Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 22

where B T is the inverse of A with elements b ij . Thus, the transformation (1.139) is
expressible as

τ i

j

= |

A | b ij τ . (1.141)

Writing out the implied summation in expression (1.34), we have, by the familiar
chain rule for partial derivatives,

∂ u j = ∂ u i

∂ u i

∂ u 1 ∂ u 1 ∂ u j + ∂ u i ∂ u 2 ∂ u 2 ∂ u j + ∂ u i ∂ u 3 ∂ u 3

i j .

∂ u j = δ

(1.142)

Choosing

b kj = ∂ u k

∂ u j ,

(1.143)

relation (1.142) can be written as

i j ,

a ki b kj = δ (1.144)

or, in matrix notation, as

A T B B T A = = I , (1.145)

where I is the unit matrix, confirming the choice (1.143) of the elements of
B T A − 1 . Taking the determinant of both sides of (1.145), =

| A || B |= 1, (1.146)

A T B T since | |=| A | and | |=| B | . The transformation (1.141) then takes the form

j J u 1

∂ u i
∂ u j τ

, u 2 , u 3

1

τ i

j

= b ij τ = . (1.147)

| B |

u 1 , u 2 , u 3

j transforms as a contravariant vector, except that in the transforma-
tion it is divided by the Jacobian. Thus, it is not a true tensor but is a tensor density .

The quantity τ

1.1.9 Cartesian tensors
Tensor analysis is considerably simplified when carried out in Cartesian co-ordinate
systems. Continuing to confine our attention to right-handed systems, a point
P ( u 1 , u 2 , u 3 ) is at Cartesian co-ordinates u 1 x 1 , u 2 x 2 , u 3 x 3 .The

unit vectors e 1 , e 2 , e 3 in the directions of increasing x 1 , x 2 , x 3 form a right-handed
system with metrical coe

= x = = y = = z =

1. Then, the distinction between
covariant and contravariant tensors vanishes as does the distinction between tensor
and physical components. In addition, both the unitary base vectors and the recip-
rocal unitary base vectors become identical to the unit vectors e 1 , e 2 , e 3 .

cients h 1 = h 2 = h 3 =