$R$ = warp bubble radius.
$\frac{1}{\sigma}$ = warp bubble thickness.
${x_s}$ = warp bubble center location along the x axis.
${{v_s}} = {\frac{\partial {x_s}}{\partial t}}$ = warp bubble velocity, as a function of t
${{r_s}} = {\sqrt{{{\left({{x}{-{{x_s}}}}\right)}^{2}} + {{y}^{2}} + {{z}^{2}}}}$ = warp bubble radial coordinate
${f\left( {r_s}\right)} = {\frac{{\tanh\left( {{{\sigma}} \cdot {{\left({{{r_s}} + {R}}\right)}}}\right)}{-{\tanh\left( {{{\sigma}} \cdot {{\left({{{r_s}}{-{R}}}\right)}}}\right)}}}{{{2}} {{\tanh\left( {{{\sigma}} \cdot {{R}}}\right)}}}}$ = shape of bubble
${\frac{\partial {r_s}}{\partial t}} = {{\frac{1}{{r_s}}} {{{{v_s}}} \cdot {{\left({{-{x}} + {{x_s}}}\right)}}}}$
${\frac{\partial {r_s}}{\partial x}} = {{\frac{1}{{r_s}}}{\left({{x}{-{{x_s}}}}\right)}}$
${\frac{\partial {r_s}}{\partial y}} = {{\frac{1}{{r_s}}} {y}}$
${\frac{\partial {r_s}}{\partial z}} = {{\frac{1}{{r_s}}} {z}}$

${\frac{\partial^ 2 {r_s}}{\partial t^ 2}} = {\frac{{{{{{r_s}}^{2}}} {{{{v_s}}^{2}}}}{-{{{{{v_s}}^{2}}} {{{x}^{2}}}}}{-{{{{{v_s}}^{2}}} {{{{x_s}}^{2}}}}}{-{{{x}} {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}}}} + {{{{x_s}}} \cdot {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}}} + {{{2}} {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial x\partial t}} = {\frac{{{{v_s}}} \cdot {{\left({{-{{{r_s}}^{2}}} + {{x}^{2}} + {{{x_s}}^{2}}{-{{{2}} {{x}} {{{x_s}}}}}}\right)}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial y\partial t}} = {\frac{{{{v_s}}} \cdot {{y}} {{\left({{x}{-{{x_s}}}}\right)}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial z\partial t}} = {\frac{{{{v_s}}} \cdot {{z}} {{\left({{x}{-{{x_s}}}}\right)}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial x^ 2}} = {\frac{{{{r_s}}^{2}}{-{{x}^{2}}}{-{{{x_s}}^{2}}} + {{{2}} {{x}} {{{x_s}}}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial y\partial x}} = {\frac{{{y}} {{\left({{-{x}} + {{x_s}}}\right)}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial z\partial x}} = {\frac{{{z}} {{\left({{-{x}} + {{x_s}}}\right)}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial y^ 2}} = {\frac{{{\left({{{r_s}}{-{y}}}\right)}} {{\left({{{r_s}} + {y}}\right)}}}{{{r_s}}^{3}}}$
${\frac{\partial^ 2 {r_s}}{\partial z\partial y}} = {-{\frac{{{y}} {{z}}}{{{r_s}}^{3}}}}$
${\frac{\partial^ 2 {r_s}}{\partial z^ 2}} = {\frac{{{\left({{{r_s}}{-{z}}}\right)}} {{\left({{{r_s}} + {z}}\right)}}}{{{r_s}}^{3}}}$

${u} = {{{{v_s}}} \cdot {{f\left( {r_s}\right)}}}$

${{ u} _{,i}} = {{{{{ {v_s}} _{,i}}} {{f\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{{ {r_s}} _{,i}}} {{f'\left( {r_s}\right)}}}}$
${\frac{\partial u}{\partial t}} = {{{{f\left( {r_s}\right)}} {{\frac{\partial {v_s}}{\partial t}}}} + {{{{v_s}}} \cdot {{\frac{\partial {r_s}}{\partial t}}} {{f'\left( {r_s}\right)}}}}$
${\frac{\partial u}{\partial x}} = {{{{v_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{\partial {r_s}}{\partial x}}}}$
${\frac{\partial u}{\partial y}} = {{{{v_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{\partial {r_s}}{\partial y}}}}$
${\frac{\partial u}{\partial z}} = {{{{v_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{\partial {r_s}}{\partial z}}}}$

${\frac{\partial u}{\partial t}} = {{\frac{1}{{r_s}}}{\left({{{{{r_s}}} \cdot {{\frac{\partial {v_s}}{\partial t}}} {{f\left( {r_s}\right)}}}{-{{{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}}\right)}}$
${\frac{\partial u}{\partial x}} = {{\frac{1}{{r_s}}} {{{{v_s}}} \cdot {{\left({{x}{-{{x_s}}}}\right)}} {{f'\left( {r_s}\right)}}}}$
${\frac{\partial u}{\partial y}} = {{\frac{1}{{r_s}}} {{{{v_s}}} \cdot {{y}} {{f'\left( {r_s}\right)}}}}$
${\frac{\partial u}{\partial z}} = {{\frac{1}{{r_s}}} {{{{v_s}}} \cdot {{z}} {{f'\left( {r_s}\right)}}}}$

