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G
−
l
=
G
JK
=
G
11
G
22
=
G
22
.
G
12
1
det[G
l
]
−
G
12
G
21
−
G
12
G
11
2
l
r
=
r
, Korn-Lichtenstein equations).
Lemma D.1 (Conformeomorphism,
M
→
M
E
Equivalent formulations of the Korn-Lichtenstein equations which produce a conformal mapping
M
2
l
r
=E
r
are the following:
→
M
d
x
=
∗
d
y,
(D.12)
∂U
I
=
e
IJ
det[G
l
]G
JK
∂y
∂x
∂U
K
,
(D.13)
1
det[G
l
]
(
x
U
=
−
G
12
y
U
+
G
11
y
V
)
,
1
det[G
l
]
(
−G
22
y
U
+
G
12
y
V
)
.
x
V
=
(D.14)
End of Lemma.
Generalizations to a conformeomorphism of higher order, namely
M
l
→
M
r
=
E
r
,leadtothe
Zund equations, and its generalizations
M
l
→
M
r
=
E
r
are referred to
Grafarend and Syf-
fus
(
1998d
).
D-2 d'Alembert-Euler (Cauchy-Riemann) Equations
2
,λ
−
2
δ
μν
}
(“sur-
face”) has been established, an
alternative isometric coordinate system
can be constructed by
solving a boundary value problem of the
d'Alembert-Euler equations
(
Cauchy-Riemann equa-
tions
) subject to the integrability conditions of harmonicity type. Here, we are going to construct
fundamental solutions of these basic equations governing conformal mapping.
Once an
isometric coordinate system
of a two-dimensional
Riemann manifold
{
M
Lemma D.2 (Fundamental solution of the d'Alembert-Euler equations subject to integrability
conditions of harmonicity, polynomial representation).
A fundamental solution of the d'Alembert-Euler equations (Cauchy-Riemann equations) subject
to the integrability conditions of harmonicity type is
x
=
α
0
+
α
1
q
+
β
1
p
+
1)
s
r
q
r−
2
s
p
2
s
+
1)
s
+1
r
q
r−
2
s
+1
p
2
s−
1
,
[
r/
2]
[(
r
+1)
/
2]
n
N
+
α
r
(
−
β
r
(
−
(D.15)
2
s
2
s −
1
r
=2
s
=0
r
=2
s
=1
y
=
β
0
+
β
1
q
+
α
1
p
+
1)
s
r
q
r−
2
s
p
2
s
+
1)
s
+1
r
q
r−
2
s
+1
p
2
s−
1
.
[
r/
2]
[(
r
+1)
/
2]
n
N
+
β
r
(
−
α
r
(
−
(D.16)
2
s
2
s −
1
r
=2
s
=0
r
=2
s
=1
End of Lemma.
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