Kronecker Delta

The simplest interpretation of the Kronecker delta is as the discrete version of the delta function defined by

(1)

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)

where is a contour corresponding to the unit circle and and are integers.

In three-space, the Kronecker delta satisfies the identities

	(3)
	(4)
	(5)
	(6)

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)

Since, by definition, the coordinates and are independent for ,

(8)

so

(9)

and is really a mixed second-rank tensor. It satisfies

			(10)
			(11)
			(12)
			(13)
			(14)