 GenerateSymmetricTensors - Maple Help

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Tensor[GenerateSymmetricTensors] - generate a list of symmetric tensors from a list of tensors

Calling Sequences

GenerateSymmetricTensors(Tlist, r)

Parameters

Tlist    - a list of tensor fields

r        - a positive integer, the number of tensors to be chosen from Tlist Description

 • With Tlist = [], the command GenerateSymmetricTensors(Tlist) will generate a list of symmetric tensors by forming all possible r-fold tensor products ${t}_{{i}_{1}}\otimes {t}_{{i}_{2}}...\otimes {t}_{{i}_{r}}$ and symmetrizing the result.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form GenerateSymmetricTensors(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-GenerateSymmetricTensors. Examples

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

Example 1.

First create a 4-tensional manifold $M$.

 > $\mathrm{DGsetup}\left(\left[w,x,y,z\right],M\right):$

Create a list $\mathrm{L1}$ of all rank 2 covariant symmetric tensors from the 1-forms .

 M > $\mathrm{T1}≔\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right]$
 ${\mathrm{T1}}{:=}\left[{\mathrm{dx}}{,}{\mathrm{dy}}{,}{\mathrm{dz}}\right]$ (2.1)
 M > $\mathrm{L1}≔\mathrm{GenerateSymmetricTensors}\left(\mathrm{T1},2\right)$
 ${\mathrm{L1}}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.2)
 M > $\mathrm{nops}\left(\mathrm{L1}\right)$
 ${6}$ (2.3)