} {{{{{ d} _r} _i} _s}}} + {{{{{\alpha}} \cdot {{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _k} _d} _e}} {{{{{ d} _s} _i} _r}}} - {{{\alpha}} \cdot {{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _i} _r} _s}} {{{{{ d} _k} _d} _e}}}} - {{{\alpha}} \cdot {{{{ \gamma} ^f} ^l}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _l} _f}}}} + {{{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}}} - {{{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^m}} {{{{ \gamma} ^l} ^n}}}} + {{{{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^m} ^n}}} - {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}}}}\right)}}$
${{{ Z} _i} _{,t}} = {{{{{ \beta} ^k}} {{{{ Z} _i} _{,k}}}} + {{{{{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _l} _i} _{,k}}}} - {{{2}} {{\alpha}} \cdot {{{{ K} _l} _i}} {{{{ \gamma} ^c} ^l}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _k} _b} _c}}}} - {{{\Theta}} \cdot {{\alpha}} \cdot {{{ a} _i}}}} - {{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _k} _l} _{,i}}}}} + {{{2}} {{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ \gamma} ^c} ^l}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _i} _b} _c}}} + {{{\alpha}} \cdot {{{ \Theta} _{,i}}}} + {{{{{{ Z} _k}} {{{{ b} ^k} _i}}} - {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _k} _d} _i}}}} - {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _i} _d} _k}}}} + {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _d} _k} _i}}} + {{{{{\alpha}} \cdot {{{{ K} _l} _i}} {{{{ \gamma} ^a} ^m}} {{{{ \gamma} ^k} ^l}} {{{{{ d} _k} _m} _a}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ K} _l} _i}} {{{{ \gamma} ^k} ^l}}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{{ S} _i}}}}}$
neglecting source terms
${{{ \beta} ^k} _{,t}} = {0}$
${{{{ b} ^k} _i} _{,t}} = {{{{ B} ^k} _{,i}} + {{{{ \beta} ^j}} {{{{{ b} ^k} _j} _{,i}}}}}$
${{{ B} ^i} _{,t}} = {{{{{ \beta} ^k}} {{{{ B} ^i} _{,k}}}} + {{\frac{1}{4}} {{{3}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,t}}} {{{\alpha}^{2}}}}} + {-{{\frac{1}{4}} {{{3}} {{{{{ \Gamma} ^i} ^j} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{{\alpha}^{2}}}}}} + {-{{\frac{1}{4}} {{{3}} {{{ \beta} ^l}} {{{{ \gamma} ^j} ^k}} {{{{{{ \Gamma} ^i} _j} _k} _{,l}}} {{{\alpha}^{2}}}}}}}$
${{ \alpha} _{,t}} = {0}$
${{{{ \gamma} _i} _j} _{,t}} = {0}$
${{{ a} _k} _{,t}} = {{{{{ \beta} ^i}} {{{{ a} _i} _{,k}}}} + {{{2}} {{\alpha}} \cdot {{f}} {{{ \Theta} _{,k}}}} + {-{{{\alpha}} \cdot {{f}} {{{{ \gamma} ^i} ^l}} {{{{{ K} _i} _l} _{,k}}}}}}$
${{{{{ d} _k} _i} _j} _{,t}} = {{-{{{\alpha}} \cdot {{{{{ K} _i} _j} _{,k}}}}} + {{{{ \beta} ^l}} {{{{{{ d} _l} _i} _j} _{,k}}}} + {{\frac{1}{2}} {{{{{ \gamma} _l} _j}} {{{{{ b} ^l} _i} _{,k}}}}} + {{\frac{1}{2}} {{{{{ \gamma} _i} _l}} {{{{{ b} ^l} _j} _{,k}}}}}}$
${{{{ K} _i} _j} _{,t}} = {{-{{{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _i} _j} _{,b}}}}} + {{{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _i} _{,j}}}} + {{{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{{{ d} _a} _b} _j} _{,i}}}} + {-{{{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{{{ d} _i} _a} _b} _{,j}}}}} + {{{\alpha}} \cdot {{{{ Z} _j} _{,i}}}} + {{{\alpha}} \cdot {{{{ Z} _i} _{,j}}}} + {{{{ \beta} ^a}} {{{{{ K} _i} _j} _{,a}}}} + {-{{\frac{1}{2}} {{{\alpha}} \cdot {{{{ a} _i} _{,j}}}}}} + {-{{\frac{1}{2}} {{{\alpha}} \cdot {{{{ a} _j} _{,i}}}}}}}$
${{ \Theta} _{,t}} = {{{{{ \Theta} _{,k}}} {{{ \beta} ^k}}} + {{{\alpha}} \cdot {{{{ Z} _a} _{,k}}} {{{{ \gamma} ^k} ^a}}} + {{{\alpha}} \cdot {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^k}} {{{{{{ d} _j} _d} _e} _{,k}}}} + {-{{{\alpha}} \cdot {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{{ d} _b} _c} _j} _{,k}}}}}}$
${{{ Z} _i} _{,t}} = {{{{{ \beta} ^k}} {{{{ Z} _i} _{,k}}}} + {{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _l} _i} _{,k}}}} + {-{{{\alpha}} \cdot {{{{ \gamma} ^k} ^l}} {{{{{ K} _k} _l} _{,i}}}}} + {{{\alpha}} \cdot {{{ \Theta} _{,i}}}}}$
...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({{{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{f}}} + {{{{{ \beta} ^i}} {{{ a} _i}}} - {{{\alpha}} \cdot {{f}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}}}}}\right)}}$
