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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

Examples

Description

•

The Kronecker delta tensor $K$ of rank $r$ is the type $(\binom{r}{r})$ tensor which is defined as follows. Let $I$ be the type $(\binom{1}{1})$ tensor whose components in any coordinate system are given by the identity matrix, that is, for any vector field $I (X) = X$ . 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

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

Example 1.

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

>	$DGsetup ([x, y, z], M) &colon;$

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

M >	$K1 ≔ KroneckerDelta (bas, 1)$

$K1 :=_DG ([[tensor, M, [[con_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], 1], [[3, 3], 1]]]),_DG ([[tensor, M, [[con_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], 1], [[3, 3], 1]]]),_DG ([[tensor, M, [[con_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], 1], [[3, 3], 1]]]),_DG ([[tensor, M, [[con_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], 1], [[3, 3], 1]]])$

(2.1)

M >	$K2 ≔ KroneckerDelta (bas, 2)$

$K2 :=_DG ([[tensor, M, [[con_bas, con_bas, cov_bas, cov_bas], []]], [[[1, 2, 1, 2], 1], [[1, 2, 2, 1], −1], [[1, 3, 1, 3], 1], [[1, 3, 3, 1], −1], [[2, 1, 1, 2], −1], [[2, 1, 2, 1], 1], [[2, 3, 2, 3], 1], [[2, 3, 3, 2], −1], [[3, 1, 1, 3], −1], [[3, 1, 3, 1], 1], [[3, 2, 2, 3], −1], [[3, 2, 3, 2], 1]]]),_DG ([[tensor, M, [[con_bas, con_bas, cov_bas, cov_bas], []]], [[[1, 2, 1, 2], 1], [[1, 2, 2, 1], −1], [[1, 3, 1, 3], 1], [[1, 3, 3, 1], −1], [[2, 1, 1, 2], −1], [[2, 1, 2, 1], 1], [[2, 3, 2, 3], 1], [[2, 3, 3, 2], −1], [[3, 1, 1, 3], −1], [[3, 1, 3, 1], 1], [[3, 2, 2, 3], −1], [[3, 2, 3, 2], 1]]]),_DG ([[tensor, M, [[con_bas, con_bas, cov_bas, cov_bas], []]], [[[1, 2, 1, 2], 1], [[1, 2, 2, 1], −1], [[1, 3, 1, 3], 1], [[1, 3, 3, 1], −1], [[2, 1, 1, 2], −1], [[2, 1, 2, 1], 1], [[2, 3, 2, 3], 1], [[2, 3, 3, 2], −1], [[3, 1, 1, 3], −1], [[3, 1, 3, 1], 1], [[3, 2, 2, 3], −1], [[3, 2, 3, 2], 1]]]),_DG ([[tensor, M, [[con_bas, con_bas, cov_bas, cov_bas], []]], [[[1, 2, 1, 2], 1], [[1, 2, 2, 1], −1], [[1, 3, 1, 3], 1], [[1, 3, 3, 1], −1], [[2, 1, 1, 2], −1], [[2, 1, 2, 1], 1], [[2, 3, 2, 3], 1], [[2, 3, 3, 2], −1], [[3, 1, 1, 3], −1], [[3, 1, 3, 1], 1], [[3, 2, 2, 3], −1], [[3, 2, 3, 2], 1]]])$

(2.2)

