Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 20

while that in the u 2 direction has components
h 2 ∂ u 1

∂ u 2 du 2 ∂ u 2 , ∂ u 2 ∂ u 2 , ∂ u 3

, (1.128)

andthatinthe u 3 direction has components
h 3 ∂ u 1

∂ u 3 du 3

∂ u 3 , ∂ u 2 ∂ u 3 , ∂ u 3

. (1.129)

Writing the triple scalar product in its determinant form (1.112), the volume ele-
ment becomes

∂ u 1 ∂ u 2 ∂ u 3

/∂ u 1 /∂ u 1 /∂ u 1

∂ u 1 /∂ u 2 ∂ u 2 /∂ u 2 ∂ u 3 /∂ u 2 h 1 h 2 h 3 du 1 du 2 du 3 . (1.130)

∂ u 1 /∂ u 3 ∂ u 2 /∂ u 3 ∂ u 3 /∂ u 3

The determinant is recognised as the familiar Jacobian and, from (1.109), the
volume element in the new co-ordinate system becomes
d v = J u 1

d v.

, u 2 , u 3

(1.131)

u 1 , u 2 , u 3

Thus, the volume element is not a true scalar but is multiplied by the
Jacobian in transforming to a new co-ordinate system. Quantities that transform
in such a fashion are called tensor capacities . The volume element is a scalar
capacity .
As an example of a tensor density, consider a quantity resembling a vector asso-
ciated with a twice covariant, antisymmetric, second order tensor,
⎝

⎠ = ⎝ ⎠

t 11 t 12 t 13 0 t 12 − t 31

t 21 t 22 t 23 − t 12 0 t 23 . (1.132)

t 31 t 32 t 33 t 31 − t 23 0

Because of its antisymmetry, there are only three independent components, and
these may be taken as the components τ

1 2 3

t 12 , of a quantity
resembling a contravariant vector, τ . In Cartesian co-ordinates, the double contrac-
tion ξ ijk t jk is called the dual vector of the second order tensor. It has components
t 23

= t 23 , τ = t 31 , τ =

t 21 , and, hence, it is equal to 2 τ . In transforming from
an old system of co-ordinates ( u 1

− t 32 , t 31 − t 13 , t 12 −

, u 2 , u 3 ) to a new system ( u 1 , u 2 , u 3 ), the twice

covariant, second-order tensor, t ij , transforms as (1.37),

∂ u k ∂ u j

∂ u i

t kl =

∂ u l t ij .

(1.133)

Since t ij is antisymmetric,

∂ u k ∂ u j

∂ u i

t kl =−

∂ u l t ji .

(1.134)