${{{ \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}}}$
${{ a} _k} _{,t}$$ + \dots = $${{{{{ a} _i}} {{{{ b} ^i} _k}}} - {{{f'}} \cdot {{{ a} _k}} {{{{ K} _i} _l}} {{{{ \gamma} ^i} ^l}} {{{\alpha}^{2}}}}} + {{{2}} {{\Theta}} \cdot {{f'}} \cdot {{{ a} _k}} {{{\alpha}^{2}}}} + {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{f}} {{{ a} _k}}} + {{{{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}}}}$
${{{{ 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}$$ + \dots = $${{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} + {{{{{ K} _a} _i}} {{{{ b} ^a} _j}}} + {{{{{{ K} _a} _j}} {{{{ b} ^a} _i}}} - {{{2}} {{\Theta}} \cdot {{\alpha}} \cdot {{{{ K} _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 {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _b} _i} _j}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _i} _b} _j}}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _a}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _j} _b} _i}}}} - {{{2}} {{\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} _i} _j}} {{{{{ d} _c} _b} _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}}}} + {{{2}} {{\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} _i} _a} _d}} {{{{{ d} _j} _b} _c}}} + {{{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _i} _a} _j}}} + {{{{{{\alpha}} \cdot {{{{ \gamma} ^a} ^b}} {{{{ \gamma} ^a} ^c}} {{{{{ d} _a} _b} _c}} {{{{{ d} _j} _a} _i}}} - {{{\alpha}} \cdot {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _a} _i} _j}} {{{{{ d} _a} _b} _c}}}} - {{{\alpha}} \cdot {{{{ \gamma} ^a} ^c}} {{{{ \gamma} ^a} ^b}} {{{{{ d} _i} _a} _b}} {{{{{ d} _j} _a} _c}}}} - {{{\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}}} - {{{\alpha}} \cdot {{{ a} _i}} {{{ a} _j}}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}}} - {{{4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}}$
${ \Theta} _{,t}$$ + \dots = $${\frac{1}{2}} {{{\alpha}} \cdot {{\left({{-{{{2}} {{\Theta}} \cdot {{{{ K} _k} _l}} {{{{ \gamma} ^k} ^l}}}} + {{{{{{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^m} ^n}}} - {{{{{ K} _k} _l}} {{{{ K} _m} _n}} {{{{ \gamma} ^k} ^m}} {{{{ \gamma} ^l} ^n}}}} - {{{4}} {{{ Z} _a}} {{{{ \gamma} ^h} ^a}} {{{{ \gamma} ^k} ^g}} {{{{{ d} _k} _g} _h}}}} - {{{2}} {{{ Z} _a}} {{{ a} _k}} {{{{ \gamma} ^k} ^a}}}} + {{{{{{2}} {{{ Z} _k}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}}} - {{{4}} {{{{ \gamma} ^d} ^g}} {{{{ \gamma} ^e} ^k}} {{{{ \gamma} ^h} ^j}} {{{{{ d} _j} _d} _e}} {{{{{ d} _k} _g} _h}}}} - {{{4}} {{{{ \gamma} ^d} ^j}} {{{{ \gamma} ^e} ^g}} {{{{ \gamma} ^h} ^k}} {{{{{ d} _j} _d} _e}} {{{{{ d} _k} _g} _h}}}} - {{{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _i} _r} _s}} {{{{{ d} _k} _d} _e}}}} + {{{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _k} _d} _e}} {{{{{ d} _r} _i} _s}}} + {{{{{{ \gamma} ^d} ^r}} {{{{ \gamma} ^e} ^s}} {{{{ \gamma} ^k} ^i}} {{{{{ d} _k} _d} _e}} {{{{{ d} _s} _i} _r}}} - {{{{{ \gamma} ^f} ^l}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _l} _f}}}} + {{{4}} {{{{ \gamma} ^h} ^b}} {{{{ \gamma} ^j} ^c}} {{{{ \gamma} ^k} ^g}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _g} _h}}} + {{{{4}} {{{{ \gamma} ^h} ^c}} {{{{ \gamma} ^j} ^g}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _b} _c} _j}} {{{{{ d} _k} _g} _h}}} - {{{16}} {{\pi}} \cdot {{\rho}}}}}\right)}}}$
${{ Z} _i} _{,t}$$ + \dots = $${{-{{{2}} {{\alpha}} \cdot {{{{ K} _l} _i}} {{{{ \gamma} ^c} ^l}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _k} _b} _c}}}} - {{{\Theta}} \cdot {{\alpha}} \cdot {{{ a} _i}}}} + {{{2}} {{\alpha}} \cdot {{{{ K} _k} _l}} {{{{ \gamma} ^c} ^l}} {{{{ \gamma} ^k} ^b}} {{{{{ d} _i} _b} _c}}} + {{{{{{ Z} _k}} {{{{ b} ^k} _i}}} - {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _k} _d} _i}}}} - {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _i} _d} _k}}}} + {{{\alpha}} \cdot {{{{ K} _l} _r}} {{{{ \gamma} ^k} ^l}} {{{{ \gamma} ^r} ^d}} {{{{{ d} _d} _k} _i}}} + {{{{{\alpha}} \cdot {{{{ K} _l} _i}} {{{{ \gamma} ^a} ^m}} {{{{ \gamma} ^k} ^l}} {{{{{ d} _k} _m} _a}}} - {{{2}} {{\alpha}} \cdot {{{ Z} _k}} {{{{ K} _l} _i}} {{{{ \gamma} ^k} ^l}}}} - {{{8}} {{\alpha}} \cdot {{\pi}} \cdot {{{ S} _i}}}}$
spelled out