${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} B& 0& 0& 0\\ 0& A& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {\left({{{r}} {{\sin\left( \theta\right)}}}\right)}^{2}\end{array}\right]}}$ ${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{1}{B}& 0& 0& 0\\ 0& \frac{1}{A}& 0& 0\\ 0& 0& {r}^{-2}& 0\\ 0& 0& 0& {\left({{{r}} {{\sin\left( \theta\right)}}}\right)}^{-2}\end{array}\right]}}$ ${{{ G} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{B}} {{\left({{A}{-{{A}^{2}}}{-{{{r}} {{\frac{\partial A}{\partial r}}}}}}\right)}}}{{{{A}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& \frac{{B}{-{{{A}} {{B}}}} + {{{r}} {{\frac{\partial B}{\partial r}}}}}{{{B}} {{{r}^{2}}}}& 0& 0\\ 0& 0& \frac{{-{{{A}} {{{r}^{2}}} {{{\frac{\partial B}{\partial r}}^{2}}}}}{-{{{2}} {{r}} {{{B}^{2}}} {{\frac{\partial A}{\partial r}}}}}{-{{{B}} {{{r}^{2}}} {{\frac{\partial B}{\partial r}}} {{\frac{\partial A}{\partial r}}}}} + {{{2}} {{A}} {{B}} {{r}} {{\frac{\partial B}{\partial r}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{\frac{\partial^ 2 B}{\partial r^ 2}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{A}} {{{r}^{2}}} {{{\frac{\partial B}{\partial r}}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{\frac{\partial B}{\partial r}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{2}} {{r}} {{{B}^{2}}} {{\frac{\partial A}{\partial r}}}}} + {{{2}} {{r}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial A}{\partial r}}}}{-{{{B}} {{{r}^{2}}} {{\frac{\partial B}{\partial r}}} {{\frac{\partial A}{\partial r}}}}} + {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial A}{\partial r}}} {{\frac{\partial B}{\partial r}}}} + {{{2}} {{A}} {{B}} {{r}} {{\frac{\partial B}{\partial r}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{\frac{\partial^ 2 B}{\partial r^ 2}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 B}{\partial r^ 2}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial B}{\partial r}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}\end{array}\right]}}$
${{{ T} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {P} + {\rho} + {{{B}} {{P}}}& 0& 0& 0\\ 0& {{A}} {{P}}& 0& 0\\ 0& 0& {{P}} {{{r}^{2}}}& 0\\ 0& 0& 0& {{P}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}\end{array}\right]}}$
${\frac{{{B}} {{\left({{A}{-{{A}^{2}}}{-{{{r}} {{\frac{\partial A}{\partial r}}}}}}\right)}}}{{{{A}^{2}}} {{{r}^{2}}}}} = {{{8}} {{π}} \cdot {{\left({{P} + {\rho} + {{{B}} {{P}}}}\right)}}}$
${\frac{{B}{-{{{A}} {{B}}}} + {{{r}} {{\frac{\partial B}{\partial r}}}}}{{{B}} {{{r}^{2}}}}} = {{{8}} {{π}} \cdot {{A}} {{P}}}$
${\frac{{-{{{A}} {{{r}^{2}}} {{{\frac{\partial B}{\partial r}}^{2}}}}}{-{{{2}} {{r}} {{{B}^{2}}} {{\frac{\partial A}{\partial r}}}}}{-{{{B}} {{{r}^{2}}} {{\frac{\partial B}{\partial r}}} {{\frac{\partial A}{\partial r}}}}} + {{{2}} {{A}} {{B}} {{r}} {{\frac{\partial B}{\partial r}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{\frac{\partial^ 2 B}{\partial r^ 2}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} = {{{8}} {{π}} \cdot {{P}} {{{r}^{2}}}}$
${\frac{{-{{{A}} {{{r}^{2}}} {{{\frac{\partial B}{\partial r}}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{\frac{\partial B}{\partial r}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{2}} {{r}} {{{B}^{2}}} {{\frac{\partial A}{\partial r}}}}} + {{{2}} {{r}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial A}{\partial r}}}}{-{{{B}} {{{r}^{2}}} {{\frac{\partial B}{\partial r}}} {{\frac{\partial A}{\partial r}}}}} + {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial A}{\partial r}}} {{\frac{\partial B}{\partial r}}}} + {{{2}} {{A}} {{B}} {{r}} {{\frac{\partial B}{\partial r}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{\frac{\partial^ 2 B}{\partial r^ 2}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 B}{\partial r^ 2}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial B}{\partial r}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} = {{{8}} {{π}} \cdot {{P}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}$
difference between 1st and 2nd...
${{{{A}} {{\frac{\partial B}{\partial r}}}} + {{{B}} {{\frac{\partial A}{\partial r}}}}} = {-{{{8}} {{r}} {{π}} \cdot {{{A}^{2}}} {{\left({{P} + {\rho}}\right)}}}}$
${{\frac{\partial}{\partial r}}\left({{{A}} {{B}}}\right)} = {{{-8}} {{π}} \cdot {{r}} {{{A}^{2}}} {{\left({{\rho} + {P}}\right)}}}$
difference between 2nd and 3rd...
${{{{A}} {{{r}^{2}}} {{{\frac{\partial B}{\partial r}}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}}} + {{{2}} {{r}} {{{B}^{2}}} {{\frac{\partial A}{\partial r}}}} + {{{B}} {{{r}^{2}}} {{\frac{\partial B}{\partial r}}} {{\frac{\partial A}{\partial r}}}} + {{{2}} {{A}} {{B}} {{r}} {{\frac{\partial B}{\partial r}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{\frac{\partial^ 2 B}{\partial r^ 2}}}}}} = {0}$
differences between 3rd and 4th...
${0} = {0}$
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