${{{ u} _{,i}} _{,j}} = {{{{{{ {v_s}} _{,i}} _{,j}}} {{f\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{{{ {r_s}} _{,i}} _{,j}}} {{f'\left( {r_s}\right)}}}}$
${{\frac{\partial^ 2 u}{\partial t^ 2}} = {{{{f\left( {r_s}\right)}} {{\frac{\partial^ 2 {v_s}}{\partial t^ 2}}}} + {{{{v_s}}} \cdot {{\frac{\partial^ 2 {r_s}}{\partial t^ 2}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{{\frac{\partial {r_s}}{\partial t}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{2}} {{f'\left( {r_s}\right)}} {{\frac{\partial {v_s}}{\partial t}}} {{\frac{\partial {r_s}}{\partial t}}}}}} = {{{{{{v_s}}^{3}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{r_s}}}}} + {{{f\left( {r_s}\right)}} {{\frac{\partial^ 2 {v_s}}{\partial t^ 2}}}} + {{{-1}} {{{{v_s}}^{3}}} {{{x}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{-1}} {{{{v_s}}^{3}}} {{{{x_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{{v_s}}^{3}}} {{{x}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{{{v_s}}^{3}}} {{{{x_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{2}} {{x}} {{{x_s}}} \cdot {{{{v_s}}^{3}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{-2}} {{x}} {{{x_s}}} \cdot {{{{v_s}}^{3}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{-3}} {{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}} {{\frac{\partial {v_s}}{\partial t}}} {{\frac{1}{{r_s}}}}} + {{{3}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{\partial {v_s}}{\partial t}}} {{\frac{1}{{r_s}}}}}}$
${{\frac{\partial^ 2 u}{\partial x\partial t}} = {{{{{v_s}}} \cdot {{\frac{\partial^ 2 {r_s}}{\partial x\partial t}}} {{f'\left( {r_s}\right)}}} + {{{\frac{\partial {r_s}}{\partial x}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{f''\left( {r_s}\right)}} {{\frac{\partial {r_s}}{\partial x}}} {{\frac{\partial {r_s}}{\partial t}}}}}} = {{{{-1}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{r_s}}}}} + {{{{{v_s}}^{2}}} {{{x}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{-1}} {{{{v_s}}^{2}}} {{{x}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{-1}} {{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{x}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{r_s}}}}} + {{{-1}} {{{x_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{\partial {v_s}}{\partial t}}} {{\frac{1}{{r_s}}}}} + {{{-2}} {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{2}} {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial y\partial t}} = {{{{{v_s}}} \cdot {{\frac{\partial^ 2 {r_s}}{\partial y\partial t}}} {{f'\left( {r_s}\right)}}} + {{{\frac{\partial {r_s}}{\partial y}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{f''\left( {r_s}\right)}} {{\frac{\partial {r_s}}{\partial y}}} {{\frac{\partial {r_s}}{\partial t}}}}}} = {{{{x}} {{y}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{-1}} {{{x_s}}} \cdot {{y}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{y}} {{f'\left( {r_s}\right)}} {{\frac{\partial {v_s}}{\partial t}}} {{\frac{1}{{r_s}}}}} + {{{-1}} {{x}} {{y}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{{x_s}}} \cdot {{y}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial z\partial t}} = {{{{{v_s}}} \cdot {{\frac{\partial^ 2 {r_s}}{\partial z\partial t}}} {{f'\left( {r_s}\right)}}} + {{{\frac{\partial {r_s}}{\partial z}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{f''\left( {r_s}\right)}} {{\frac{\partial {r_s}}{\partial z}}} {{\frac{\partial {r_s}}{\partial t}}}}}} = {{{{x}} {{z}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{-1}} {{{x_s}}} \cdot {{z}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{z}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{r_s}}}}} + {{{-1}} {{x}} {{z}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{{x_s}}} \cdot {{z}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial x^ 2}} = {{{{v_s}}} \cdot {{\left({{{{f'\left( {r_s}\right)}} {{\frac{\partial^ 2 {r_s}}{\partial x^ 2}}}} + {{{{\frac{\partial {r_s}}{\partial x}}^{2}}} {{f''\left( {r_s}\right)}}}}\right)}}}} = {{{{{v_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{1}{{r_s}}}}} + {{{-1}} {{{v_s}}} \cdot {{{x}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{-1}} {{{v_s}}} \cdot {{{{x_s}}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{{x}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{{v_s}}} \cdot {{{{x_s}}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{2}} {{{v_s}}} \cdot {{x}} {{{x_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{-2}} {{{v_s}}} \cdot {{x}} {{{x_s}}} \cdot {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial x\partial y}} = {{{{v_s}}} \cdot {{\left({{{{f'\left( {r_s}\right)}} {{\frac{\partial^ 2 {r_s}}{\partial x\partial y}}}} + {{{\frac{\partial {r_s}}{\partial y}}} {{\frac{\partial {r_s}}{\partial x}}} {{f''\left( {r_s}\right)}}}}\right)}}}} = {{{{-1}} {{{v_s}}} \cdot {{x}} {{y}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{{x_s}}} \cdot {{y}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{x}} {{y}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{-1}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{y}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial x\partial z}} = {{{{v_s}}} \cdot {{\left({{{{f'\left( {r_s}\right)}} {{\frac{\partial^ 2 {r_s}}{\partial x\partial z}}}} + {{{\frac{\partial {r_s}}{\partial z}}} {{\frac{\partial {r_s}}{\partial x}}} {{f''\left( {r_s}\right)}}}}\right)}}}} = {{{{-1}} {{{v_s}}} \cdot {{x}} {{z}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{{x_s}}} \cdot {{z}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{x}} {{z}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}} + {{{-1}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{z}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial y^ 2}} = {{{{v_s}}} \cdot {{\left({{{{f'\left( {r_s}\right)}} {{\frac{\partial^ 2 {r_s}}{\partial y^ 2}}}} + {{{{\frac{\partial {r_s}}{\partial y}}^{2}}} {{f''\left( {r_s}\right)}}}}\right)}}}} = {{{{{v_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{1}{{r_s}}}}} + {{{-1}} {{{v_s}}} \cdot {{{y}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{{y}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial z\partial y}} = {{{{v_s}}} \cdot {{\left({{{{f'\left( {r_s}\right)}} {{\frac{\partial^ 2 {r_s}}{\partial z\partial y}}}} + {{{\frac{\partial {r_s}}{\partial z}}} {{\frac{\partial {r_s}}{\partial y}}} {{f''\left( {r_s}\right)}}}}\right)}}}} = {{{{-1}} {{{v_s}}} \cdot {{y}} {{z}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{y}} {{z}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$
${{\frac{\partial^ 2 u}{\partial z^ 2}} = {{{{v_s}}} \cdot {{\left({{{{f'\left( {r_s}\right)}} {{\frac{\partial^ 2 {r_s}}{\partial z^ 2}}}} + {{{{\frac{\partial {r_s}}{\partial z}}^{2}}} {{f''\left( {r_s}\right)}}}}\right)}}}} = {{{{{v_s}}} \cdot {{f'\left( {r_s}\right)}} {{\frac{1}{{r_s}}}}} + {{{-1}} {{{v_s}}} \cdot {{{z}^{2}}} {{f'\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{3}}}}} + {{{{v_s}}} \cdot {{{z}^{2}}} {{f''\left( {r_s}\right)}} {{\frac{1}{{{r_s}}^{2}}}}}}$

