Geography Reference
In-Depth Information
⎡
⎤
∂
X
=[
E
1
,
E
2
,
E
3
]
X
Λ
X
Φ
Y
Λ
Y
Φ
Z
Λ
Z
Φ
∂Λ
,
∂
X
⎣
⎦
=
∂Φ
⎡
⎤
A
1
(1
−
E
2
)sin
Φ
cos
Λ
(1
−E
2
sin
2
Φ
)
3
/
2
√
1
−E
2
sin
2
Φ
A
1
cos
Φ
sin
Λ
−
−
⎣
⎦
A
1
(1
−
E
2
)sin
Φ
sin
Λ
(1
+
A
1
cos
Φ
sin
Λ
√
1
−E
2
sin
2
Φ
=[
E
1
,
E
2
,
E
3
]
−
,
(1.19)
−E
2
sin
2
Φ
)
3
/
2
+
A
1
(1
−
E
2
)cos
Φ
0
(1
−E
2
sin
2
Φ
)
3
/
2
∂
x
∂λ
,
∂
∂φ
}
as well as the vector field
x
(
λ, φ
) is locally characterized by the
field of tangent vectors
,
the
Jacobi map
with respect to the “spherical longitude
λ
” and the “spherical latitude
φ
”, namely
{
⎡
⎤
∂
x
=[
e
1
,
e
2
,
e
3
]
x
λ
x
φ
∂λ
,
∂
x
⎣
⎦
=
y
λ
y
φ
∂φ
z
λ
z
φ
(1.20)
⎡
⎤
−r
cos
φ
sin
λ
−r
sin
φ
cos
λ
⎣
⎦
.
=[
e
1
,
e
2
,
e
3
]
+
r
cos
φ
cos
λ
−
r
sin
φ
sin
λ
0
r
cos
φ
Next, we are going to identify the coordinates of the left metric tensor G
l
and of the right
metric tensor G
r
, in particular, from the inner products
∂
X
∂Λ
=
∂
x
∂λ
=
r
2
cos
2
φ
=:
g
11
,
A
1
cos
2
Φ
∂
X
∂Λ
∂
x
∂λ
E
2
sin
2
Φ
=:
G
11
,
1
−
∂
X
∂Λ
=
∂
X
∂Φ
=:
G
12
=0
,
∂
x
∂λ
=
∂
x
∂φ
=:
g
12
=0
,
(1.21)
∂
X
∂Φ
∂
X
∂Λ
∂
x
∂φ
∂
x
∂λ
∂
X
∂Φ
=
∂
x
∂φ
=
r
2
=:
g
22
,
∂
X
∂Φ
A
1
(1
− E
2
)
2
(1
∂
x
∂φ
E
2
sin
2
Φ
)
3
=:
G
22
,
−
A
1
cos
2
Φ
A
1
(1
E
2
)
2
−
d
S
2
=
1
− E
2
sin
2
Φ
d
Λ
2
+
(1
− E
2
sin
2
Φ
)
3
d
Φ
2
,
d
s
2
=
r
2
cos
2
φ
d
λ
2
+
r
2
d
φ
2
.
Resorting to this identification, we obtain the left metric tensor, i.e. G
l
, and the right metric
tensor, i.e. G
r
, according to
G
l
:=
G
11
=
G
r
:=
g
11
=
G
12
g
12
{
G
MN
}
=
{
g
μν
}
=
G
12
G
22
g
12
g
22
=
A
1
cos
2
Φ
,
=
r
2
cos
2
φ
r
2
.
0
0
1
−E
2
sin
2
Φ
(1.22)
(1
−E
2
)
2
(1
−E
2
sin
2
Φ
)
3
A
1
0
0
Finally, we implement the
isoparametric mapping
f
= id. Applying the summation convention
over repeated indices, this is realized by
u
μ
=
f
μ
(
U
μ
)
,u
μ
=
δ
M
U
M
,u
1
=
U
1
,u
2
=
U
2
,λ
=
Λ, φ
=
Φ,
J
l
=I
2
=J
r
,
U
M
→
(1.23)
∂U
M
/∂u
μ
∂u
μ
/∂U
M
|
|
=1
>
0
,
|
|
=1
>
0
,
f
∗
:d
u
μ
=
δ
M
d
U
M
,
f
∗
:d
U
M
=
δ
μ
d
u
μ
,
(1.24)
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