Geology Reference
In-Depth Information
The following theorem, known as the
Green
'
s
third identity
, is probably the most important
result for potential functions, in particular for
harmonic fields.
Green's Third Identity
Let
§
D
§(
r
)
be a scalar field
,
defined in a closed
region
R
with boundary S
(
R
),
and let P be a fixed
point in
R
.
If r is the distance of any other point
Q
2
R
from P
,
then
:
Fig. 4.20
Geometry of problem for Green's third identity
Z
I
1
4
1
r
r
1
4
1
r
@§
@n
dS
2
§dV
C
§.P/
D
element versor
n
is directed
inwards
, because
the sphere itself belongs to the complement of
E
. Therefore, for any scalar field
f
defined on
the surface of †, the directional derivative in
the direction
n
is the opposite of a directional
derivative in the direction of
Q
, so that it results:
R
S
.
R
/
1
r
dS
I
1
4
§
@
@n
S.
R
/
(4.68)
Proof
Let us consider the scalar field ¥
D
1/
r
.
In this instance, the Green's second identity as-
sumes the form:
I
@f
@r
Furthermore, when we integrate over the sur-
face of †, the points
Q
have a fixed distance
r
D
R
from
P
,
R
being the radius of †, and the spatial
average of
§
over the surface of this sphere is:
@f
@n
D
1
r
1
r
dS
@§
@n
§
@
@n
S.
R
/
1
r
r
2
1
r
dV
Z
I
2
§
§
r
D
1
4 R
2
h
§
i
†
D
§dS
R
†
2
(1/
r
)
D
0 for any point
Q
¤
P
. Therefore, let us consider a small sphere,
†, about
P
(Fig.
4.20
)andtheset
E
D
R
†.
In
E
, the previous identity can be rewritten as
follows:
Z
We
know
that
r
Clearly, if we take the limit as
R
!
0of
this expression, we obtain simply §(
P
). At the
same time, the integral at left-hand side of (
4.69
)
will be extended to the whole region
R
.Thelast
integral at the right-hand side of (
4.69
) can be
evaluated as follows:
I
1
r
1
r
dS
I
1
r
r
@§
@n
§
@
2
§dV
D
1
r
1
r
dS
@n
@§
@n
§
@
E
S
.
E
/
I
1
r
1
r
dS
@n
@§
@n
§
@
†
D
I
1
r
dS
D
@n
1
r
@§
@r
C
§
@
D
S.
R
/
@r
I
1
r
1
r
dS
@§
@n
§
@
†
C
I
r
2
§
dS
D
@n
1
r
@§
@r
1
D
†
(4.69)
†
I
1
R
@§
@r
dS
In evaluating the integral over the surface of
†, we must take into account that any surface
D
4
h
§
i
†
†