${\alpha} = {1}$ = metric lapse
${{ \beta} ^i} = {\overset{i\downarrow}{\left[\begin{matrix} -{u} \\ 0 \\ 0\end{matrix}\right]}}$ = metric shift
spatial metric:
${{{ \gamma} _i} _j} = {\overset{i\downarrow j\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$
${{{ \gamma} ^i} ^j} = {\overset{i\downarrow j\rightarrow}{\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}$

${{{ e} _a} ^I} = {\overset{a\downarrow I\rightarrow}{\left[\begin{array}{cccc} \sqrt{{1}{-{{u}^{2}}}}& 0& 0& 0\\ \frac{u}{\sqrt{{1}{-{{u}^{2}}}}}& \frac{1}{\sqrt{{1}{-{{u}^{2}}}}}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$

4-metric:
${{{ g} _a} _b} = {{{{{ e} _a} ^I}} {{{{ e} _b} ^J}} {{{{ \eta} _I} _J}}}$

${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {-{1}} + {{u}^{2}}& -{u}& 0& 0\\ -{u}& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${g} = {-{1}}$
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{1}& -{u}& 0& 0\\ -{u}& {1}{-{{u}^{2}}}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
hypersurface normal:
${{ n} _u} = {\overset{u\downarrow}{\left[\begin{matrix} -1 \\ 0 \\ 0 \\ 0\end{matrix}\right]}}$
${{ n} ^u} = {\overset{u\downarrow}{\left[\begin{matrix} 1 \\ u \\ 0 \\ 0\end{matrix}\right]}}$
connection:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} {{u}} {{\frac{\partial u}{\partial t}}}& {{u}} {{\frac{\partial u}{\partial x}}}& {{u}} {{\frac{\partial u}{\partial y}}}& {{u}} {{\frac{\partial u}{\partial z}}}\\ {{u}} {{\frac{\partial u}{\partial x}}}& -{\frac{\partial u}{\partial x}}& -{{\frac{1}{2}} {\frac{\partial u}{\partial y}}}& -{{\frac{1}{2}} {\frac{\partial u}{\partial z}}}\\ {{u}} {{\frac{\partial u}{\partial y}}}& -{{\frac{1}{2}} {\frac{\partial u}{\partial y}}}& 0& 0\\ {{u}} {{\frac{\partial u}{\partial z}}}& -{{\frac{1}{2}} {\frac{\partial u}{\partial z}}}& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}}}\right)}& 0& -{{\frac{1}{2}} {\frac{\partial u}{\partial y}}}& -{{\frac{1}{2}} {\frac{\partial u}{\partial z}}}\\ 0& 0& 0& 0\\ -{{\frac{1}{2}} {\frac{\partial u}{\partial y}}}& 0& 0& 0\\ -{{\frac{1}{2}} {\frac{\partial u}{\partial z}}}& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{{{u}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{2}} {\frac{\partial u}{\partial y}}& 0& 0\\ {\frac{1}{2}} {\frac{\partial u}{\partial y}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{{{u}} {{\frac{\partial u}{\partial z}}}}& {\frac{1}{2}} {\frac{\partial u}{\partial z}}& 0& 0\\ {\frac{1}{2}} {\frac{\partial u}{\partial z}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} {{{u}^{2}}} {{\frac{\partial u}{\partial x}}}& -{{{u}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}}}}\\ -{{{u}} {{\frac{\partial u}{\partial x}}}}& \frac{\partial u}{\partial x}& {\frac{1}{2}} {\frac{\partial u}{\partial y}}& {\frac{1}{2}} {\frac{\partial u}{\partial z}}\\ -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}}}}& {\frac{1}{2}} {\frac{\partial u}{\partial y}}& 0& 0\\ -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}}}}& {\frac{1}{2}} {\frac{\partial u}{\partial z}}& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} {-{\frac{\partial u}{\partial t}}}{-{{{u}} {{\frac{\partial u}{\partial x}}}}} + {{{{u}^{3}}} {{\frac{\partial u}{\partial x}}}}& -{{{{u}^{2}}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\left({{1} + {{u}^{2}}}\right)}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\left({{1} + {{u}^{2}}}\right)}}}}\\ -{{{{u}^{2}}} {{\frac{\partial u}{\partial x}}}}& {{u}} {{\frac{\partial u}{\partial x}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}}}\\ -{{\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\left({{1} + {{u}^{2}}}\right)}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}}}& 0& 0\\ -{{\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\left({{1} + {{u}^{2}}}\right)}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}}}& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} -{{{u}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{2}} {\frac{\partial u}{\partial y}}& 0& 0\\ {\frac{1}{2}} {\frac{\partial u}{\partial y}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} -{{{u}} {{\frac{\partial u}{\partial z}}}}& {\frac{1}{2}} {\frac{\partial u}{\partial z}}& 0& 0\\ {\frac{1}{2}} {\frac{\partial u}{\partial z}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{matrix}\right]}}$
geodesic:
${{ \overset{u\downarrow}{\left[\begin{matrix} {\ddot{x}^t} \\ {\ddot{x}^x} \\ {\ddot{x}^y} \\ {\ddot{x}^z}\end{matrix}\right]}} ^a} = {\overset{a\downarrow}{\left[\begin{matrix} {-{{{{{\dot{x}^x}}^{2}}} {{\frac{\partial u}{\partial x}}}}}{-{{{{{\dot{x}^t}}^{2}}} {{{u}^{2}}} {{\frac{\partial u}{\partial x}}}}}{-{{{{\dot{x}^x}}} \cdot {{{\dot{x}^y}}} \cdot {{\frac{\partial u}{\partial y}}}}}{-{{{{\dot{x}^x}}} \cdot {{{\dot{x}^z}}} \cdot {{\frac{\partial u}{\partial z}}}}} + {{{{\dot{x}^t}}} \cdot {{{\dot{x}^y}}} \cdot {{u}} {{\frac{\partial u}{\partial y}}}} + {{{{\dot{x}^t}}} \cdot {{{\dot{x}^z}}} \cdot {{u}} {{\frac{\partial u}{\partial z}}}} + {{{2}} {{{\dot{x}^t}}} \cdot {{{\dot{x}^x}}} \cdot {{u}} {{\frac{\partial u}{\partial x}}}} \\ {{{{{\dot{x}^t}}^{2}}} {{\frac{\partial u}{\partial t}}}} + {{{{\dot{x}^t}}} \cdot {{{\dot{x}^y}}} \cdot {{\frac{\partial u}{\partial y}}}} + {{{{\dot{x}^t}}} \cdot {{{\dot{x}^z}}} \cdot {{\frac{\partial u}{\partial z}}}} + {{{u}} {{{{\dot{x}^t}}^{2}}} {{\frac{\partial u}{\partial x}}}}{-{{{u}} {{{{\dot{x}^x}}^{2}}} {{\frac{\partial u}{\partial x}}}}}{-{{{{{\dot{x}^t}}^{2}}} {{{u}^{3}}} {{\frac{\partial u}{\partial x}}}}} + {{{{\dot{x}^t}}} \cdot {{{\dot{x}^y}}} \cdot {{{u}^{2}}} {{\frac{\partial u}{\partial y}}}} + {{{{\dot{x}^t}}} \cdot {{{\dot{x}^z}}} \cdot {{{u}^{2}}} {{\frac{\partial u}{\partial z}}}}{-{{{{\dot{x}^x}}} \cdot {{{\dot{x}^y}}} \cdot {{u}} {{\frac{\partial u}{\partial y}}}}}{-{{{{\dot{x}^x}}} \cdot {{{\dot{x}^z}}} \cdot {{u}} {{\frac{\partial u}{\partial z}}}}} + {{{2}} {{{\dot{x}^t}}} \cdot {{{\dot{x}^x}}} \cdot {{{u}^{2}}} {{\frac{\partial u}{\partial x}}}} \\ {{{\dot{x}^t}}} \cdot {{\frac{\partial u}{\partial y}}} {{\left({{-{{\dot{x}^x}}} + {{{{\dot{x}^t}}} \cdot {{u}}}}\right)}} \\ {{{\dot{x}^t}}} \cdot {{\frac{\partial u}{\partial z}}} {{\left({{-{{\dot{x}^x}}} + {{{{\dot{x}^t}}} \cdot {{u}}}}\right)}}\end{matrix}\right]}}$

