Geography Reference
In-Depth Information
A
1
cos
Φ
r
cos
φ
1
Λ
1
=
λ
−
1
1
.
2
E
2
sin
2
Φ
−
In the light of the
equivalence theorem
1.14 (areomorphism), we are now prepared to solve the
following problems. (i) Can we prove the first equivalence given by (
1.285
) and (ii) can we prove
the third equivalence given by (
1.286
)?
det[C
l
]=det[G
l
]ord [C
1
]=det[G
r
]
,
(1.285)
det[G
l
]
det[G
r
]
=
G
11
G
22
−
G
12
u
U
v
V
−
u
V
u
U
=
g
11
g
22
− g
12
.
(1.286)
Solution (the first problem).
Start from the equiareal map of Box
1.41
in order to prove det[C
l
]=det[G
l
], where the left
Cauchy-Green matrix C
l
as well as the left matrix G
l
of the metric is given by means of Box
1.40
.
Here, again we collect all deviational items in Box
1.43
.Assoonasweimplement
f
(
Φ
)intothe
determinantal identity, the proof is closed.
End of Solution (the first problem).
Solution (the second problem).
By means of Box
1.44
, let us work out the partial differential equation which governs an equiareal
mapping. Note that we here specify
{
u
=
λ, v
=
φ
}
and
{
U
=
Λ, V
=
Φ
}
subject to the “Ansatz”
{λ
=
Λ, φ
=
f
(
Φ
)
}
. Indeed, we find
f
(
Φ
) as given already in Box
1.42
.
End of Solution (the second problem).
Note that the second or canonical equivalence
Λ
1
Λ
2
= 1 has already been used to construct the
equiareal map
2
2
E
A
1
,A
1
,A
2
→
S
r
.
A
1
,A
1
,A
2
→
S
r
,
det[C
l
]=det[G
l
]).
Box 1.43 (Equiareal mapping:
E
,
G
l
=
A
1
cos
2
Φ
,
C
l
=
r
2
cos
2
φ
0
0
1
−E
2
sin
2
Φ
(1.287)
r
2
f
2
(
φ
)
A
1
(1
−E
2
)
2
(1
0
0
E
2
sin
2
Φ
)
3
−
det[C
l
]=
r
4
cos
2
φf
2
(
φ
)
f
2
(
Φ
)=
A
1
(1
− E
2
)
2
(1
E
2
sin
2
Φ
)
4
cos
2
Φ
=det[G
l
]
⇒
det[C
l
]=
A
1
E
2
)
2
(1
−
1
r
4
cos
2
φ
(1
−E
2
sin
2
Φ
)
4
cos
2
Φ
−
q
.
e
.
d
.
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