Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 15

1.1.7 Orthogonal co-ordinates
Up to this point, we have not imposed any requirement on the unitary base vectors
other than that they must not be coplanar. Our applications of scalar, vector and
tensor analysis will be, without exception, carried out in orthogonal co-ordinate
systems. Thus, from now onwards, we will assume that we are dealing with ortho-
gonal co-ordinate systems.
The squared magnitudes of the reciprocal unitary base vectors, defined in terms
of the unitary base vectors by (1.6), are given by

1
V 2 ( b 2 × b 3 )

b 1 · b 1

= · ( b 2 × b 3 )

V 2 ( b 2

b 2 ) ,

1

= · b 2 )( b 3 · b 3 ) − ( b 2 · b 3 )( b 3 · (1.99)

using the vector identity (1.57). Permuting subscripts in the vector identity yields,

1
V 2 ( b 3 ×

b 2 b 2 · = b 1 ) · ( b 3 × b 1 )

V 2 ( b 3

b 3 ) ,

1

= · b 3 )( b 1 · b 1 ) − ( b 3 · b 1 )( b 1 · (1.100)

and

1
V 2 ( b 1

b 3 b 3 · = × b 2 ) · ( b 1 × b 2 )

V 2 ( b 1 ·

b 1 ) .

1

= b 1 )( b 2 · b 2 ) − ( b 1 · b 2 )( b 2 · (1.101)

In an orthogonal co-ordinate system, the metric tensor has only three non-vanishing
components,

g 11 = b 1 · b 1 , 22 = b 2 · b 2 , 33 = b 3 · b 3 . (1.102)

Then, from (1.66) and (1.67), the three non-vanishing components of the metric
tensor yield

V 2 = ( b 1 · b 1 )( b 2 · b 2 )( b 3 · b 3 ). (1.103)

Thus, from (1.99), (1.100) and (1.101), the squared magnitudes of the reciprocal
unitary base vectors become

1 1 1

b 1 b 1

b 1 , b 2

b 2

b 2 , b 3

b 3

· = · = · =

b 3 ,

(1.104)

b 1 · b 2 · b 3 ·

the reciprocals of the squared magnitudes of the original unitary base vectors. In an
orthogonal co-ordinate system, the reciprocal unitary base vectors defined by (1.6)
are parallel to the unitary base vectors but are subject to the scalings