Environmental Engineering Reference
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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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