M >	$K3 ≔ KroneckerDelta (bas, 3)$

$K3 :=_DG ([[tensor, M, [[con_bas, con_bas, con_bas, cov_bas, cov_bas, cov_bas], []]], [[[1, 2, 3, 1, 2, 3], 1], [[1, 2, 3, 1, 3, 2], −1], [[1, 2, 3, 2, 1, 3], −1], [[1, 2, 3, 2, 3, 1], 1], [[1, 2, 3, 3, 1, 2], 1], [[1, 2, 3, 3, 2, 1], −1], [[1, 3, 2, 1, 2, 3], −1], [[1, 3, 2, 1, 3, 2], 1], [[1, 3, 2, 2, 1, 3], 1], [[1, 3, 2, 2, 3, 1], −1], [[1, 3, 2, 3, 1, 2], −1], [[1, 3, 2, 3, 2, 1], 1], [[2, 1, 3, 1, 2, 3], −1], [[2, 1, 3, 1, 3, 2], 1], [[2, 1, 3, 2, 1, 3], 1], [[2, 1, 3, 2, 3, 1], −1], [[2, 1, 3, 3, 1, 2], −1], [[2, 1, 3, 3, 2, 1], 1], [[2, 3, 1, 1, 2, 3], 1], [[2, 3, 1, 1, 3, 2], −1], [[2, 3, 1, 2, 1, 3], −1], [[2, 3, 1, 2, 3, 1], 1], [[2, 3, 1, 3, 1, 2], 1], [[2, 3, 1, 3, 2, 1], −1], [[3, 1, 2, 1, 2, 3], 1], [[3, 1, 2, 1, 3, 2], −1], [[3, 1, 2, 2, 1, 3], −1], [[3, 1, 2, 2, 3, 1], 1], [[3, 1, 2, 3, 1, 2], 1], [[3, 1, 2, 3, 2, 1], −1], [[3, 2, 1, 1, 2, 3], −1], [[3, 2, 1, 1, 3, 2], 1], [[3, 2, 1, 2, 1, 3], 1], [[3, 2, 1, 2, 3, 1], −1], [[3, 2, 1, 3, 1, 2], −1], [[3, 2, 1, 3, 2, 1], 1]]]),_DG ([[tensor, M, [[con_bas, con_bas, con_bas, cov_bas, cov_bas, cov_bas], []]], [[[1, 2, 3, 1, 2, 3], 1], [[1, 2, 3, 1, 3, 2], −1], [[1, 2, 3, 2, 1, 3], −1], [[1, 2, 3, 2, 3, 1], 1], [[1, 2, 3, 3, 1, 2], 1], [[1, 2, 3, 3, 2, 1], −1], [[1, 3, 2, 1, 2, 3], −1], [[1, 3, 2, 1, 3, 2], 1], [[1, 3, 2, 2, 1, 3], 1], [[1, 3, 2, 2, 3, 1], −1], [[1, 3, 2, 3, 1, 2], −1], [[1, 3, 2, 3, 2, 1], 1], [[2, 1, 3, 1, 2, 3], −1], [[2, 1, 3, 1, 3, 2], 1], [[2, 1, 3, 2, 1, 3], 1], [[2, 1, 3, 2, 3, 1], −1], [[2, 1, 3, 3, 1, 2], −1], [[2, 1, 3, 3, 2, 1], 1], [[2, 3, 1, 1, 2, 3], 1], [[2, 3, 1, 1, 3, 2], −1], [[2, 3, 1, 2, 1, 3], −1], [[2, 3, 1, 2, 3, 1], 1], [[2, 3, 1, 3, 1, 2], 1], [[2, 3, 1, 3, 2, 1], −1], [[3, 1, 2, 1, 2, 3], 1], [[3, 1, 2, 1, 3, 2], −1], [[3, 1, 2, 2, 1, 3], −1], [[3, 1, 2, 2, 3, 1], 1], [[3, 1, 2, 3, 1, 2], 1], [[3, 1, 2, 3, 2, 1], −1], [[3, 2, 1, 1, 2, 3], −1], [[3, 2, 1, 1, 3, 2], 1], [[3, 2, 1, 2, 1, 3], 1], [[3, 2, 1, 2, 3, 1], −1], [[3, 2, 1, 3, 1, 2], −1], [[3, 2, 1, 3, 2, 1], 1]]]),_DG ([[tensor, M, [[con_bas, con_bas, con_bas, cov_bas, cov_bas, cov_bas], []]], [[[1, 2, 3, 1, 2, 3], 1], [[1, 2, 3, 1, 3, 2], −1], [[1, 2, 3, 2, 1, 3], −1], [[1, 2, 3, 2, 3, 1], 1], [[1, 2, 3, 3, 1, 2], 1], [[1, 2, 3, 3, 2, 1], −1], [[1, 3, 2, 1, 2, 3], −1], [[1, 3, 2, 1, 3, 2], 1], [[1, 3, 2, 2, 1, 3], 1], [[1, 3, 2, 2, 3, 1], −1], [[1, 3, 2, 3, 1, 2], −1], [[1, 3, 2, 3, 2, 1], 1], [[2, 1, 3, 1, 2, 3], −1], [[2, 1, 3, 1, 3, 2], 1], [[2, 1, 3, 2, 1, 3], 1], [[2, 1, 3, 2, 3, 1], −1], [[2, 1, 3, 3, 1, 2], −1], [[2, 1, 3, 3, 2, 1], 1], [[2, 3, 1, 1, 2, 3], 1], [[2, 3, 1, 1, 3, 2], −1], [[2, 3, 1, 2, 1, 3], −1], [[2, 3, 1, 2, 3, 1], 1], [[2, 3, 1, 3, 1, 2], 1], [[2, 3, 1, 3, 2, 1], −1], [[3, 1, 2, 1, 2, 3], 1], [[3, 1, 2, 1, 3, 2], −1], [[3, 1, 2, 2, 1, 3], −1], [[3, 1, 2, 2, 3, 1], 1], [[3, 1, 2, 3, 1, 2], 1], [[3, 1, 2, 3, 2, 1], −1], [[3, 2, 1, 1, 2, 3], −1], [[3, 2, 1, 1, 3, 2], 1], [[3, 2, 1, 2, 1, 3], 1], [[3, 2, 1, 2, 3, 1], −1], [[3, 2, 1, 3, 1, 2], −1], [[3, 2, 1, 3, 2, 1], 1]]]),_DG ([[tensor, M, [[con_bas, con_bas, con_bas, cov_bas, cov_bas, cov_bas], []]], [[[1, 2, 3, 1, 2, 3], 1], [[1, 2, 3, 1, 3, 2], −1], [[1, 2, 3, 2, 1, 3], −1], [[1, 2, 3, 2, 3, 1], 1], [[1, 2, 3, 3, 1, 2], 1], [[1, 2, 3, 3, 2, 1], −1], [[1, 3, 2, 1, 2, 3], −1], [[1, 3, 2, 1, 3, 2], 1], [[1, 3, 2, 2, 1, 3], 1], [[1, 3, 2, 2, 3, 1], −1], [[1, 3, 2, 3, 1, 2], −1], [[1, 3, 2, 3, 2, 1], 1], [[2, 1, 3, 1, 2, 3], −1], [[2, 1, 3, 1, 3, 2], 1], [[2, 1, 3, 2, 1, 3], 1], [[2, 1, 3, 2, 3, 1], −1], [[2, 1, 3, 3, 1, 2], −1], [[2, 1, 3, 3, 2, 1], 1], [[2, 3, 1, 1, 2, 3], 1], [[2, 3, 1, 1, 3, 2], −1], [[2, 3, 1, 2, 1, 3], −1], [[2, 3, 1, 2, 3, 1], 1], [[2, 3, 1, 3, 1, 2], 1], [[2, 3, 1, 3, 2, 1], −1], [[3, 1, 2, 1, 2, 3], 1], [[3, 1, 2, 1, 3, 2], −1], [[3, 1, 2, 2, 1, 3], −1], [[3, 1, 2, 2, 3,</]]]])$

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