The array ijk of order 333 defined by is known as the alternating tensor (Ames & Mumaghan,

The array ϵijk of order 3×3×3 defined by is known as the alternating tensor (Ames & Mumaghan, 1929, p. 440). For any 3 × 3 matrix a i

r

, show that

ϵijka i

ra j

sa k

t = ϵrst det(A).

Hence show that ϵijk is an isotropic tensor under O

+, the orthogonal group with positive determinant (Jeffreys & Jeffreys, 1956, Sections 2.07, 3.03).

Write down the generalization of the alternating tensor appropriate for a p × p × p array.

ϵ123 = ϵ231 = ϵ321 = 1

ϵ2