Environmental Engineering Reference
In-Depth Information
a surface
S
δ
of containing
M
0
on
S
such that, for all
M
∈
V
,
P
∈
S
δ
,
C
r
2
−
δ
|
F
(
M
,
P
)
|≤
PM
,
(7.148)
the integral in Eq. (7.147) converges uniformly at
M
0
so
v
(
M
)
is continuous at
M
0
.
In Eq. (7.148),
C
is a constant and 0
<
δ
≤
1.
Proof.
Without loss of generality and for convenience, take
M
0
as the origin of the
coordinate system
(
ξ
,
η
,
ζ
)
, and the tangent plane of
S
at
M
0
as the
ξη
-plane. The
surface
S
can thus be expressed by, in the neighborhood of
M
0
,
ζ
=
ϕ
(
ξ
,
η
)
.
Also,
ξ
=
0
ξ
=
0
,
∂ϕ
∂ξ
,
∂ϕ
∂η
ϕ
(
0
,
0
)=
0
η
=
0
=
0
η
=
0
=
0
.
2
2
h
2
,andlet
S
h
be its
Consider a sufficiently small circle on
ξη
-plane:
ξ
+
η
≤
corresponding surface on
S
.On
S
h
, both
and
∂ϕ
∂ξ
∂ϕ
∂η
are smaller than a constant
and Eq. (7.148) holds. Let the coordinates of
M
and
P
be
(
x
,
y
,
z
)
and
(
ξ
,
η
,
ζ
)
,
respectively. Thus,
1
+(
∂
∂ξ
)
+(
∂
∂η
)
2
2
F
(
M
,
P
)
d
S
≤
C
d
ξ
d
η
2
−
δ
2
[(
ξ
−
x
)
2
+(
η
−
y
)
2
+(
ζ
−
z
)
2
]
S
h
2
2
h
2
ξ
+
η
≤
d
ξ
d
η
≤
C
1
2
−
δ
2
[(
ξ
−
x
)
2
+(
η
−
y
)
2
]
2
2
h
2
ξ
+
η
≤
d
μ
d
γ
=
C
1
,
(7.149)
2
−
δ
2
2
2
[
μ
+
γ
]
2
2
≤
h
2
(
μ
−
x
)
+(
γ
−
y
)
where
C
and
C
1
are constants.
For a sufficiently small
V
,
M
V
satisfies
x
2
y
2
h
2
(
x
,
y
,
z
)
∈
+
≤
so the circular
2
2
h
2
is inside the circular region
2
2
2
. Thus
region
(
μ
−
x
)
+(
γ
−
y
)
≤
μ
+
γ
≤
(
2
h
)
we have
d
μ
d
γ
d
μ
d
γ
−
δ
2
≤
2
2
−
δ
2
(
μ
2
+
γ
2
)
(
μ
2
+
γ
2
)
2
2
h
2
2
2
2
(
μ
−
x
)
+(
γ
−
y
)
≤
μ
+
γ
≤
(
2
h
)
⎧
⎨
4
π
h
,
δ
=
1
,
2
h
d
r
r
1
−
δ
=
=
2
π
(7.150)
1
δ
(
⎩
)
δ
,
2
π
2
h
0
<
δ
<
1
.
0
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