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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
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Description
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The Kronecker delta tensor K of rank r is the type (r, r) tensor which is defined as follows. Let I be the type (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.
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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 explicited specified.
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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.
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Examples
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Example 1.
We create a 3 dimensional manifold M with coordinates [x, y, z].
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![DGsetup([x, y, z], M)](/support/helpjp/helpview.aspx?si=5552/file05886/math89.png)
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Define the 3 different Kronecker delta tensors on M.
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M >
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M >
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| (2.2) |
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M >
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We check that the contraction of K3 gives a multiple of K2 and that the contraction of K2 gives a multiple of K1.
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M >
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![`&minus`(ContractIndices(K3, [[3, 6]]), K2)](/support/helpjp/helpview.aspx?si=5552/file05886/math132.png)
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| (2.4) |
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M >
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![`&minus`(ContractIndices(K2, [[2, 4]]), `&mult`(2, K1))](/support/helpjp/helpview.aspx?si=5552/file05886/math139.png)
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| (2.5) |
We check that K2 can be constructed from K1 tensor K1 by rearranging the indices and by skew-symmetrization.
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M >
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![T0 := RearrangeIndices(`&tensor`(K1, K1), [1, 3, 2, 4])](/support/helpjp/helpview.aspx?si=5552/file05886/math154.png)
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| (2.6) |
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M >
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![T := `&mult`(2, SymmetrizeIndices(T0, [1, 2], "SkewSymmetric"))](/support/helpjp/helpview.aspx?si=5552/file05886/math161.png)
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| (2.7) |
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M >
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| (2.8) |
Example 2.
We create a 2 dimensional vector bundle over E with fiber coordinates [p, q].
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M >
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![DGsetup([x, y, z], [p, q], E)](/support/helpjp/helpview.aspx?si=5552/file05886/math185.png)
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| (2.9) |
Define the possible Kronecker delta tensors on the fibers of E.
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E >
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| (2.10) |
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E >
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| (2.11) |
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