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1
E/
2
tan(
4
−
1
1+
E
sin
B
1
E/
2
B
2
)
E
2
2
A
1
E
1+
E
−
−
:=
−
√
1
=
cos
B
E
2
sin
2
B
−
E
sin
B
1
−
−
E
2
E
2
1
−
=
−
E
2
sin
2
B
)
f
(
B
)
.
cos
B
(1
−
Korn-Lichtenstein equations:
x
L
=
G
11
G
22
G
11
y
L
,
L
=
G
11
G
22
x
B
,
B
=
G
22
G
22
y
B
,
B
=
−
−
G
11
x
L
,
(15.21)
G
11
G
22
=cos
B
1
−
E
2
sin
2
B
1
−
E
2
⇒
E
2
1
−
E
2
sin
2
B
)
f
(
B
)sin
L
=
f
(
B
)sin
L,
y
B
=
−
(15.22)
cos
B
(1
−
E
2
sin
2
B
)
cosB
(1
−
f
(
B
)cos
L
=
f
(
B
)cos
L.
y
L
=
−
1
−
E
2
End of Example.
15-2 A Fundamental Solution for the Korn-Lichtenstein Equations
A fundamental solution for the Korn-Lichtenstein equations of conformal mapping. The
ellipsoidal Korn-Lichtenstein equations, the ellipsoidal Laplace-Beltrami equations.
A
1
,A
1
,A
2
For the biaxial ellipsoid
, we shall construct a fundamental solution for the Korn-
Lichtenstein equations of conformal mapping (
15.5
) subject to the vectorial Laplace-Beltrami
equations (
15.7
). The condition of orientation conservation (
15.8
) is automatically fulfilled.
x
L
=
G
11
/G
22
y
B
,
E
G
11
/G
22
x
B
,
y
L
=
−
or
(15.23)
x
B
=
−
G
22
/G
11
y
L
, yB
=
G
22
/G
11
x
L
,
G
22
/G
11
x
L
L
+
G
11
/G
22
x
B
B
=0
,
(15.24)
G
22
/G
11
y
L
L
+
G
11
/G
22
y
B
B
=0
,
x
L
y
B
− x
B
y
L
=
G
22
/G
11
x
L
+
G
11
/G
22
x
2
B
>
0
,
(15.25)
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