Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 38

The dual vector of a second-order tensor T jk is given by the double contraction

v i = ξ ijk T jk . Thus v 1 = T 23 − T 32 , v 2 = T 31 − T 13 , v 3 = T 12 − T 21 . Only the antisym-

metric part of T jk contributes to the dual vector.
One-half of the dual vector of the vector gradient ∂ u k /∂ x j is therefore

2 ξ ijk ω jk + e jk

1
2 ξ ijk ω jk =

1 1

2 ξ ijk ∂ u k

1
2 (

Ω i = =

∂ x j =

∇× u ) i .

(1.230)

If T jk is an antisymmetric tensor, it may be recovered from its dual vector v i
since

ξ ijk v i = ξ ijk ξ ilm T lm = T jk − T kj = 2 T jk . (1.231)

Thus,

1
2 ξ ijk v i .

T jk = (1.232)

Applied to ω ji ,wefindω ji = ξ kji Ω k .Then

ω ji dx j = ξ kji Ω k dx j = ( Ω × d r ) i , (1.233)

where ( d r ) i = dx i .

The displacement of the material particle P , originally at x k + dx k ,relativeto
the material particle P , originally at x k ,is

du i = e ij dx j + ( Ω × d r ) i . (1.234)

Hence, ω ij represents a rigid-body rotation of P with respect to P through the
angle Ω . It does not, therefore, represent a true deformation of the medium.
If, on the other hand, the displacement field is entirely rigid-body in the neigh-
bourhood of P and P ,andif P is a third neighbouring material particle, originally
at x k + δ x k , the angle P PP = θmust be preserved and the lengths PP , PP must

remain unchanged in the deformation. The scalar product d r · δ r before and after

deformation is invariant. Thus,

dx k

∂ x j dx j ∂ x l δ x l

∂ u k ∂ u k

dx k δ x k = + δ x k + . (1.235)

To lowest order in small quantities, we have

∂ u k
∂ x j dx j δ x k

∂ u k
∂ x l δ x l dx k ,

dx k δ x k ≈ dx k δ x k + + (1.236)