${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{2}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}}& {{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}} + {{{2}} {{{\frac{\partial u}{\partial x}}^{2}}}}}\right)}}& {{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& {{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}& -{\left({{{\frac{\partial u}{\partial x}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{\frac{\partial u}{\partial z}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}& -{\left({{{\frac{\partial u}{\partial x}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ \frac{\partial^ 2 u}{\partial x\partial t}& \frac{\partial^ 2 u}{\partial x^ 2}& \frac{\partial^ 2 u}{\partial x\partial y}& \frac{\partial^ 2 u}{\partial x\partial z}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial y}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y^ 2}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z^ 2}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial y}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y^ 2}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{{\frac{\partial u}{\partial z}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z^ 2}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {-{\frac{\partial^ 2 u}{\partial t^ 2}}}{-{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}}}{-{{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}} + {{{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{3}} {{{u}^{2}}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial t}}}}& {-{\frac{\partial^ 2 u}{\partial x\partial t}}}{-{{\frac{\partial u}{\partial x}}^{2}}} + {{{3}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}}{-{{{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}} + {{{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}& {-{\frac{\partial^ 2 u}{\partial y\partial t}}}{-{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}}}{-{{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}} + {{{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{3}} {{{u}^{2}}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}& {-{\frac{\partial^ 2 u}{\partial z\partial t}}}{-{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}}}{-{{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}} + {{{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{3}} {{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}\\ -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{2}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}}}& -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}} + {{{2}} {{{\frac{\partial u}{\partial x}}^{2}}}}}\right)}}}& -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}& -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial t}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial y}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial z}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial t}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial z}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial z}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{2}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}}}& -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}} + {{{2}} {{{\frac{\partial u}{\partial x}}^{2}}}}}\right)}}}& -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}& -{{{u}} {{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}\\ {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial t}}}}& {{\frac{\partial u}{\partial x}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}& {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}& {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}\\ {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{{\frac{\partial u}{\partial z}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial t}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial y}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial z}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial t}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial z}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial z}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}\\ {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{{\frac{\partial u}{\partial z}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial y}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& -{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial y}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y^ 2}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial y}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y^ 2}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial z}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& -{\left({{{{u}} {{\frac{\partial^ 2 u}{\partial y\partial z}}}} + {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& -{\left({{{\frac{\partial u}{\partial z}}^{2}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z^ 2}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial t}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial x\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial z}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z^ 2}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

