Environmental Engineering Reference
In-Depth Information
The integral on the left-hand side of Eq. (7.150) tends to zero as
h
0suchthat
the integral in the right-hand side of Eq. (7.149) also tends to zero. Therefore, for
any small
→
0, there always exists a neighboring region
V
of
M
0
and a surface
S
h
containing
M
0
on
S
such that, for all
M
ε
>
∈
V
,
F
(
M
,
P
)
d
S
<
ε
.
S
h
Thus the integral in Eq. (7.147) converges uniformly at
M
0
,and
v
(
M
)
is continuous
at
M
0
.
Remark 1.
The uniform convergence of the integral in Eq. (7.147) at
M
0
is a suffi-
cient condition for the continuity of
v
at
M
0
. But, it is not a necessary condition.
Remark 2.
If the condition in Eq. (7.148) is only valid at
M
0
,i.e.
(
M
)
C
r
2
−
δ
|
F
(
M
0
,
P
)
|≤
M
0
P
,
0
<
δ
≤
1
.
we can only have the convergence of the integral in Eq. (7.147) at
M
0
, but not nec-
essarily the continuity of
v
(
M
)
at
M
0
.
7.8.3 Solid Angle and Russin Surface
Consider a surface
S
with fixed normal direction. Its normal
n
isshowninFig.7.7.
The solid angle of surface
S
viewing from point
O
refers to the angle occupied by its
projection on the spherical surface
K
of a sphere of center
O
and unit radius. If the
viewing direction is along
n
, the solid angle is defined as positive (negative).
For the case shown in Fig. 7.7, the solid angle is positive. If we reverse the normal,
the solid angle becomes negative. For a closed surface, we always refer the external
normal as the normal of surface. Therefore, the solid angle of a closed surface is 4
(
−
n
)
π
if the
O
is inside
S
, 0 if the
O
is outside
S
and 2
π
if the
O
is on the
S
.
If we use
r
2
to measure the area of a spherical surface of radius
r
, we obtain
r
2
.The4
4
π
so that the spherical area is 4
π
π
is called the
solid angle of spherical
surfaces
.Let
d
ω
be the solid angle of area d
σ
on a spherical surface of radius
r
.
Then
r
2
d
r
2
d
σ
=
ω
or
d
ω
=
d
σ
/
.
be the solid
angle of d
S
viewing from point
O
. The projection of d
S
on the spherical surface
S
r
OP
Consider an area element d
S
at
P
on a general surface
S
.Letd
ω
is denoted by d
σ
. Thus
(
,
)=
σ
,
d
S
cos
OP
n
d
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