Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 17

e 1 / h 1 e 2 / h 2 e 3 / h 3
V 1 / h 1 V 2 / h 2 V 3 / h 3
W 1 / h 1 W 2 / h 2 W 3 / h 3

V × W = h 1 h 2 h 3

e 1 e 2 e 3
V 1 V 2 V 3
W 1 W 2 W 3

= . (1.111)

The cross product takes its familiar form in all orthogonal co-ordinate systems.
From this, it follows that the triple scalar product (A.1) can be written in the form
of a determinant as

a 1 a 2 a 3
b 1 b 2 b 3
c 1

a · ( b × c ) = . (1.112)

c 2 c 3

Substitution in formula (1.79) for the curl gives

h 1 e 1 h 2 e 2 h 3 e 3

1
h 1 h 2 h 3

∂/∂ u 1 ∂/∂ u 2 ∂/∂ u 3 ∇× V = , (1.113)

h 1 V 1 h 2 V 2 h 3 V 3

while substitution in formula (1.90) for the divergence gives

∂
∂ u 1 ( h 2 h 3 V 1 ) ∂ u 3 ( h 1 h 2 V 3 )

1
h 1 h 2 h 3

+ ∂ + ∂ ∇· V =

∂ u 2 ( h 3 h 1 V 2 )

. (1.114)

The gradient (1.96) of a scalar function φ becomes

h 1 ∂φ

1

h 2 ∂φ

1

h 3 ∂φ

1

∇ φ =

∂ u 1 e 1 + ∂ u 2 e 2 + ∂ u 3 e 3 ,

(1.115)

b j by 1/ h i .
The Laplacian (1.98) of φ, in an orthogonal system, takes the form

ij

b i

on replacing g = ·

∂

∂ u 1 h 2 h 3 ∂ u 1 ∂ u 2 h 3 h 1 ∂ u 2 ∂ u 3 h 1 h 2 ∂ u 3

1
h 1 h 2 h 3 h 1 ∂φ

∂

h 2 ∂φ

∂

h 3 ∂φ

2

∇ φ = + + . (1.116)

In Cartesian co-ordinates, the vector identity (A.14),

2 V ,

∇× ( ∇× V ) =∇ ( ∇· V ) −∇ (1.117)

is easily established. In Cartesian co-ordinates the unit vectors are independent of
position, and the operator

2 V can be interpreted as the Laplacian operating on
each of the Cartesian components of V ,or

∇

2 V 2 V j .

∇ = e j ∇ (1.118)