${{{{{ {(\Gamma^2)}} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}}{\left({{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{2}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}& -{{\frac{1}{4}} {{{u}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}\\ -{{\frac{1}{2}}{\left({{{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{2}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}}& {\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\\ -{{\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}& 0& -{{\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}}& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\\ -{{\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}& 0& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}& -{{\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{4}} {{{u}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}}}}& 0& 0& 0\\ {\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}& 0& 0& 0\\ -{{\frac{1}{2}} {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}& 0& 0& 0\\ -{{\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}}& 0& 0& 0\\ -{{\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}& 0& 0& 0\\ -{{\frac{1}{2}} {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}& 0& 0& 0\\ -{{\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}& 0& 0& 0\\ -{{\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {{{u}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}}& {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}}} + {{{4}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}& {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial u}{\partial t}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}}}}\right)}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial u}{\partial t}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}}}}\right)}} {{\frac{\partial u}{\partial z}}}}\\ -{{\frac{1}{2}}{\left({{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{2}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}}& {\frac{1}{4}} {{{u}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}}\\ -{{\frac{1}{2}} {{{u}} {{\left({{\frac{\partial u}{\partial t}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}}}}\right)}} {{\frac{\partial u}{\partial y}}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}& 0& 0\\ -{{\frac{1}{2}} {{{u}} {{\left({{\frac{\partial u}{\partial t}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}}}}\right)}} {{\frac{\partial u}{\partial z}}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}}} + {{{4}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial u}{\partial t}}} {{\frac{\partial u}{\partial x}}}}}\right)}& -{{{\frac{\partial u}{\partial x}}} {{\left({{{{u}} {{\frac{\partial u}{\partial x}}}} + {\frac{\partial u}{\partial t}}}\right)}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}\\ {\frac{1}{4}} {{{u}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}}}& 0& 0& 0\\ {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}}& -{{\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}}& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}\\ {\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}}& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}& -{{\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial u}{\partial t}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}}}}\right)}} {{\frac{\partial u}{\partial y}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}& 0& 0\\ -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}}& 0& 0& 0\\ 0& -{{\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}}& 0& 0\\ 0& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial u}{\partial t}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}}}}\right)}} {{\frac{\partial u}{\partial z}}}}& -{{\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}& 0& 0\\ -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}}& 0& 0& 0\\ 0& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}& 0& 0\\ 0& -{{\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}} + {{{{u}^{3}}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}& {\frac{1}{2}} {{{{u}^{2}}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}& {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}& {\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}} {{\left({{-{1}} + {{u}^{2}}}\right)}}}\\ {\frac{1}{2}} {{{{u}^{2}}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}& -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}& -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {{{{u}^{2}}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}& -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}& -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\\ -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}& {\frac{1}{2}} {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}& {\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}& -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}& 0& 0\\ -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}& {\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}} {{\left({{-{1}} + {{u}^{2}}}\right)}}}& -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}& 0& 0\\ -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}& {\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\left({{\frac{\partial u}{\partial t}} + {{{u}} {{\frac{\partial u}{\partial x}}}} + {{{{u}^{3}}} {{\frac{\partial u}{\partial x}}}}}\right)}}}}& {\frac{1}{2}} {{{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}& {\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}} {{\left({{-{1}} + {{u}^{2}}}\right)}}}& {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial z}}^{2}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}\\ {\frac{1}{2}} {{{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}& -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}& -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} {{{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}& -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}& -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\\ -{{\frac{1}{2}} {{{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}& {\frac{1}{2}} {{{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}& {\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}& {\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}} {{\left({{-{1}} + {{u}^{2}}}\right)}}}& -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}& 0& 0\\ -{{\frac{1}{4}} {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}& {\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial z}}^{2}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}& -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}& 0& 0\\ -{{\frac{1}{4}} {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}& {\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

extrinsic curvature
${{{ K} _i} _j} = {\overset{i\downarrow j\rightarrow}{\left[\begin{array}{ccc} -{\frac{\partial u}{\partial x}}& -{{\frac{1}{2}} {\frac{\partial u}{\partial y}}}& -{{\frac{1}{2}} {\frac{\partial u}{\partial z}}}\\ -{{\frac{1}{2}} {\frac{\partial u}{\partial y}}}& 0& 0\\ -{{\frac{1}{2}} {\frac{\partial u}{\partial z}}}& 0& 0\end{array}\right]}}$

