Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 46

r | from expression (1.173), where R . The volume integrals can be simplified

using the theorem of Gauss, generalized to a tensor field T ijk ··· ,
∂ T ijk ···
∂ x r

=| r −

T ijk ··· ν r d

d V= S , (1.290)

the surface integral being taken over the surface S with unit outward normal vector

V

ν i enclosing the volume .If T ijk ··· is the vector field u j ,

∂ u j
∂ x i d

u j ν i d

V= S .

(1.291)

Contracting both sides, we find
∂ u i
∂ x i d V=

u i ν i d S ,

(1.292)

or

u

∇· u d V= · ν d S , (1.293)

the familiar, elementary form of Gauss's theorem. Taking the dual vector of the
vector gradient as the tensor field, we get

ξ ijk ∂ u k

∂ x j d

V= ξ ijk u k ν j d S , (1.294)

or

( ν

∇× u d V= × u ) d S . (1.295)

Using the two vector calculus identities,

F ( r )
R

·∇ 1
R

= ∇ ·

F ( r )
R +

∇ · F ( r ) , (1.296)

and

F ( r )
R R +∇ 1

= ∇ ×

F ( r )

∇ × F ( r ), × (1.297)

R

together with the forms (1.293) and (1.295) of Gauss's theorem, the potentials for
the body force density can be converted to

·∇ 1
R

d V −

F ( r ) F ( r )

R ·

1
4π

1
4π

ν d S ,

L F = (1.298)

and

F ( r )

×∇ 1
R

d F ( r )

R ×

1
4π

1
4π

V − S . A F = ν d (1.299)