Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 3

i j is the Kronecker delta , which is equal to unity for i

where δ j , zero otherwise.

The original three unitary base vectors can be recovered from the new triplet of
base vectors, using the properties (A.4) of the quadruple vector product, giving

=

V b 2 b 3 , b 2 V b 3 b 1 , b 3 V b 1 b 2 . b 1 = × = × = × (1.8)

The triplet, b 1 , b 2 and b 3 , are called reciprocal unitary vectors . They may be used
as a base system of the vector space as an alternative to the original three unitary
base vectors. In this base system, the di

ff erential d r becomes expressible as

b 1 du 1 b 2 du 2 b 3 du 3 . d r = + + (1.9)

Adopting the summation convention , whereby a repeated superscript or subscript
implies summation over that index, equating the expressions (1.4) and (1.9) for the
di ff erential d r gives

b i du i b j du j . d r = = (1.10)

Taking the scalar product of this equation, first with b k ,thenwith b k , and using the
orthogonality relation (1.7), produces

du k b k b j du j , du k b i du i = · = b k · . (1.11)

Replacing the superscript k by i in the first relation, and replacing the subscript k by
j in the second, relates the components of d r in the unitary and reciprocal unitary
base systems by

du i

ij du j , du j = g ji du i

= g , (1.12)

with

ij

b i b j

ji

g = · = g , g ji = b j · b i = g ij . (1.13)

An arbitrary vector V may be expressed as a linear combination of its compon-
ents in the unitary base system b 1 , b 2 , b 3 , or in the reciprocal unitary base system
b 1 , b 2 , b 3 ,as

i b i

= v j b j V = v , (1.14)

where, by (1.7), the components in each system are

i

b i v = V · , j = V · b j . (1.15)

Again, with scalar multiplication and using the orthogonality relation (1.7), the
components in the unitary base system and in the reciprocal unitary base system
are found to be related by

i ij i

v = g v j , j = g ji v . (1.16)