Riemann curvature:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{4}} {{{u}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}}}& -{{\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}}& -{{\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}}\\ {\frac{1}{4}} {{{u}} {{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}} + {{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}} + {{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}}}& 0& {\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}}\\ {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}}}& 0& 0\\ {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}} + {{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}} + {{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ {\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}& 0& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial x}}}& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial x}}}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial x}}& 0& 0\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial x}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{4}}{\left({{{{3}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}& {\frac{1}{4}}{\left({{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& 0& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y^ 2}}}& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial y}}}\\ -{{\frac{1}{4}}{\left({{{{3}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y^ 2}}& 0& 0\\ -{{\frac{1}{4}}{\left({{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial y}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{4}}{\left({{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{4}}{\left({{{{3}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& 0& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial y}}}& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z^ 2}}}\\ -{{\frac{1}{4}}{\left({{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial y}}& 0& 0\\ -{{\frac{1}{4}}{\left({{{{3}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}}& {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z^ 2}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}} + {{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}{-{{{4}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}{-{{{4}} {{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{4}} {{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}}{-{{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial t\partial y}}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}{-{{{2}} {{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}}}{-{{{2}} {{{u}^{2}}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial x}}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}}{-{{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial t\partial z}}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}{-{{{2}} {{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}}}{-{{{2}} {{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial x}}}}}}\right)}\\ {\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}} + {{{4}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}} + {{{4}} {{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}} + {{{4}} {{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}& 0& {\frac{1}{2}} {{{\frac{\partial^ 2 u}{\partial y\partial x}}} {{\left({{-{1}} + {{u}^{2}}}\right)}}}& {\frac{1}{2}} {{{\frac{\partial^ 2 u}{\partial z\partial x}}} {{\left({{-{1}} + {{u}^{2}}}\right)}}}\\ {\frac{1}{2}}{\left({{-{\frac{\partial^ 2 u}{\partial t\partial y}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial t\partial y}}}}{-{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}}}{-{{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}} + {{{2}} {{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{{u}^{2}}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}} {{{\frac{\partial^ 2 u}{\partial y\partial x}}} {{\left({{1}{-{{u}^{2}}}}\right)}}}& 0& 0\\ {\frac{1}{2}}{\left({{-{\frac{\partial^ 2 u}{\partial t\partial z}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial t\partial z}}}}{-{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}}}{-{{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}} + {{{2}} {{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{{u}^{2}}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}} {{{\frac{\partial^ 2 u}{\partial z\partial x}}} {{\left({{1}{-{{u}^{2}}}}\right)}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{4}} {{{u}} {{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}} + {{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}} + {{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}}}& {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}& {\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}\\ {\frac{1}{4}} {{{u}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}}}& 0& -{{\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}}}& -{{\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}}}\\ -{{\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}}& 0& 0\\ -{{\frac{1}{2}} {{{u}} {{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}}& {\frac{1}{2}} {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial y}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& 0& -{{\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}}& -{{\frac{1}{4}}{\left({{{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}}}\right)}}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}}& {\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}& 0& 0\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial y}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& {\frac{1}{4}}{\left({{{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}}}\right)}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial z}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial y}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial z}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& 0& -{{\frac{1}{4}}{\left({{{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}}}\right)}}& -{{\frac{1}{4}}{\left({{{\frac{\partial u}{\partial z}}^{2}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial y}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& {\frac{1}{4}}{\left({{{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}}}\right)}& 0& 0\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z^ 2}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{2}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}& {\frac{1}{4}}{\left({{{\frac{\partial u}{\partial z}}^{2}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{4}}{\left({{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}& {\frac{1}{4}}{\left({{{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& 0& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}}& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ -{{\frac{1}{4}}{\left({{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}\right)}}& {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}& 0& 0\\ -{{\frac{1}{4}}{\left({{{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial x}}}& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}}& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial y\partial x}}& 0& {\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}& {\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}\\ {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}}}\right)}& -{{\frac{1}{4}} {{\frac{\partial u}{\partial y}}^{2}}}& 0& 0\\ {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{4}}{\left({{{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{4}}{\left({{{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{3}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& 0& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}\\ -{{\frac{1}{4}}{\left({{{{3}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& 0& 0\\ -{{\frac{1}{4}}{\left({{{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{3}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}}& {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{{\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial x}}}& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& -{{\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}\\ {\frac{1}{2}} {\frac{\partial^ 2 u}{\partial z\partial x}}& 0& {\frac{1}{4}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}\\ {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z\partial y}}}} + {{{u}} {{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}\right)}& -{{\frac{1}{4}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}}& 0& 0\\ {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}& -{{\frac{1}{4}} {{\frac{\partial u}{\partial z}}^{2}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$

Ricci curvature:
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}}}{-{{{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}}}{-{{{2}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{2}} {{{\frac{\partial u}{\partial x}}^{2}}}}} + {{{2}} {{{u}^{2}}} {{{\frac{\partial u}{\partial x}}^{2}}}}{-{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}{-{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}}}{-{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}} + {{{2}} {{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}} + {{{2}} {{{u}^{3}}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y^ 2}} + {\frac{\partial^ 2 u}{\partial z^ 2}} + {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}{-{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{2}} {{u}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{2}} {{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial z}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y^ 2}} + {\frac{\partial^ 2 u}{\partial z^ 2}} + {{{u}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}{-{{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}}}{-{{{2}} {{u}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{2}} {{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}& {\frac{1}{2}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}}{-{{\frac{\partial u}{\partial z}}^{2}}} + {{{2}} {{\frac{\partial^ 2 u}{\partial t\partial x}}}} + {{{2}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{2}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial y}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}} {{\frac{\partial u}{\partial y}}^{2}}& {\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial x}} + {{{u}} {{\frac{\partial^ 2 u}{\partial t\partial z}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial t\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial x}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}} {{{\frac{\partial u}{\partial y}}} {{\frac{\partial u}{\partial z}}}}& {\frac{1}{2}} {{\frac{\partial u}{\partial z}}^{2}}\end{array}\right]}}$

Gaussian curvature:
${R} = {{\frac{1}{2}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}} + {{{4}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}} + {{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}\right)}}$

Einstein curvature:
${{{ G} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}} + {{{3}} {{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}}& {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}\\ {\frac{1}{4}}{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}& -{{\frac{1}{4}} {{{3}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}}}}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial y\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}& {\frac{1}{4}}{\left({{{\frac{\partial u}{\partial y}}^{2}}{-{{\frac{\partial u}{\partial z}}^{2}}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}& {\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}\\ -{{\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial x\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}& {\frac{1}{2}}{\left({{\frac{\partial^ 2 u}{\partial z\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}& {\frac{1}{2}} {{{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}& {\frac{1}{4}}{\left({{-{{\frac{\partial u}{\partial y}}^{2}}} + {{\frac{\partial u}{\partial z}}^{2}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}\end{array}\right]}}$

