Environmental Engineering Reference
In-Depth Information
Thus the counterparts of Eqs. (7.140), (7.141) and (7.143) in two-dimensional cases
are
1
r
PM
u
(
M
)=
ρ
(
P
)
ln
d
σ
,
(7.144)
D
1
r
PM
u
(
M
)=
M
(
p
)
ln
d
s
,
(7.145)
C
ln
d
s
)
∂
∂
1
r
PM
u
(
M
)=
C
τ
(
p
,
(7.146)
n
where
D
is a plane domain,
C
is a plane curve, and
n
is the normal of curve
C
from
the negative charge side to the positive charge side.
The integral in Eq. (7.144) is called the
logarithmic potential
. The integrals in
Eqs. (7.145) and (7.146) are called the
single-layer and the double-layer potential
in a plane
, respectively.
7.8.2 Generalized Integrals with Parameters
The point
M
is a parameter in the potentials that is defined by the integrals. If
M
is outside the integral domain, the integral is a normal integration provided that all
densities of electric-charges or dipole moments are continuous. When
M
is inside
the integral domain, however, the integral is a generalized one because
P
→
M
(
lim
1
/
r
PM
)=
∞
.
To examine properties of potentials, we must first discuss some properties of gen-
eralized integrals with parameters, which we do here by using surface integrals as
examples.
Let
M
0
be a point on surface
S
.
F
is a function of three variables (spatial
coordinates of
P
) with coordinates of
M
as the parameter. Let d
S
be the area element
at point
P
. Consider a surface integral
(
M
,
P
)
v
(
M
)=
F
(
M
,
P
)
d
S
.
(7.147)
S
When
M
is outside of
S
,
v
(
M
)
is continuous at
M
provided that
F
(
M
,
P
)
is a contin-
uous function of
P
.When
M
=
M
0
∈
S
and lim
P
→
M
F
(
M
,
P
)=
∞
,however,the
v
(
M
)
in
Eq. (7.147) is a generalized integral.
Definition.
Let
M
0
be a point on
S
. If, for any
0, there always exists a neigh-
boring region
V
of
M
0
and a surface
S
δ
containing
M
0
on
S
such that, for all
M
ε
>
∈
V
,
F
(
M
,
P
)
d
S
(spatial coordinates of
P
are the integral variables,
M
is a parame-
S
δ
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