The simplest interpretation of the Kronecker delta is as the discrete version of the delta function defined by
(1)
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The Kronecker delta is implemented in the Wolfram Language as KroneckerDelta[i, j], as well as in a generalized form KroneckerDelta[i, j, ...] that returns 1 iff all arguments are equal and 0 otherwise.
It has the contour integral representation
(2)
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where is a contour corresponding to the unit circle and and are integers.
In three-space, the Kronecker delta satisfies the identities
(3)
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(4)
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(5)
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(6)
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where Einstein summation is implicitly assumed, , 2, 3, and is the permutation symbol.
Technically, the Kronecker delta is a tensor defined by the relationship
(7)
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Since, by definition, the coordinates and are independent for ,
(8)
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so
(9)
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and is really a mixed second-rank tensor. It satisfies
(10)
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(11)
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(12)
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(13)
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(14)
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