Geography Reference
In-Depth Information
Table 22.16
Harmonic map, “
Northing
”, first boundary value problem,
q
representation
y
=
η
(
q,
0) =
γ
0
+
γ
1
q
+
γ
2
q
2
+
γ
3
q
3
+
γ
4
q
4
+
O
q
5
(22.97)
“
symmetry
”
η
(
q,l
)=
η
(
q,
−
l
)
(22.98)
γ
0
+
γ
1
q
+
δ
1
l
+
γ
2
q
2
−
l
2
+2
δ
2
ql
+
γ
3
q
3
−
3
ql
2
+
δ
3
3
q
2
l
l
3
−
+
γ
4
q
4
−
6
q
2
l
2
+
l
4
+
δ
4
4
q
3
l
4
ql
3
+
O
(5) =
−
δ
1
l
+
γ
2
q
2
−
l
2
−
2
δ
2
ql
+
γ
3
q
3
−
3
ql
2
−
δ
3
3
q
2
l
l
3
γ
0
+
γ
1
q
−
−
+
γ
4
q
4
−
6
q
2
l
2
+
l
4
−
δ
4
4
q
3
l
4
ql
3
+
O
(5)
−
↔
δ
1
=0
,δ
2
=0
,δ
3
=0
,δ
4
=0
etc.
(22.99)
y
(
q,l
)=
γ
0
+
γ
1
q
+
γ
2
q
2
−
l
2
+
γ
3
q
3
−
3
ql
2
+
1)
s
r
2
s
q
r−
2
s
l
2
s
+
r
=4
γ
r
[
r/
2]
(22.100)
(
−
s
=0
l
:=
L
−
L
0
(22.101)
Q
−
Q
0
=
lamQ
(
B
)
−
lamQ
(
B
0
)=:
q
(22.102)
“
Taylor ser ies
”
q
=
N
η
=1
q
η
b
η
(22.103)
q
2
=
N
r
1
,r
2
=1
q
r
1
q
r
2
b
r
1
b
r
2
,q
r
1
r
2
:=
q
r
1
q
r
2
(22.104)
q
3
=
N
r
1
,r
2
,r
3
=1
q
r
1
q
r
2
q
r
3
b
r
1
b
r
2
b
r
3
,q
r
1
r
2
r
3
:=
q
r
1
q
r
2
q
r
3
(22.105)
“
derivatives
”
⎧
⎨
q
1
:=
1!
dQ
dB
(
B
0
)=
1!
Q
(
B
0
)
d
2
Q
q
2
:=
2!
dB
2
(
B
0
)=
2!
Q
(
B
0
)
(22.106)
⎩
d
3
Q
q
3
:=
3!
dB
3
(
B
0
)=
3!
Q
(
B
0
)
d
r
Q
q
r
:=
1
r
!
dB
r
(
B
0
)=
1
r
!
Q
(
r
)
(
B
0
)
(22.107)
for the “
Eastern function
”
x
=
ξ
(
q,l
). In terms of the fundamental solution of
Laplace-
Beltrami equation
presented to you in Table
22.11
we have been able to determine the coe-
cients
β
1
and
{
α
0
=0
,α
2
=0
,α
2
=0
,β
3
=0
,α
4
=0
,β
5
=0
,α
6
=0
,etc.
}
, but
not
the coecients
{
α
1
,β
2
,α
3
,β
4
,α
5
,β
6
,α
7
,β
8
,etc
.
.
How to determine the coecients
}
{
α
1
,β
2
,α
3
,β
4
,α
5
,β
6
,α
7
,β
8
,etc
.
}
?
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