Environmental Engineering Reference
In-Depth Information
where
A
is a constant, constant
δ
satisfies 0
<
δ
≤
1, and
r
P
1
P
2
is the distance
between
P
1
and
P
2
;
4. The absolute solid angle of any part
S
Δ
of
S
viewing from any point
O
in the
space is bounded.
Theorem 3
.Let
S
be a
Ляпунов
surface. For any two points
P
and
M
on
S
,wehave
≤
cos
(
PM
,
n
)
C
r
2
−
δ
PM
,
r
PM
where constant
C
does not depend on points on
S
, constant
δ
satisfies 0
<
δ
≤
1,
r
PM
is the distance between
P
and
M
,and
n
is the external normal of
S
at
P
.
This theorem plays an important role in potential theory. Hereafter, surfaces in
single-layer and double-layer potentials are all assumed to be
Ляпунов
surfaces.
The surfaces in real applications normally satisfy the four conditions for
Ляпунов
surfaces.
7.8.4 Properties of Surface Potentials
in single-layer potentials
(Eq. (7.141)) and double-layer potentials (Eq. (7.143)) are continuous on
S
, the po-
tentials must be harmonic functions for all
M
not on
S
.
Theorem 4.
If density functions
μ
(
P
)
and
τ
(
P
)
Proof.
When
M
is not on
S
, integrands in Eqs. (7.141) and (7.143) are continu-
ous and differentiable with respect to
M
up to any order. We can thus take deriva-
tives with respect to
M
(
x
,
y
,
z
)
inside the integration. For the single-layer potential
(Eq. (7.141)), we have
1
r
PM
d
S
Δ
u
(
M
)=
μ
(
P
)
Δ
=
0
,
S
1
r
PM
=
because
Δ
0 . For the double-layer potential Eq. (7.143), we also have
d
S
)
∂
∂
1
r
PM
Δ
u
(
M
)=
τ
(
P
Δ
=
0
,
n
S
because
1
r
PM
cos
(
PM
,
n
)
=
∂
∂
∂
∂
n
=
∂
,
Δ
n
Δ
.
r
PM
n
∂
Theorem 5.
Let
S
be a bounded smooth surface. If
μ
(
P
)
is continuous on
S
,the
single-layer potential (Eq. (7.141)) is convergent when
M
∈
S
and continuous on
S
.
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