Einstein field equation:
${{{ G} _a} _b} = {{{\frac{{{8}} {{π}} \cdot {{G}}}{{c}^{4}}}} {{{{ T} _a} _b}}}$
${{{ T} _a} _b} = {\frac{{{{{ G} _a} _b}} {{{c}^{4}}}}{{{8}} {{G}} {{π}}}}$
${{{ T} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{\frac{{{{c}^{4}}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}} + {{{3}} {{{u}^{2}}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{{u}^{2}}} {{{\frac{\partial u}{\partial z}}^{2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}}}\right)}}}{{{32}} {{G}} {{π}}}}& \frac{{{{c}^{4}}} {{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}}{{{32}} {{G}} {{π}}}& -{\frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial x\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}{{{16}} {{G}} {{π}}}}& -{\frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial x\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}{{{16}} {{G}} {{π}}}}\\ \frac{{{{c}^{4}}} {{\left({{{{2}} {{\frac{\partial^ 2 u}{\partial y^ 2}}}} + {{{2}} {{\frac{\partial^ 2 u}{\partial z^ 2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial y}}^{2}}}} + {{{3}} {{u}} {{{\frac{\partial u}{\partial z}}^{2}}}}}\right)}}}{{{32}} {{G}} {{π}}}& -{\frac{{{3}} {{{c}^{4}}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}}}{{{32}} {{G}} {{π}}}}& \frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial y\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}{{{16}} {{G}} {{π}}}& \frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial z\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}{{{16}} {{G}} {{π}}}\\ -{\frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial x\partial y}} + {{{u}} {{\frac{\partial^ 2 u}{\partial y\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}{{{16}} {{G}} {{π}}}}& \frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial y\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial y}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial y}}}}}\right)}}}{{{16}} {{G}} {{π}}}& \frac{{{{c}^{4}}} {{\left({{{\frac{\partial u}{\partial y}}^{2}}{-{{\frac{\partial u}{\partial z}}^{2}}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}}}{{{32}} {{G}} {{π}}}& \frac{{{{c}^{4}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}{{{16}} {{G}} {{π}}}\\ -{\frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial x\partial z}} + {{{u}} {{\frac{\partial^ 2 u}{\partial z\partial t}}}} + {{{{u}^{2}}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{u}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}{{{16}} {{G}} {{π}}}}& \frac{{{{c}^{4}}} {{\left({{\frac{\partial^ 2 u}{\partial z\partial t}} + {{{u}} {{\frac{\partial^ 2 u}{\partial x\partial z}}}} + {{{2}} {{\frac{\partial u}{\partial x}}} {{\frac{\partial u}{\partial z}}}}}\right)}}}{{{16}} {{G}} {{π}}}& \frac{{{{c}^{4}}} {{\frac{\partial u}{\partial z}}} {{\frac{\partial u}{\partial y}}}}{{{16}} {{G}} {{π}}}& \frac{{{{c}^{4}}} {{\left({{-{{\frac{\partial u}{\partial y}}^{2}}} + {{\frac{\partial u}{\partial z}}^{2}}{-{{{4}} {{\frac{\partial^ 2 u}{\partial x\partial t}}}}}{-{{{4}} {{{\frac{\partial u}{\partial x}}^{2}}}}}{-{{{4}} {{u}} {{\frac{\partial^ 2 u}{\partial x^ 2}}}}}}\right)}}}{{{32}} {{G}} {{π}}}\end{array}\right]}}$

