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Tensor[IndependentKillingTensors] - given two lists R and S of rank p Killing tensors, find a maximal sublist T of R such that the combined list of tensors [T, S] are linearly independent (over the real numbers)

Calling Sequences

IndependentKillingTensors(R, S)

Parameters

R, S    - two lists of Killing tensors, the tensors in S must be linearly independent over the real numbers

Description

 • The suggested use of this program is as follows. Let $K=\left[{K}_{1},{K}_{2},...,{K}_{p-1}\right]$ be a list of bases for the Killing tensors of a metric $g$, from rank $1$ to rank $p-1$. Let $S$ be the Killing tensors of rank $p$ which are symmetric tensor products of the elements of $K$. This list of Killing tensors $S$ can be generated by the command SymmetricProductsOfKillingTensors. Now use the command KillingTensors to calculate the Killing tensors $T$ of rank $p$.  Then the program IndependentKillingTensors(T, S) will return the list of rank $p$ Killing tensors in $T$ not in the symmetric algebra of Killing tensors generated by $S$.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form IndependentKillingTensors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-IndependentKillingTensors.

Examples

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

Example 1.

We consider a 2 dimensional manifold with a metric of constant negative curvature. For such metrics it is known that the Killing 1-forms algebraically generate all higher rank Killing tensors. We check this for rank $2$ and rank 3 Killing tensors using the programs IndependentKillingTensors and SymmetricProductsOfKillingTensors.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$
 M > $g≔\mathrm{evalDG}\left(\left(\frac{1}{{y}^{2}}\right)&mult\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}\right)\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right)$ (2.1)

Calculate the rank 1 Killing tensors.

 M > $\mathrm{K1}≔\mathrm{KillingTensors}\left(g,1\right)$
 ${\mathrm{K1}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}\right]{,}\frac{{x}}{{y}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}\right]{,}\frac{{x}}{{y}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{x}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}\right]{,}\frac{{1}}{{y}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{x}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}\right]{,}\frac{{1}}{{y}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right)\right]$ (2.2)

Calculate the rank 2 Killing tensors which are symmetric products of the rank 1 Killing tensors.

 M > $\mathrm{K2}≔\mathrm{SymmetricProductsOfKillingTensors}\left(\left[\mathrm{K1}\right],2\right)$
 ${\mathrm{K2}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right)}^{{2}}}{{4}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right)}^{{2}}}{{4}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{4}}}\right]\right]\right]\right)\right]$ (2.3)

Calculate the rank 2 Killing tensors by directly solving the Killing tensor equations.

 M > $\mathrm{T2}≔\mathrm{KillingTensors}\left(g,2\right)$
 ${\mathrm{T2}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{8}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{2}{}{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{8}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{2}{}{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{4}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{4}}}\right]\right]\right]\right)\right]$ (2.4)

Use the IndependentKillingTensors command to deduce that all of the Killing tensors of rank 2 are algebraically generated by the Killing vectors.

 M > $\mathrm{IndependentKillingTensors}\left(\mathrm{T2},\mathrm{K2}\right)$
 $\left[{}\right]$ (2.5)

Calculate the rank 3 Killing tensors which are symmetric products of the rank 1 Killing tensors.

 M > $\mathrm{K3}≔\mathrm{SymmetricProductsOfKillingTensors}\left(\left[\mathrm{K1}\right],3\right):$

Calculate the rank 3 Killing tensors by directly solving the Killing tensor equations.

