KroneckerDelta - Maple Help

Tensor[KroneckerDelta] - find the Kronecker delta tensor of rank r

Calling Sequences

KroneckerDelta(spatialType, r, fr)

Parameters

spatialType  - a string, either "bas" or "vrt"

r            - a non-negative integer

fr           - (optional) the name of a defined frame

Description

 • The Kronecker delta tensor $K$ of rank $r$ is the type $\left(\genfrac{}{}{0}{}{r}{r}\right)$ tensor which is defined as follows. Let $I$ be the type $\left(\genfrac{}{}{0}{}{1}{1}\right)$ tensor whose components in any coordinate system are given by the identity matrix, that is, for any vector field $I\left(X\right)=X$. Then $K$ is obtained from the $r$-fold tensor product of $I$ fully skew-symmetrizing over all the covariant indices.
 • The command KroneckerDelta(spatialType, r) returns the rank r Kronecker delta tensor $K$ 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 KroneckerDelta(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KroneckerDelta.

Examples

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

Example 1.

We create a 3 dimensional manifold $M$ with coordinates $\left(x,y,z\right)$.

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

Define the 3 different Kronecker delta tensors on $M$.

 M > $\mathrm{K1}≔\mathrm{KroneckerDelta}\left("bas",1\right)$
 ${\mathrm{K1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (2.1)
 M > $\mathrm{K2}≔\mathrm{KroneckerDelta}\left("bas",2\right)$
 ${\mathrm{K2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (2.2)
 M > $\mathrm{K3}≔\mathrm{KroneckerDelta}\left("bas",3\right)$
 ${\mathrm{K3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{2}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{1}{,}{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{3}{,}{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{2}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{1}{,}{3}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{2}{,}{3}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{3}{,}{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)$ (2.3)

We check that the contraction of $\mathrm{K3}$ gives a multiple of $\mathrm{K2}$ and that the contraction of $\mathrm{K2}$ gives a multiple of $\mathrm{K1}$.

 M > $\left(\mathrm{ContractIndices}\left(\mathrm{K3},\left[\left[3,6\right]\right]\right)\right)&minus\mathrm{K2}$
 ${0}{}{\mathrm{D_x}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dx}}$ (2.4)
 M > $\left(\mathrm{ContractIndices}\left(\mathrm{K2},\left[\left[2,4\right]\right]\right)\right)&minus\left(2&mult\mathrm{K1}\right)$
 ${0}{}{\mathrm{D_x}}{}{\mathrm{dx}}$ (2.5)

We check that $\mathrm{K2}$ can be constructed from $\mathrm{K1}\otimes \mathrm{K1}$ by rearranging the indices and by skew-symmetrization.

 M > $\mathrm{T0}≔\mathrm{RearrangeIndices}\left(\mathrm{K1}&tensor\mathrm{K1},\left[1,3,2,4\right]\right)$
 ${\mathrm{T0}}{≔}{\mathrm{D_x}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{\mathrm{D_y}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{\mathrm{D_z}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.6)
 M > $T≔2&mult\left(\mathrm{SymmetrizeIndices}\left(\mathrm{T0},\left[1,2\right],"SkewSymmetric"\right)\right)$
 ${T}{≔}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dy}}{-}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dy}}$ (2.7)
 M > $T&minus\mathrm{K2}$
 ${0}{}{\mathrm{D_x}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dx}}$ (2.8)

Example 2.

We create a 2 dimensional vector bundle over $E$ with fiber coordinates $\left(p,q\right)$.

 M > $\mathrm{DGsetup}\left(\left[x,y,z\right],\left[p,q\right],E\right)$
 ${\mathrm{frame name: E}}$ (2.9)

Define the possible Kronecker delta tensors on the fibers of $E$.

 E > $\mathrm{K1}≔\mathrm{KroneckerDelta}\left("vrt",1\right)$
 ${\mathrm{K1}}{≔}{\mathrm{D_p}}{}{\mathrm{dp}}{+}{\mathrm{D_q}}{}{\mathrm{dq}}$ (2.10)
 E > $\mathrm{K2}≔\mathrm{KroneckerDelta}\left("vrt",2\right)$
 ${\mathrm{K2}}{≔}{\mathrm{D_p}}{}{\mathrm{D_q}}{}{\mathrm{dp}}{}{\mathrm{dq}}{-}{\mathrm{D_p}}{}{\mathrm{D_q}}{}{\mathrm{dq}}{}{\mathrm{dp}}{-}{\mathrm{D_q}}{}{\mathrm{D_p}}{}{\mathrm{dp}}{}{\mathrm{dq}}{+}{\mathrm{D_q}}{}{\mathrm{D_p}}{}{\mathrm{dq}}{}{\mathrm{dp}}$ (2.11)