Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 9

and

= √ g 22 du 2 = √ g 33 du 3 ds 2 , ds 3 , (1.54)

represent infinitesimal displacements in the u 2 and u 3 directions, respectively.
Next, we consider an element of surface area da 1 , on the surface defined by u 1
constant, contained by the parallelogram formed by the infinitesimal displacements
d s 2 , d s 3 along the directions u 2 , u 3 . The area of the parallelogram is

du 2 du 3

= |

d s 2

×

d s 3 | = |

b 2 × b 3

da 1 | , (1.55)

with

( b 2

b 3 )

( b 2

b 3 ) . | b 2 × b 3 |= × · × (1.56)

Using the vector identity (A.3),

( b 2 × b 3 ) · ( b 2 × b 3 ) = ( b 2 · b 2 )( b 3 · b 3 ) − ( b 2 · b 3 )( b 3 · b 2 ), (1.57)

we find that

2

23 du 2 du 3

da 1 = g 22 g 33 − g . (1.58)

Similarly, for elements on the u 2 and u 3 surfaces their areas are

2

31 du 3 du 1

da 2 = g 33 g 11 − g (1.59)

and

12 du 1 du 2

da 3 = g 11 g 22 − g . (1.60)

An element of volume, bounded by the three co-ordinate surfaces, is given by

( b 2 × b 3 ) du 1 du 2 du 3

d v = d s 1 ·

( d s 2 × d s 3 ) = b 1 ·

. (1.61)

If we set V = b 2 × b 3 in the expression (1.17), we get

b i

b 3 ) b i . b 2 × b 3 = · ( b 2 × (1.62)

Replacing b i by its expressions (1.6) and taking the scalar product with b 1 ,wehave

( b 2 ×

b 1

b 1 · ( b 2 × b 3 ) =

b 3 ) ·

b 3 ) · ( b 2 × b 3 ) b 1 + ( b 3 × b 1 ) · ( b 2 × b 3 ) b 2

b 1 · ( b 2 ×

b 3 ) b 3 + ( b 1 × b 2 ) · ( b 2 × . (1.63)