PermutationSymbol - Maple Help

Tensor[PermutationSymbol] - create a permutation symbol

Calling Sequences

PermutationSymbol(indexType, fr)

Parameters

indexType    - a string, one of the 4 possible index types "cov_bas", "con_bas", "cov_vrt", or "con_vrt"

fr           - (optional) the name of a defined frame

Description

 • A permutation symbol is a tensor density which is fully skew-symmetric and whose component values are +1 or -1. The rank of the permutation symbol is the dimension of the manifold $M$, or the base or fiber dimension of a vector bundle $E\to M$. The covariant permutation symbol is a tensor density of weight -1 while the contravariant permutation symbol is a tensor density of weight +1.
 • The command PermutionSymbol(indexType) returns the permutation symbol of the type specified by indexType in the current frame unless the frame is explicitly specified.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form PermutationSymbol(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-PermutationSymbol.

Examples

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

Example 1.

Create a 2-dimensional manifold $M$.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Here are the 2 different permutation symbols one can define on $M$. For each we use the DGinfo command to display the tensor type -- recall that the first list characterizes the index type and the second list the tensor density weight.

 M > $\mathrm{P1}≔\mathrm{PermutationSymbol}\left("cov_bas"\right)$
 ${\mathrm{P1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\left[{"bas"}{,}{-1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\left[{"bas"}{,}{-1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right)$ (2.2)
 M > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{P1},"TensorType"\right)$
 $\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\left[{"bas"}{,}{-}{1}\right]\right]\right]$ (2.3)
 M > $\mathrm{P2}≔\mathrm{PermutationSymbol}\left("con_bas"\right)$
 ${\mathrm{P2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right)$ (2.4)
 M > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{P2},"TensorType"\right)$
 $\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]$ (2.5)

Example 2.

Create a rank 2 vector bundle $E$ over a 3-dimensional base.

 M > $\mathrm{DGsetup}\left(\left[x,y,z\right],\left[u,v\right],E\right)$
 ${\mathrm{frame name: E}}$ (2.6)

Here are the 4 different permutation symbols one can define on $E$ and their tensor types.

 E > $\mathrm{P1}≔\mathrm{PermutationSymbol}\left("cov_bas"\right)$
 ${\mathrm{P1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\left[{"bas"}{,}{-1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\left[{"bas"}{,}{-1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right)$ (2.7)
 E > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{P1},"TensorType"\right)$
 $\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\left[{"bas"}{,}{-}{1}\right]\right]\right]$ (2.8)
 E > $\mathrm{P2}≔\mathrm{PermutationSymbol}\left("con_bas"\right)$
 ${\mathrm{P2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}\right]{,}{-1}\right]\right]\right]\right)$ (2.9)
 E > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{P2},"TensorType"\right)$
 $\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]$ (2.10)
 E > $\mathrm{P3}≔\mathrm{PermutationSymbol}\left("cov_vrt"\right)$
 ${\mathrm{P3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\left[{"vrt"}{,}{-1}\right]\right]\right]\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\left[{"vrt"}{,}{-1}\right]\right]\right]\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{4}\right]{,}{-1}\right]\right]\right]\right)$ (2.11)
 E > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{P3},"TensorType"\right)$
 $\left[\left[{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\left[{"vrt"}{,}{-}{1}\right]\right]\right]$ (2.12)
 E > $\mathrm{P4}≔\mathrm{PermutationSymbol}\left("con_vrt"\right)$
 ${\mathrm{P4}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\left[{"vrt"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\left[{"vrt"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{4}\right]{,}{-1}\right]\right]\right]\right)$ (2.13)
 E > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{P4},"TensorType"\right)$
 $\left[\left[{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\left[{"vrt"}{,}{1}\right]\right]\right]$ (2.14)