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(orientation conserving conformeomorphism).
Δ
LB
x
=0and
Δ
LB
y
= 0, respectively, are called
vectorial Laplace-Beltrami equations
.The
matrix of the metric of the first fundamental form of
2
A
1
,A
1
,A
2
E
is defined by
⎡
⎤
G
11
G
12
⎣
⎦
∀
G
MN
=
M, N
∈{
1
,
2
}
.
(15.9)
G
12
G
22
A derivation of the Korn-Lichtenstein equations is given in Appendix
D
. Here, we are inter-
ested in some examples of the Korn-Lichtenstein equations (
15.5
) subject to the integrability
conditions (
15.7
) and the condition of orientation conservation (
15.8
).
Example 15.1 (Universal Mercator Projection (UMP)).
x
=
A
1
L,
y
=
A
1
ln
tan
π
E/
2
.
1
4
+
B
E
sin
B
1+
E
sin
B
−
(15.10)
2
The matrix of the metric of the ellipsoid-of-revolution
E
A
1
,A
1
,A
2
is represented by
⎡
⎤
⎦
=
A
1
cos
2
B
.
G
11
G
12
0
⎣
1
−E
2
sin
2
B
G
MN
=
(15.11)
(1
−E
2
)
2
(1
−E
2
sin
B
)
3
A
1
0
G
12
G
22
The mapping equations of type UMP imply
E
2
)
A
1
(1
−
x
L
=
A
1
,
B
=0
,
L
=0
,
B
=
E
2
sin
2
B
)cos
B
.
(15.12)
(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
11
x
L
,
G
22
y
B
,
B
=
−
−
−
(15.13)
G
11
E
2
sin
2
B
1
− E
2
E
2
)
G
22
=
1
−
A
1
(1
−
cos
B
⇒
y
B
=
E
2
sin
2
B
)cos
B
.
(1
−
Integrability conditions:
Δ
LB
x
=
G
11
G
22
x
B
+
G
22
G
11
x
L
=0
,Δ
LB
y
=
G
11
G
22
y
B
+
G
22
G
11
y
L
=0
,
B
L
B
L
(15.14)
E
2
sin
2
B
)cos
B
,
G
22
G
11
G
22
x
B
=0
,
G
22
G
11
x
L
=
E
2
)
A
1
(1
−
G
11
x
L
=0
,
(15.15)
(1
−
L
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