${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cc} {g_{xx}}& {g_{xy}}\\ {g_{xy}}& {g_{yy}}\end{array}\right]}}$
${g} = {{-{{{g_{xy}}}^{2}}} + {{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}$
${{{ g} ^u} ^v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cc} {\frac{1}{g}} {{g_{yy}}}& \frac{{g_{xy}}}{{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\\ \frac{{g_{xy}}}{{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}& {\frac{1}{g}} {{g_{xx}}}\end{array}\right]}}$
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cc} {\frac{1}{2}} { {g_{xx}}_{,{{x}}}}& {\frac{1}{2}} { {g_{xx}}_{,{{y}}}}\\ {\frac{1}{2}} { {g_{xx}}_{,{{y}}}}& {\frac{1}{2}}{\left({{-{ {g_{yy}}_{,{{x}}}}} + {{{2}} {{ {g_{xy}}_{,{{y}}}}}}}\right)}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cc} {\frac{1}{2}}{\left({{-{ {g_{xx}}_{,{{y}}}}} + {{{2}} {{ {g_{xy}}_{,{{x}}}}}}}\right)}& {\frac{1}{2}} { {g_{yy}}_{,{{x}}}}\\ {\frac{1}{2}} { {g_{yy}}_{,{{x}}}}& {\frac{1}{2}} { {g_{yy}}_{,{{y}}}}\end{array}\right]}\end{matrix}\right]}}$
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cc} \frac{{-{{{g}} {{{g_{xy}}}} \cdot {{ {g_{xx}}_{,{{y}}}}}}}{-{{{{g_{xx}}}} \cdot {{{{g_{yy}}}^{2}}} {{ {g_{xx}}_{,{{x}}}}}}} + {{{{g_{yy}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{xx}}_{,{{x}}}}}} + {{{2}} {{g}} {{{g_{xy}}}} \cdot {{ {g_{xy}}_{,{{x}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}& \frac{{{{g}} {{{g_{xy}}}} \cdot {{ {g_{yy}}_{,{{x}}}}}}{-{{{{g_{xx}}}} \cdot {{{{g_{yy}}}^{2}}} {{ {g_{xx}}_{,{{y}}}}}}} + {{{{g_{yy}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{xx}}_{,{{y}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}\\ \frac{{{{g}} {{{g_{xy}}}} \cdot {{ {g_{yy}}_{,{{x}}}}}}{-{{{{g_{xx}}}} \cdot {{{{g_{yy}}}^{2}}} {{ {g_{xx}}_{,{{y}}}}}}} + {{{{g_{yy}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{xx}}_{,{{y}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}& \frac{{{{g}} {{{g_{xy}}}} \cdot {{ {g_{yy}}_{,{{y}}}}}} + {{{{g_{xx}}}} \cdot {{{{g_{yy}}}^{2}}} {{ {g_{yy}}_{,{{x}}}}}}{-{{{{g_{yy}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{yy}}_{,{{x}}}}}}}{-{{{2}} {{{g_{xx}}}} \cdot {{{{g_{yy}}}^{2}}} {{ {g_{xy}}_{,{{y}}}}}}} + {{{2}} {{{g_{yy}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{xy}}_{,{{y}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cc} \frac{{{{g}} {{{g_{xy}}}} \cdot {{ {g_{xx}}_{,{{x}}}}}}{-{{{{g_{xx}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{xx}}_{,{{y}}}}}}} + {{{{g_{yy}}}} \cdot {{{{g_{xx}}}^{2}}} {{ {g_{xx}}_{,{{y}}}}}} + {{{2}} {{{g_{xx}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{xy}}_{,{{x}}}}}}{-{{{2}} {{{g_{yy}}}} \cdot {{{{g_{xx}}}^{2}}} {{ {g_{xy}}_{,{{x}}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}& \frac{{{{g}} {{{g_{xy}}}} \cdot {{ {g_{xx}}_{,{{y}}}}}} + {{{{g_{xx}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{yy}}_{,{{x}}}}}}{-{{{{g_{yy}}}} \cdot {{{{g_{xx}}}^{2}}} {{ {g_{yy}}_{,{{x}}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}\\ \frac{{{{g}} {{{g_{xy}}}} \cdot {{ {g_{xx}}_{,{{y}}}}}} + {{{{g_{xx}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{yy}}_{,{{x}}}}}}{-{{{{g_{yy}}}} \cdot {{{{g_{xx}}}^{2}}} {{ {g_{yy}}_{,{{x}}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}& \frac{{-{{{g}} {{{g_{xy}}}} \cdot {{ {g_{yy}}_{,{{x}}}}}}} + {{{{g_{xx}}}} \cdot {{{{g_{xy}}}^{2}}} {{ {g_{yy}}_{,{{y}}}}}}{-{{{{g_{yy}}}} \cdot {{{{g_{xx}}}^{2}}} {{ {g_{yy}}_{,{{y}}}}}}} + {{{2}} {{g}} {{{g_{xy}}}} \cdot {{ {g_{xy}}_{,{{y}}}}}}}{{{2}} {{g}} {{\left({{{{g_{xy}}}^{2}}{-{{{{g_{xx}}}} \cdot {{{g_{yy}}}}}}}\right)}}}\end{array}\right]}\end{matrix}\right]}}$