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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 > $g≔\mathrm{evalDG}\left(\left(\frac{1}{{y}^{2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]\right]\right]\right)$ (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 > $\mathrm{T3}≔\mathrm{KillingTensors}\left(g,3\right)$
 ${\mathrm{T3}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}\frac{{{x}}^{{6}}{-}{3}{}{{y}}^{{2}}{}{{x}}^{{4}}{+}{3}{}{{x}}^{{2}}{}{{y}}^{{4}}{-}{{y}}^{{6}}}{{48}{}{{y}}^{{6}}}\right]{,}\left[\left[{1}{,}{1}{,}{2}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}\right]{,}\frac{{x}{}\left({{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right)}{{24}{}{{y}}^{{5}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}\frac{\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{x}}^{{2}}}{{12}{}{{y}}^{{4}}}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}\frac{{{x}}^{{3}}}{{6}{}{{y}}^{{3}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\right]\right]\right)\right]$