Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 10

Permuting subscripts in the first factor on the left of the vector identity (1.57),
two additional vector identities emerge,

( b 3 × b 1 ) · ( b 2 × b 3 ) = ( b 3 · b 2 )( b 1 · b 3 ) − ( b 3 · b 3 )( b 1 · b 2 ) (1.64)

and

( b 1 × b 2 ) · ( b 2 × b 3 ) = ( b 1 · b 2 )( b 2 · b 3 ) − ( b 1 · b 3 )( b 2 · b 2 ). (1.65)

Using the three vector identities, expression (1.63) can be reduced to
b 1

b 3 ) 2 b 1 ) ( b 2 b 2 ) · ( b 2 × = ( b 1 · · b 2 )( b 3 · b 3 ) − ( b 2 · b 3 )( b 3 ·

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

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

b 1 · b 1 b 1 · b 2 b 1 · b 3

= b 2 b 3

b 3 · b 1 b 3 · b 2 b 3 · b 3

· b 1 b 2 · b 2 b 2 ·

. (1.66)

Replacing the scalar products with components of the covariant metric tensor
defined in the second of relations (1.13), the determinant in expression (1.66) is
identical to that defined by (1.42). Hence,
b 1

b 3 ) 2 V 2 · ( b 2 × = = g. (1.67)

Then, the expression (1.61) for the volume element becomes
d v = √ g du 1 du 2 du 3 . (1.68)

1.1.6 The cross product and di ff erential operators
The three unitary base vectors define a parallelepiped with volume V givenbyany
of the three scalar triple products as expressed in (1.5). On taking the square root
of (1.67) this volume becomes

= √ g. V (1.69)

Suppose that, in addition to the arbitrary vector V , expressed by (1.14) as a linear
combination of its contravariant components and the unitary base vectors, we have
a second arbitrary vector W similarly expressed. Then, the cross product of the
two vectors is