Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 14

To obtain the divergence as expressed by (1.84), this must be divided by the volume
of the parallelepiped given by (1.68), leading to

∂ u i v

√ g . 1

√ g

∂

i

∇· V = (1.90)

∇ φ, is a vector representing the max-
imum rate of change of φ in direction and magnitude. A di

The gradient of a scalar functionφ, written

ff erential displacement

d r results in a di ff erential change in φ given by

∂φ

∂ u i du i

d φ =∇ φ · d r

=

. (1.91)

The contravariant components of d r , by the first of relations (1.15), are

du i b i = · d r , (1.92)

leading to

∂ u i

b i ∂φ ∇ φ − · d r = 0. (1.93)

Since the displacement d r is arbitrary, the gradient of an arbitrary scalar function
φ is

b i ∂φ ∇ φ =

∂ u i ,

(1.94)

expressed in terms of the reciprocal base vectors. Substituting b i for the vector V

in (1.17) gives

b i b i b j b j

ij b j ,

= · = g

(1.95)

so that, in terms of the base vectors,

ij ∂φ

∇ φ = b j g

∂ u i .

(1.96)

2

φ, and is considered to be the
divergence of the gradient. The contravariant components of the gradient from
(1.96) are

Finally, the Laplacian of a scalar field φ is ∇

ji ∂φ ij ∂φ

∂ u j = g

g

∂ u j ,

(1.97)

and direct substitution in (1.90) yields

∂ u i √ gg ∂ u j

1
√ g

∂

ij ∂φ 2

∇·∇ φ =∇ φ = . (1.98)