On isotropic tensors

Source abstract

1. An isotropic tensor is one the values of whose components are unaltered by any rotation of rectangular axes (with metric σ i ( dx i ) 2 ). Those up to order 4 in 2 and 3 dimensions have many applications. The results suggest a general theorem for tensors of order m in n dimensions, that any isotropic tensor can be expressed as a linear combination of products of δ and є tensors, where δ ij = 1 if i = j and 0 otherwise, and is 0 if any two of the i 1 to i n are equal, 1 if i 1 … i n is an even permutation of 1, 2, 3, …, n , and – 1 if it is an odd permutation.