 M > $\mathrm{T3}≔\mathrm{KillingTensors}\left(g,3\right)$
 ${\mathrm{T3}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{{x}}^{{6}}{-}{3}{}{{y}}^{{2}}{}{{x}}^{{4}}{+}{3}{}{{x}}^{{2}}{}{{y}}^{{4}}{-}{{y}}^{{6}}}{{48}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{{x}}^{{3}}}{{6}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{{x}}^{{6}}{-}{3}{}{{y}}^{{2}}{}{{x}}^{{4}}{+}{3}{}{{x}}^{{2}}{}{{y}}^{{4}}{-}{{y}}^{{6}}}{{48}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{{x}}^{{3}}}{{6}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{8}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{5}{}{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{5}{}{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{5}{}{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{8}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{5}{}{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{5}{}{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{5}{}{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{2}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{-}\frac{{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{5}{}{{y}}^{{4}}}{{8}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}{-}\frac{{1}}{{6}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}{-}\frac{{1}}{{6}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}{-}\frac{{1}}{{6}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{-}\frac{{{x}}^{{4}}{-}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{5}{}{{y}}^{{4}}}{{8}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}{-}\frac{{1}}{{6}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}{-}\frac{{1}}{{6}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}{-}\frac{{1}}{{6}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{x}}{{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{2}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}{-}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}\left({{x}}^{{2}}{-}{3}{}{{y}}^{{2}}\right)}{{2}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}{-}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{8}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}{{8}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{x}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{x}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{x}}{{3}{}{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{x}}{{2}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{x}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{3}{}{{x}}^{{2}}{-}{{y}}^{{2}}}{{6}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{x}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{x}}{{3}{}{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{1}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{1}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{1}}{{3}{}{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{4}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{{1}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{{1}}{{3}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{{1}}{{3}{}{{y}}^{{4}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{x}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{x}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{x}}{{3}{}{{y}}^{{5}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{{x}}^{{2}}{-}{{y}}^{{2}}}{{2}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{x}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{x}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{x}}{{3}{}{{y}}^{{5}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{1}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{1}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{1}}{{3}{}{{y}}^{{5}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{x}}{{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{1}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{1}}{{3}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{1}}{{3}{}{{y}}^{{5}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{6}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{6}}}\right]\right]\right]\right)\right]$ (2.6)

There are no "new" Killing tensors in T3.

 M > $\mathrm{IndependentKillingTensors}\left(\mathrm{T3},\mathrm{K3}\right)$
 $\left[{}\right]$ (2.7)

Example 2.

In this example we find that there are 4 Killing 1-forms and 4 rank 2 Killing tensors which are not symmetric products of the Killing 1-forms.

 M > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 M > $g≔\mathrm{evalDG}\left(x\mathrm{dx}&s\mathrm{dy}+x\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{≔}\frac{{x}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{x}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{x}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.8)
 M > $\mathrm{T1}≔\mathrm{KillingTensors}\left(g,1\right)$
 ${\mathrm{T1}}{≔}\left[{-}{x}{}{z}{}{\mathrm{dx}}{+}{{x}}^{{2}}{}{\mathrm{dz}}{,}{x}{}{\mathrm{dz}}{,}{2}{}{y}{}{x}{}{\mathrm{dx}}{-}{{x}}^{{2}}{}{\mathrm{dy}}{+}{x}{}{z}{}{\mathrm{dz}}{,}{x}{}{\mathrm{dx}}\right]$ (2.9)
 M > $\mathrm{T2}≔\mathrm{KillingTensors}\left(g,2\right):$
 M > $\mathrm{nops}\left(\mathrm{T1}\right),\mathrm{nops}\left(\mathrm{T2}\right)$
 ${4}{,}{14}$ (2.10)

Calculate the rank 2 Killing tensors which are symmetric products of the rank 1 Killing tensors. There are 10 such Killing tensors.

 M > $\mathrm{S2}≔\mathrm{SymmetricProductsOfKillingTensors}\left(\left[\mathrm{T1}\right],2\right):$
 M > $\mathrm{nops}\left(\mathrm{S2}\right)$
 ${10}$ (2.11)

Calculate the rank 2 Killing tensors which are not symmetric products of the rank 1 Killing tensors.

 M > $\mathrm{K2}≔\mathrm{IndependentKillingTensors}\left(\mathrm{T2},\mathrm{S2}\right)$
 ${\mathrm{K2}}{≔}\left[\frac{{y}{}{x}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{y}{}{x}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{{x}}^{{2}}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{y}{}{x}{}{\mathrm{dz}}{}{\mathrm{dz}}{,}\frac{{x}{}{{z}}^{{2}}}{{4}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{x}{}{{z}}^{{2}}}{{4}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{{x}}^{{2}}{}{z}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}\frac{{{x}}^{{2}}{}{z}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}\frac{{x}{}\left({2}{}{y}{}{x}{+}{{z}}^{{2}}\right)}{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}{,}\frac{{x}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{x}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{x}{}{\mathrm{dz}}{}{\mathrm{dz}}{,}\frac{{x}{}{z}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{x}{}{z}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{{x}}^{{2}}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}\frac{{{x}}^{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{x}{}{z}{}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.12)