Geology Reference
In-Depth Information
'.¥
§/
C
“
@¥
region, admits a unique solution. The following
corollary of Green's first identity is related to
another boundary-value problem, the
Neumann
boundary
-
value problem
, which is to find a
function that solves Laplace's equation in the
interior of
R
given the values of the normal
derivatives on the boundary of this region.
@§
@n
@n
D
0
@¥
@n
@§
@n
'
“
.¥
§/
D
Thus, by the Green first identity we have:
I
.¥
§/
@¥
dS
D
@§
@n
Corollary 5
Any function that is harmonic in a region
R
is
determined
,
up to an additive constant
,
by the
values of its normal derivative on the boundary
S
(
R
).
@n
S
.
R
/
I
Z
'
“
.¥
§/
2
dS
D
Œ
r
.¥
§/
2
dV
S.
R
/
R
Proof
If
f
D
f
(
r
) is harmonic in a closed region
R
, then the Green first identity yields:
I
This identity can be satisfied only if all terms
are zero, because '/“>0. Therefore,
I
Z
Z
f.r/
@f
'
“
.¥
§/
2
dS
D
.
r
f/
2
dV
@n
dS
D
Œ
r
.¥
§/
2
dV
D
0
S
.
R
/
R
S
.
R
/
R
Let ¥
D
¥(
r
)and§
D
§(
r
) be two harmonic
functions in
R
, such that
f
(
r
)
D
¥(
r
)
§(
r
), and
let us assume that @¥/@
n
D
@§/@
n
. Then:
Z
Consequently, ¥
D
§
C
const
in
R
and ¥
D
§
on
S
(
R
). By continuity, we must have ¥
D
§ also
in
R
and the uniqueness is proved.
I
Another important set of properties for the
potential arises from the second Green's identity,
which can be obtained easily from (
4.61
).
Œ
r
.
¥
§
/
2
dV
D
.
¥
§
/
R
S.
R
/
@¥
@n
dS
D
0
@§
@n
Green's Second Identity
Let
¥
D
¥(
r
)
and
§
D
§(
r
)
be scalar fields
,
defined in a closed region
R
with boundary S
(
R
).
Then
:
I
Therefore,
r
(
¥
§
)
D
0in
R
and
¥
§
D
const
. This proves the theorem.
¥.r/
@§
dS
The last corollary is a uniqueness theorem for
the
mixed boundary
-
value problem
.
@n
§.r/
@¥
@n
S.
R
/
Z
Corollary 6
Let
§
be a harmonic function in a region
R
,
and
let
', “,
and g be continuous functions on S
(
R
),
with
'/“>0,
such that
:
¥.r/
r
2
¥
dV (4.66)
2
§
§.r/
r
D
R
Proof
If we interchange ¥ and § in (
4.61
)and
subtract the result from this identity, the identity
(
4.66
) immediately follows.
'§
C
“
@§
@n
D
g
(4.65)
As a corollary, when both
¥
and
§
are har-
monic in
R
, the second Green's identity becomes:
on S
(
R
).
Then
§
is uniquely determined in
R
.
Proof
Let ¥
D
¥(
r
)and§
D
§(
r
)betwohar-
monic functions in
R
, such that (
4.65
) is satisfied
on
S
(
R
). Then:
¥.r/
@§
dS
D
0 (4.67)
I
@n
§.r/
@¥
@n
S
.
R
/
