Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 8

more commonly known as vectors. The transformation laws (1.37) and (1.47) for
the metric tensors are representative of those defining second-order, twice covariant
and twice contravariant tensors. These transformation laws are easily generalised
to define tensors of arbitrary order and even tensors of mixed contravariance and
covariance.
When two tensors are multiplied and summed over one or more indices, such as
in (1.43), the result is called the contracted product . The contracted product can be
used to generate a test for tensor character . Suppose we have a quantity t ( i , j , k )
with three indices i , j , k . Further suppose we have two contravariant vectors, u and
w , and a covariant vector, v , and we form the triple contracted product

= t ik ,

t ( i , j , k ) u i

k

v j w (1.48)

and find that it is invariant in all co-ordinate systems. Then t ik is a third-order tensor,
once contravariant in the index j , twice covariant in the indices i and k .
Such tests for tensor character can take many forms. For example, when we
introduced the Kronecker delta in relation (1.7), we wrote it as though it were a
second-order mixed tensor. If this were so, it would transform as
δ l = ∂ u k

∂ u i ∂ u j

i j ,

∂ u l δ

(1.49)

whereδ l is the Kronecker delta in the new co-ordinate system. Taking the implied
summation over j , the right side reduces to
∂ u k

∂ u i ∂ u i

∂ u k

∂ u l = δ l , ∂ u l =

(1.50)

as required by the assumed transformation law (1.49). This identifies the Kronecker
delta as a second-order mixed tensor, as assumed by our adopted notation.

1.1.5 Metric tensors and elements of arc, surface and volume
An infinitesimal displacement d s 1 , along the u 1 direction from a point P with
co-ordinates ( u 1

, u 2 , u 3 ), from (1.4), is

d s 1 = b 1 du 1 . (1.51)

The magnitude of this infinitesimal displacement is

b 1

= √ g 11 du 1 du 1 b 1 du 1 ds 1 =| b 1 | = · , (1.52)

using the second of relations (1.13). Similarly,

b 2 du 2 b 3 du 3 d s 2 = , d s 3 = , (1.53)