Geography Reference
In-Depth Information
•
Fifth, by means of Box
1.35
, we aim at representing the factors of conformality,
λ
l
(
φ
)and
λ
r
(
φ
). in terms of left conformal coordinates
{P,Q}
and right conformal coordinates
{p, q}
.
We begin with transforming the right factor of conformality,
λ
r
(
φ
)
→ λ
r
(
p, q
), since it is
available in closed form. In contrast, the transformation
of the le
ft factor of conformality,
λ
l
(
Φ
)
λ
l
(
P, Q
), is only symbolically written since
f
−
1
(
P
2
+
Q
2
) is not available in closed
form. Note the beautiful transformations
→
{
sin
φ,
cos
φ
}
and
{
sin
λ,
cos
λ
}
as functions of the
“UPS coordinates”
p
and
q
.
Sixth, Box
1.36
outlines that
λ
r
and
λ
l
, respectively, fulfill the conformal representation of
the right and the left Gaussian curvature, here written in two versions as a special Helmholtz
differential equation. The simple representation is performed First, followed by the left rep-
resentation. For a given Gaussian curvature
k
r
=1
/r
2
•
2
= constant of the sphere
S
r
and
E
2
sin
2
Φ
)
2
/
[
A
1
(1
E
2
)] of the ellipsoid-of-revolution
2
A
1
,A
1
,A
2
k
l
=(1
−
−
E
being transformed into
left and right conformal coordinates, we succeed to prove
Δ
ln
λ
2
+2
kλ
2
=0
of type “right” and “left”.
{
p, q
}
and
{
P,Q
}
End of Solution (the second problem).
Box 1.32 (Left Korn-Lichtenstein equations, UPS of
E
A
1
,A
1
,A
2
, harmonicity, orientation).
Left Korn-Lichtenstein equations:
(1st)
P
Λ
=+
G
11
G
22
G
11
Q
Λ
(2nd)
,
G
22
Q
Φ
, P
Φ
=
−
(1.229)
G
11
G
22
P
Φ
, Q
Φ
=+
G
22
(1st)
Q
Λ
=
−
G
11
P
Λ
(2nd);
G
11
G
22
E
2
sin
2
Φ
1
− E
2
E
2
G
22
=cos
Φ
1
−
1
cos
Φ
1
−
⇔
G
11
=
E
2
sin
2
Φ
;
(1.230)
1
−
⎧
⎨
⎫
⎬
1
−E
2
1
−E
2
sin
2
Φ
Q
Φ
=
f
(
Φ
)sin
Λ
=
1
−
cos
Φ
f
(
Φ
)sin
Λ
(UPS left)
(KL 2nd)
,
P
Λ
=
f
(
Φ
)sin
Λ
Q
Λ
=
f
(
Φ
)cos
Λ
−
⎩
⎭
Q
Λ
=
f
(
Φ
)cos
Λ
P
Φ
=
f
(
Φ
)cos
Λ
=
(KL 1st)
.
(UPS left)
(1.231)
1
−E
2
1
−
cos
Φ
f
(
Φ
)cos
Λ
−E
2
sin
2
Φ
1
Left integrability conditions:
Δ
Λ,Φ
P
=
G
11
G
22
P
Φ
G
11
P
Λ
+
G
22
=0
,
(1.232)
Φ
Λ
G
22
Q
Φ
G
11
Q
Λ
Δ
Λ,Φ
Q
=
G
11
+
G
22
=0
.
Φ
Λ
(1st)
G
11
f
(
Φ
)cos
Λ
=
E
2
sin
2
Φ
1
E
2
G
22
P
Φ
=cos
Φ
1
−
1
−
1
cos
Φ
−
−
E
2
E
2
sin
2
Φ
1
−
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