Environmental Engineering Reference
In-Depth Information
Fig. 7.7
Positive solid angle
where
n
is the normal of the surface
S
at
P
.
OP
is the normal of
S
r
OP
at
P
. Therefore
the solid angle of d
S
viewing from point
O
reads
σ
r
OP
=
d
cos
(
OP
,
n
)
d
ω
=
d
S
.
r
OP
Here
r
OP
is the distance between
O
and
P
. The solid angle of surface
S
viewing from
O
is thus
cos
(
OP
,
n
)
ω
=
d
S
.
r
OP
S
cos
(
OP
,
n
)
d
S
is called the
absolute solid angle
of surface
S
viewing from
O
.
r
OP
S
There are some constraints on the boundary surface
S
in studying boundary value
problems of elliptic equations. It is normally required to be a
Ляпунов
surface
defined as follows.
Definition.
A surface
S
is called a
Ляпунов
(Russin) surface
if it satisfies:
1. There exists a tangent plane at any point on
S
;
2. For any point
P
0
on
S
, there exists a sphere
V
P
0
R
of center
P
0
and radius
R
such
that the part
S
P
0
of
S
inside
V
P
0
R
can be expressed by a unique-valued function
z
.Here
R
normally depends on the position of
P
0
;
3. The unit normals
n
1
and
n
2
at any two points
P
1
and
P
2
on
S
must satisfy
=
ϕ
(
x
,
y
)
Ar
P
1
P
2
,
(
n
1
,
n
2
)
≤
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