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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