Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 25

1.2 Separation of vector fields
A vector field u may be represented in terms of its curl and divergence. If we write
u as the sum of the gradient of a scalar potential φand the curl of a vector potential
ψ , as in the classical Helmholtz separation, we have

u =∇ φ +∇× ψ . (1.162)

Since the divergence of a curl vanishes identically, as does the curl of a gradient,
we find Poisson equations for the scalar and vector potentials,

2

∇ φ =∇· u , (1.163)

2 ψ , valid in Cartesian co-

and, using the identity −∇× ( ∇× ψ ) +∇ ( ∇· ψ ) = ∇

ordinates,

2 ψ

∇ =−∇× u +∇ ( ∇· ψ ). (1.164)

The potentials in the Helmholtz separation are not unique, since the
potentials

φ = φ + C and ψ = ψ +∇ χ, (1.165)

where C is any constant and χ is an arbitrary scalar function of position, will yield
the identical vector field u . Removing the uncertainty ∇ χ in ψ is called choosing

the gauge . If we choose the gauge so that ∇· ψ = 0, then for another choice of

vector potential, ψ ,wehave

ψ = ∇·

2 2

∇· ψ +∇

χ = ∇ χ, and we have another

Poisson equation for χ,

2

ψ . ∇ χ =∇· (1.166)

In seismology, it is usual to choose the gauge so that ∇· ψ = 0. The vector

potential ψ then obeys the vector Poisson equation

2 ψ

∇ =−∇× u . (1.167)

It is easy to verify that the solution of the Poisson equation for the scalar poten-
tial is

∇ ·

u ( r )

1
4π

V , φ( r ) =− d (1.168)

| r − r |

where the potential at the field point at r is found by integrating over all source
points at r . The distance from the source point with co-ordinates ( x 1 , x 2 , x 3 )tothe
field point with co-ordinates ( x 1 , x 2 , x 3 ) is

x 1 −

r . x 1 2 + x 2 − x 2 2 + x 3 − x 3 2 R = = r − (1.169)