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Double Contraction Relation

The double contraction relation for a tensor t is

t_(ij)^m^_t_(ij)^n=delta_(mn).
(1)

This equation is satisfied by

t^^^0=(2z^^z^^-x^^x^^-y^^y^^)/(sqrt(6))
(2)
t^^^(+/-1)=∓1/2(x^^z^^+z^^x^^)-1/2i(y^^z^^-z^^y^^)
(3)
t^^^(+/-2)=∓1/2(x^^x^^+y^^y^^)-1/2i(x^^y^^-y^^x^^),
(4)

where the hat denotes zero trace, symmetric unit tensors. These tensors are used to define the spherical harmonic tensor.

See also

Spherical Harmonic Tensor, Tensor

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References

Arfken, G. "Alternating Series." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 140, 1985.

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Double Contraction Relation

Cite this as:

Weisstein, Eric W. "Double Contraction Relation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DoubleContractionRelation.html

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