tensor(deprecated)/Killing_eqns - Maple Help

tensor

 Killing_eqns
 compute component expressions for Killings equations

 Calling Sequence Killing_eqns( T, coord, Cf2)

Parameters

 T - symmetric covariant tensor coord - list of coordinate names Cf2 - Christoffel symbols of the second kind

Description

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

 • The function Killing_eqns(T, coord, Cf2 ) computes the expressions for Killing's equations for each component of the totally symmetric covariant tensor T.  Specifically, the symmetric part of the covariant derivative of T is computed and returned as a tensor_type. The components of T satisfy Killing's equations if all of the components of the result are zero.  Note that the rank of the result is one more than that of T.
 • This routine is useful in two ways:  first, as a means of verifying that a tensor satisfies Killing's equations, and second, as a way of generating the differential equations for any unknown components of a symmetric tensor which is to satisfy Killing's equations.
 • T must be of rank 1 or greater.  If T is of second rank or more, the component array of T must use Maple's symmetric indexing function (since T must be symmetric).
 • Cf2 should be indexed using the cf2 indexing function provided by the tensor package.  It can be computed using the Christoffel2 routine.
 • Simplification:  This routine uses the tensor/cov_diff/simp and tensor/lin_com/simp routines for simplification purposes.  The simplification routines are used indirectly by the symmetrize and cov_diff procedures as they are called by Killing_eqns.  By default, tensor/cov_diff/simp and tensor/lin_com/simp are initialized to the tensor/simp routine.  It is recommended that these routines be customized to suit the needs of the particular problem.

Examples

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

 > $\mathrm{with}\left(\mathrm{tensor}\right):$

Generate the Killing equation expressions for an arbitrary vector in the geometry of Euclidean 3-space using polar coordinates: First, compute the Christoffel symbols of the second kind:

 > $\mathrm{coord}≔\left[r,\mathrm{θ},\mathrm{φ}\right]:$
 > $\mathrm{g_compts}≔\mathrm{array}\left(\mathrm{symmetric},\mathrm{sparse},1..3,1..3,\left[\left(1,1\right)=1,\left(2,2\right)={r}^{2},\left(3,3\right)={r}^{2}{\mathrm{sin}\left(\mathrm{θ}\right)}^{2}\right]\right):$
 > $g≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(\mathrm{g_compts}\right)\right)$
 ${g}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {{r}}^{{2}}& {0}\\ {0}& {0}& {{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (1)
 > $\mathrm{ginv}≔\mathrm{invert}\left(g,'\mathrm{detg}'\right):$
 > $\mathrm{d1g}≔\mathrm{d1metric}\left(g,\mathrm{coord}\right):$$\mathrm{d2g}≔\mathrm{d2metric}\left(\mathrm{d1g},\mathrm{coord}\right):$
 > $\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{d1g}\right):$
 > $\mathrm{Cf2}≔\mathrm{Christoffel2}\left(\mathrm{ginv},\mathrm{Cf1}\right):$

Next, define the arbitrary vector field:

 > $V≔\mathrm{create}\left(\left[-1\right],\mathrm{array}\left(\left[\mathrm{v1}\left(r,\mathrm{θ},\mathrm{φ}\right),\mathrm{v2}\left(r,\mathrm{θ},\mathrm{φ}\right),\mathrm{v3}\left(r,\mathrm{θ},\mathrm{φ}\right)\right]\right)\right)$
 ${V}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}\right]\right]\right)$ (2)

Now compute the Killing equation expressions:

 > $\mathrm{KV}≔\mathrm{Killing_eqns}\left(V,\mathrm{coord},\mathrm{Cf2}\right)$
 ${\mathrm{KV}}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& \frac{\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{2}{}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}{}{r}}& \frac{\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{2}{}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}{}{r}}\\ \frac{\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{2}{}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}{}{r}}& \frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{r}{}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& \frac{\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}}{-}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}}\\ \frac{\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{2}{}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}{}{r}}& \frac{\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}}{-}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{2}}& \frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{v3}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{v1}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{v2}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (3)

Now try it for an arbitrary symmetric 0, 2-tensor:

 > $t≔\mathrm{array}\left(\mathrm{symmetric},1..3,1..3\right):$
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}3\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}j\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{from}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}3\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{t}_{i,j}≔\mathrm{cat}\left('t',i,j\right)\left(r,\mathrm{θ},\mathrm{φ}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do};$$T≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(t\right)\right)$
 ${\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)$
 ${T}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\\ {\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\\ {\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)& {\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (4)
 > $\mathrm{KT}≔\mathrm{Killing_eqns}\left(T,\mathrm{coord},\mathrm{Cf2}\right)$
 ${\mathrm{KT}}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}{array}{}\left({\mathrm{symmetric}}{,}{1}{..}{3}{,}{1}{..}{3}{,}{1}{..}{3}{,}\left[\left({1}{,}{1}{,}{1}\right){=}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){,}\left({1}{,}{1}{,}{2}\right){=}\frac{{2}{}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({1}{,}{1}{,}{3}\right){=}\frac{{2}{}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({1}{,}{2}{,}{1}\right){=}\frac{{2}{}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({1}{,}{2}{,}{2}\right){=}\frac{{2}{}{{r}}^{{2}}{}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}{2}{}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({1}{,}{2}{,}{3}\right){=}\frac{{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({1}{,}{3}{,}{1}\right){=}\frac{{2}{}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({1}{,}{3}{,}{2}\right){=}\frac{{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({1}{,}{3}{,}{3}\right){=}\frac{{2}{}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}{2}{}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({2}{,}{1}{,}{1}\right){=}\frac{{2}{}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({2}{,}{1}{,}{2}\right){=}\frac{{2}{}{{r}}^{{2}}{}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}{2}{}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({2}{,}{1}{,}{3}\right){=}\frac{{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({2}{,}{2}{,}{1}\right){=}\frac{{2}{}{{r}}^{{2}}{}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}{2}{}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({2}{,}{2}{,}{2}\right){=}\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{2}{}{r}{}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){,}\left({2}{,}{2}{,}{3}\right){=}\frac{\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{-}\frac{{4}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{r}{}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{,}\left({2}{,}{3}{,}{1}\right){=}\frac{{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({2}{,}{3}{,}{2}\right){=}\frac{\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{-}\frac{{4}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{r}{}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{,}\left({2}{,}{3}{,}{3}\right){=}\frac{{2}{}\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{-}\frac{{4}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{,}\left({3}{,}{1}{,}{1}\right){=}\frac{{2}{}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({3}{,}{1}{,}{2}\right){=}\frac{{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({3}{,}{1}{,}{3}\right){=}\frac{{2}{}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}{2}{}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({3}{,}{2}{,}{1}\right){=}\frac{{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({3}{,}{2}{,}{2}\right){=}\frac{\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{-}\frac{{4}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{r}{}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{,}\left({3}{,}{2}{,}{3}\right){=}\frac{{2}{}\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{-}\frac{{4}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{,}\left({3}{,}{3}{,}{1}\right){=}\frac{{2}{}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{t11}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){}{r}{+}\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{+}{2}{}\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right){}{r}{-}{4}{}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}{}{r}}{,}\left({3}{,}{3}{,}{2}\right){=}\frac{{2}{}\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{-}\frac{{4}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{t12}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t22}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{+}\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)}{{3}}{,}\left({3}{,}{3}{,}{3}\right){=}\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t33}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{2}{}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{\mathrm{t13}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){+}{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{t23}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right)\right]\right){,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}{,}{-1}\right]\right]\right)$ (5)