Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 26

ϑ

r r

r

S

r

r | Figure 1.1 Small sphere of radius = , surrounding the source point

r , as the source point and field point, r , approach coincidence.

R =| r −

For r r , the gradient of 1/ R is

1
R

1
|

r
R 3 .

r −

∇ =∇ =− (1.170)

r | r −

Taking the divergence yields

2 1
R

2 1
|

∇ =∇

r − r |

2 x 1

x 3 2 x 1 2 2 x 2 x 2 2 2 x 3

− + − + −

3
2

3
R 3 =

= − 0. (1.171)

R 5

Now suppose the source point is close to the field point so that R is the very small
quantity . Surround the source point with a sphere of radius , as illustrated in
Figure 1.1.
Now take the Laplacian of expression (1.168). As the source point and field point
approach coincidence, the right side approaches

u ( r )

2 1
R

d

1
4π ∇·

V . − ∇ (1.172)

By the divergence theorem of Gauss, the volume integral throughout the small
sphere is the integral over its surface of the outward normal component of ∇ (1/ R ),

3 . The surface integral is then 3 2

equal to 4π. Thus, the volume

integral (1.168) is the general solution of the Poisson equation (1.163).
An arbitrary vector field u can then be broken into a lamellar part, determined
within a constant by the scalar potential φ given by (1.168), and a solenoidal part

− / − / × 4π =−