Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 24

The transformation (1.149) for a vector may be described as associating a scalar
with each spatial direction by a linear, homogeneous relation in the direction
cosines. Then a second-order tensor associates a vector with each spatial direc-
tion by a linear, homogeneous relation in the direction cosines. This leads to a
generalisation that a tensor of order n associates a tensor of order n

−

1 with each
spatial direction by a linear, homogeneous relation in the direction cosines. For a
second-order tensor T ij , the transformation law (1.149) generalises to
T kl = c ki c lj T ij , (1.155)

giving the tensor T kl in the new co-ordinate system. For a tensor of arbitrary order,
the transformation generalises to
T kl ··· = c ki c lj ··· T ij ··· . (1.156)

The double contraction

v i = ξ ijk T jk (1.157)

of the second-order tensor T jk leads to the vector v i , called the dual vector of T jk ,
with components

v 1 = T 23 − T 32 , 2 = T 31 − T 13 , 3 = T 12 − T 21 , (1.158)

using the properties of the alternating tensor. The dual vector therefore depends
only on the antisymmetric part of T jk . Conversely, if the dual vector vanishes, the
tensor T jk is symmetric. The contraction

ξ ijk v i = ξ ijk ξ ilm T lm (1.159)

of the dual vector leads to a second-order tensor determined by evaluating the
product ξ ijk ξ ilm of alternating tensors. If ( i , j , k ) is a cyclic permutation of indices,
the permutation ( i , l , m ) will be cyclic if j = l and k = m , giving the product the

value + 1. If ( i , j , k )and( i , l , m ) have opposite cyclicity, the product has the value

− 1. Otherwise, the product vanishes. The product can then be written in terms of

Kronecker deltas as ξ ijk ξ ilm = δ lj δ mk − δ lk δ mj and the contraction of the dual vector

becomes

δ lj δ mk

− δ lk δ mj T lm ξ ijk v i = = T jk − T kj . (1.160)

For the antisymmetric part of T jk we have T jk 2 T jk .

Hence, the antisymmetric part of T jk can be recovered from the dual vector as

= − T kj , giving T jk − T kj =

1
2 ξ ijk v i .

T jk = (1.161)

This tensor is called the dual antisymmetric tensor of the vector v i .