Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 2

In general, a vector can be defined in a space of arbitrary dimensions numbering
two or greater. Our applications will be confined to a space of three dimensions
and we will adopt this limitation. Let r be the radius vector to a point P specified
by the three curvilinear co-ordinates u 1 , u 2 , u 3 .Then r is described by the function

r u 1 , u 3 . , u 2 r = (1.1)

The change in r due to di ff erential displacements along the co-ordinate curves

is then

= ∂ r + ∂ r + ∂ r

∂ u 1 du 1 ∂ u 2 du 2 ∂ u 3 du 3

d r . (1.2)

If two of the curvilinear co-ordinates are held fixed, the third describes a curve in
space. Moving along u j a unit distance from the point P , the change in r is equal
to b j = ∂ r /∂ u j .The unitary vectors

∂ r
∂ u 1 , b 2

∂ r
∂ u 2 , b 3

∂ r
∂ u 3 ,

b 1 = = = (1.3)

associated with the point P , form a base system for all vectors there. Any vector
at that point can be expressed as a linear, homogeneous combination of the unitary
base vectors. In particular,

b 1 du 1 b 2 du 2 b 3 du 3 d r = + + . (1.4)

erent units measured along the
co-ordinate directions. For instance, in thermodynamics the co-ordinates may rep-
resent pressure, volume and temperature, all in di

The vector space, thus defined, may have di ff

ff erent units. This is an example

of a ne geometry . More commonly, we will be concerned with vectors in metric

geometry , where the unitary vectors can be referred to a common unit of length.
This allows measurement of the absolute value of a vector of arbitrary orientation
and the distance between neighbouring points.
The three unitary base vectors b 1 , b 2 , b 3 define a parallelepiped with volume

V = b 1 · ( b 2 × b 3 ) = b 2 · ( b 3 × b 1 ) = b 3 · ( b 1 × b 2 ), (1.5)

using the properties (A.1) of the triple scalar product. A new triplet of base vectors,
defined as

1
V ( b 2

1
V ( b 3

1
V ( b 1

b 1 b 3 ), b 2 b 1 ), b 3 = × = × = × b 2 ), (1.6)

are, in turn, orthogonal to the planes formed by the pairs ( b 2

b 1 ) and
( b 1 × b 2 ) . Adopting the range convention , whereby superscripts and subscripts are
implied to range over the values 1, 2, 3, the two triplets of base vectors are found
to obey

× b 3 ) , ( b 3 ×

b i

i

· b j = δ

j ,

(1.7)