${{{ T} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{{c}^{4}}} {{\left({{-{{{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{y}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{z}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{3}} {{{r_s}}} \cdot {{{u}^{2}}} {{{{v_s}}^{2}}} {{{y}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{3}} {{{r_s}}} \cdot {{{u}^{2}}} {{{{v_s}}^{2}}} {{{z}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{4}} {{u}} {{{v_s}}} \cdot {{{y}^{2}}} {{f'\left( {r_s}\right)}}} + {{{4}} {{u}} {{{v_s}}} \cdot {{{z}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{8}} {{u}} {{{v_s}}} \cdot {{{{r_s}}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{4}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{y}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{4}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{z}^{2}}} {{f''\left( {r_s}\right)}}}}}\right)}}}{{{32}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{{c}^{4}}} {{\left({{-{{{2}} {{{v_s}}} \cdot {{{y}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{2}} {{{v_s}}} \cdot {{{z}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{{v_s}}} \cdot {{{{r_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{{v_s}}} \cdot {{{y}^{2}}} {{f''\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{{v_s}}} \cdot {{{z}^{2}}} {{f''\left( {r_s}\right)}}} + {{{3}} {{{r_s}}} \cdot {{u}} {{{{v_s}}^{2}}} {{{y}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}} + {{{3}} {{{r_s}}} \cdot {{u}} {{{{v_s}}^{2}}} {{{z}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}\right)}}}{{{32}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{y}} {{{c}^{4}}} {{\left({{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}} + {{{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{u}} {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}}}{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}} + {{{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{2}} {{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{2}} {{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{z}} {{{c}^{4}}} {{\left({{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}} + {{{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{u}} {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}}}{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}} + {{{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{2}} {{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{2}} {{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}\\ \frac{{{{c}^{4}}} {{\left({{-{{{2}} {{{v_s}}} \cdot {{{y}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{2}} {{{v_s}}} \cdot {{{z}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{{v_s}}} \cdot {{{{r_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{{v_s}}} \cdot {{{y}^{2}}} {{f''\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{{v_s}}} \cdot {{{z}^{2}}} {{f''\left( {r_s}\right)}}} + {{{3}} {{{r_s}}} \cdot {{u}} {{{{v_s}}^{2}}} {{{y}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}} + {{{3}} {{{r_s}}} \cdot {{u}} {{{{v_s}}^{2}}} {{{z}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}\right)}}}{{{32}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& -{\frac{{{3}} {{{c}^{4}}} {{{f'\left( {r_s}\right)}^{2}}} {{\left({{{{{{v_s}}^{2}}} {{{y}^{2}}}} + {{{{{v_s}}^{2}}} {{{z}^{2}}}}}\right)}}}{{{32}} {{G}} {{π}} \cdot {{{{r_s}}^{2}}}}}& \frac{{{y}} {{{c}^{4}}} {{\left({{-{{{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{2}} {{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{z}} {{{c}^{4}}} {{\left({{-{{{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{2}} {{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}\\ \frac{{{y}} {{{c}^{4}}} {{\left({{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}} + {{{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{u}} {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}}}{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}} + {{{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{2}} {{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{2}} {{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{y}} {{{c}^{4}}} {{\left({{-{{{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{2}} {{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{{c}^{4}}} {{\left({{{{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{y}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{z}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{4}} {{{{r_s}}^{2}}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{4}} {{{{v_s}}^{2}}} {{{x}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{4}} {{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{x}^{2}}} {{f''\left( {r_s}\right)}}}{-{{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{x}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{f''\left( {r_s}\right)}}}{-{{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{4}} {{u}} {{{v_s}}} \cdot {{{{r_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{u}} {{{v_s}}} \cdot {{{x}^{2}}} {{f'\left( {r_s}\right)}}} + {{{4}} {{u}} {{{v_s}}} \cdot {{{{x_s}}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{4}} {{x}} {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{{x_s}}} \cdot {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{8}} {{{r_s}}} \cdot {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}} + {{{8}} {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{4}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{x}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{4}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{{x_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{8}} {{{r_s}}} \cdot {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{8}} {{u}} {{{v_s}}} \cdot {{x}} {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}} + {{{8}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{x}} {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}}}\right)}}}{{{32}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{y}} {{z}} {{{c}^{4}}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{2}}}}\\ \frac{{{z}} {{{c}^{4}}} {{\left({{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}} + {{{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{u}} {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}}}{-{{{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}} + {{{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{2}} {{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{2}} {{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{{r_s}}} \cdot {{u}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{{r_s}}} \cdot {{{v_s}}} \cdot {{x}} {{{u}^{2}}} {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{{r_s}}} \cdot {{{v_s}}} \cdot {{{x_s}}} \cdot {{{u}^{2}}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{z}} {{{c}^{4}}} {{\left({{-{{{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{x}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}} + {{{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{2}} {{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{-{{{2}} {{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{{r_s}}} \cdot {{x}} {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{u}} {{{v_s}}} \cdot {{x}} {{f'\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}} + {{{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}{-{{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}}} + {{{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{x}} {{f''\left( {r_s}\right)}}}}\right)}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}& \frac{{{y}} {{z}} {{{c}^{4}}} {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}{{{16}} {{G}} {{π}} \cdot {{{{r_s}}^{2}}}}& \frac{{{{c}^{4}}} {{\left({{-{{{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{y}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{z}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}} + {{{4}} {{{{r_s}}^{2}}} {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{4}} {{{{v_s}}^{2}}} {{{x}^{2}}} {{f'\left( {r_s}\right)}}}}{-{{{4}} {{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{x}^{2}}} {{f''\left( {r_s}\right)}}}{-{{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{x}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}} + {{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{f''\left( {r_s}\right)}}}{-{{{4}} {{{r_s}}} \cdot {{{{v_s}}^{2}}} {{{{x_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}}}{-{{{4}} {{u}} {{{v_s}}} \cdot {{{{r_s}}^{2}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{u}} {{{v_s}}} \cdot {{{x}^{2}}} {{f'\left( {r_s}\right)}}} + {{{4}} {{u}} {{{v_s}}} \cdot {{{{x_s}}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{4}} {{x}} {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}}} + {{{4}} {{{x_s}}} \cdot {{{{r_s}}^{2}}} {{\frac{\partial {v_s}}{\partial t}}} {{f'\left( {r_s}\right)}}} + {{{8}} {{{r_s}}} \cdot {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{{f'\left( {r_s}\right)}^{2}}}} + {{{8}} {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f'\left( {r_s}\right)}}}{-{{{4}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{x}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{4}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{{{x_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{8}} {{{r_s}}} \cdot {{x}} {{{x_s}}} \cdot {{{{v_s}}^{2}}} {{f''\left( {r_s}\right)}}}}{-{{{8}} {{u}} {{{v_s}}} \cdot {{x}} {{{x_s}}} \cdot {{f'\left( {r_s}\right)}}}} + {{{8}} {{{r_s}}} \cdot {{u}} {{{v_s}}} \cdot {{x}} {{{x_s}}} \cdot {{f''\left( {r_s}\right)}}}}\right)}}}{{{32}} {{G}} {{π}} \cdot {{{{r_s}}^{3}}}}\end{array}\right]}}$

density:
${\rho} = {{{{{ T} _a} _b}} {{{ n} ^a}} {{{ n} ^b}}}$

${\rho} = {-{\frac{{{{c}^{4}}} {{\left({{{\frac{\partial u}{\partial y}}^{2}} + {{\frac{\partial u}{\partial z}}^{2}}}\right)}}}{{{32}} {{G}} {{π}}}}}$
using ${\frac{\partial u}{\partial y}} = {{\frac{1}{{r_s}}} {{{{v_s}}} \cdot {{y}} {{f'\left( {r_s}\right)}}}}$ , ${\frac{\partial u}{\partial z}} = {{\frac{1}{{r_s}}} {{{{v_s}}} \cdot {{z}} {{f'\left( {r_s}\right)}}}}$
${\rho} = {-{\frac{{{{c}^{4}}} {{{f'\left( {r_s}\right)}^{2}}} {{\left({{{{{{v_s}}^{2}}} {{{y}^{2}}}} + {{{{{v_s}}^{2}}} {{{z}^{2}}}}}\right)}}}{{{32}} {{G}} {{π}} \cdot {{{{r_s}}^{2}}}}}}$