Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 7

Comparing the left and right sides of this equation, we find that

ik i

g g kj = δ

j .

(1.40)

ik multiplied by the 3

3matrixof
components of g kj yields the unit matrix. Hence, the matrix representing g

Thus, the 3 × 3 matrix of components of g ×

ik is the
inverse of the matrix representing g kj . Since the inverse of a matrix is equal to the
matrix of cofactors divided by the determinant, we have

ik
g ,

= Δ

ik

g (1.41)

ik

Δ ×

where is the cofactor of the element g ik of the 3 3 matrix of components of

g kj and g is the determinant given by

g 11 g 12 g 13
g 21 g 22 g 23
g 31 g 32 g 33

g = . (1.42)

ik ik

Symmetry of g follows directly from the symmetry of g ik . Now, suppose ¯g is a

ik

second-order, twice contravariant tensor with components equal to those of g in

one particular co-ordinate system. Then, by (1.40), in this co-ordinate system,

ik i

= δ

¯g g kj

j .

(1.43)

This is now a tensor equation valid in all co-ordinate systems. In another system of
co-ordinates it becomes

¯g ik g kj = δ

i

j .

(1.44)

Given g kj , we can take the inverse of the 3 3 matrix of its components as g ik , ×

obeying

g ik g kj = δ

i j .

(1.45)

Comparing (1.44) and (1.45) it is found that
g ik

¯g ik = (1.46)

in the new co-ordinate system as well, and thus they are identical in all co-ordinate
systems. The metrical coe cient of the reciprocal unitary base system is then a

second-order, twice contravariant tensor obeying the transformation law

∂ u k

∂ u i ∂ u l

g kl

ij

=

∂ u j g

. (1.47)

The transformation laws (1.28), for first-order contravariant tensor components,
and (1.36), for first-order covariant tensor components, apply to physical quantities