Geography Reference
In-Depth Information
(i) Special KL:
⎡
⎤
−
q
u
q
υ
,
√
g
11
g
22
−g
12
1
p
u
p
υ
=
p
u
=
(
−
g
12
q
u
+
g
11
q
υ
)
g
12
g
11
⎣
⎦
.
√
g
11
g
22
−g
12
1
(1.173)
√
g
11
g
22
−g
12
1
−
g
22
g
12
p
υ
=
(
−
g
12
q
u
+
g
12
q
υ
)
(ii)Right integrability :
p
uυ
=
p
υu
and
q
uυ
=
q
υu
or (in termsof theLaplace
−−
Beltrami operator)
(1.174)
Δ
uυ
p
:=
g
11
p
υ
−g
12
p
u
+
g
22
p
u
−g
12
p
υ
⎡
⎤
√
g
11
g
22
−g
12
√
g
11
g
22
−g
12
=0
⎣
⎦
Δ
uυ
q
:=
g
11
p
υ
−g
12
p
u
+
g
22
q
u
−g
12
q
υ
υ
u
.
(1.175)
√
g
11
g
22
−g
12
√
g
11
g
22
−g
12
=0
υ
u
(iii) Right orientation conservation:
p
u
p
υ
q
u
q
υ
=
p
u
q
υ
− p
υ
q
u
>
0
.
(1.176)
3rd step (left-right).
The left Riemann manifold
M
l
(
P,Q|Λ
2
I
2
) which here is called
left surface
and is parameter-
ized in left conformal coordinates
{P,Q}
, is orientation preserving conformally mapped onto the
right Riemann manifold
M
|
2
), which here is called
right surface
and is parameterized
in right conformal coordinates
{p, q}
, if the following special Korn-Lichtenstein equations (i)
(called Cauchy-Riemann (or d'Alembert-Euler) equations) subject to the following integrability
conditions of harmonicity (ii) and orientation conservation (iii) are solved.
r
(
p, q|λ
2
(i) Special KL: (Cauchy-Riemann, d'Alembert-Euler):
p
P
p
Q
=
01
−
10
q
P
q
Q
,p
P
=
q
Q
and
p
Q
=
1
g
11
g
22
−
q
P
.
(1.177)
g
12
−
(ii) Right integrability:
p
PQ
=
p
QP
and
q
PQ
=
q
QP
(1.178)
or (in terms of the Laplace-Beltrami operator)
⎡
Δ
PQ
p
:=
p
PP
+
p
QQ
∂
2
p
(
P,Q
)=0
⎤
∂
2
∂Q
2
∂P
2
+
⎣
Δ
PQ
q
:=
q
PP
+
q
QQ
∂
2
q
(
P,Q
)=0
⎦
.
(1.179)
∂
2
∂Q
2
∂P
2
+
(iii) Left orientation conservation:
p
P
p
Q
q
P
q
Q
=
p
P
q
Q
−
p
Q
q
P
=0
.
(1.180)
The special Korn-Lichtenstein equations, which govern as Cauchy-Riemann (or d'Alembert-
Euler) equations any harmonic, orientation preserving conformal mapping
M
l
(
P,Q
)
→
M
r
(
p, q
),
are uniquely solvable if a proper boundary value problem is formulated.
End of Theorem.
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