Introductions to Kroneckerdelta Introduction to the Tensor Functions
Introductions to KroneckerDelta Introduction to the tensor functions General The tensor functions discrete delta and Kronecker delta first appeared in the works L. Kronecker (1866, 1903) and T. Levi–Civita (1896). Definitions of the tensor functions For all possible values of their arguments, the discrete delta functions d n and d n1, n2, ¼ , Kronecker delta functions dn and dn1,n2,¼, and signature (Levi–Civita symbol) ¶n1, n2,¼ , nd are defined by the formulas: 1 n 0 H L H L d n 0 True d n1, n2, ¼ 1 ; n1 n2 ¼ 0 H L d n1, n2, ¼ 0 ; Ø n1 n2 ¼ 0 H L 1 n 0 dn H 0 TrueL d ¼ Î Î ¼ n1,n2,¼ 1 ; n1 n2 n1 Q n2 Q d Ø ¼ n1,n2,¼ 0 ; n1 n2 . ì ì ì In other words, the Kronecker delta function is equal to 1 if all its arguments are equal. In the case of one variable, the discrete delta function d n coincides with the Kronecker delta function dn. In the case of several variables, the discrete delta function d n1, n2, ¼, nm coincides with Kronecker delta function dn ,n ,¼,n ,0: 1 2 m H L d n dn H L d n , n , ¼, n d 1 2 m n1,n2,¼,nm,0 H L t ¶n1, n2,¼ , nd -1 H L ¶n1, n2,¼ , nd 0 ; ni nj 1 £ i £ d 1 £ j £ d, H L where t is the number of permutations needed to go from the sorted version of n1, n2, ¼, nd to n1, n2, ¼, nd . ì ì Connections within the group of tensor functions and with other function groups 8 < 8 < Representations through equivalent functions http://functions.wolfram.com 2 The tensor functions d n , d n1, n2, ¼ , dn, and dn1,n2,¼ have the following representations through equivalent functions: H L H L d n and d n1,
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