Geography Reference
In-Depth Information
Table 22.30
Variational calculus:
δ
dS
l
2
tr
C
l
G
−
l
= 0 (first part)
E
2
A
1
,A
2
:
Surface element
”
“
dS
l
=
det
G
l
dLdB
=
N
cos
BdLdB
(22.191)
A
1
1
E
2
−
dS
l
=
cos
BdLdB
(22.192)
1
E
2
sin
2
B
2
−
“
Hilbert invariant
”
1
G
11
1
G
22
1
G
11
1
G
22
tr
C
l
G
−
l
=
x
L
+
x
2
B
+
y
L
+
y
B
(22.193)
1
N
2
cos
2
B
x
L
+
1
M
2
1
N
2
cos
2
B
y
L
+
1
M
2
tr
C
l
G
−
1
l
x
2
B
+
y
B
=
(22.194)
“
integral functional
”
G
11
x
2
B
+
dS
l
2
G
11
y
B
dS
l
tr
C
l
G
−
1
l
=
dS
l
2
J
:=
2
x
L
+
G
22
y
L
+
G
22
(22.195)
=
|
+
||
1
G
11
x
2
B
=
dS
l
1
:=
dS
l
1
2
1
G
22
2
x
L
+
|
2
Grad x
(22.196)
:=
dS
l
1
2
1
G
11
y
B
=
dS
l
1
1
G
22
2
y
L
+
||
2
Grad y
(22.197)
“
E
A
1
,A
2
:
Cartan
2
−
leg
”
∂L
=
∂X
C
1
=
∂X
1
√
G
11
∂X
∂L
÷
(22.198)
∂L
∂B
=
∂X
C
2
=
∂X
1
√
G
22
∂X
∂B
÷
(22.199)
∂B
1
√
G
11
∂X
∂L
+
C
2
1
√
G
22
∂X
∂B
Grad x
=
C
1
(22.200)
∂y
∂L
+
C
2
∂y
∂B
1
√
G
11
1
√
G
22
Grad y
=
C
1
(22.201)
1
G
11
1
G
22
2
=
x
L
+
x
2
B
Grad x
Grad x
|
Grad x
=
(22.202)
1
G
11
1
G
22
2
=
y
L
+
y
B
Grad y
Grad y
|
Grad y
=
(22.203)
“1st variation”
δ|
= 0
(22.204)
δ
||
= 0
(22.205)
=
dS
l
1
G
11
x
B
δx
B
= 0
=
dS
l
1
G
22
δ
|
Grad x
|
Grad x
x
L
δx
L
+
(22.206)
=
dS
l
=
dS
l
1
G
11
y
B
δy
B
= 0
1
G
22
δ
||
Grad y
|
Grad y
y
L
δy
L
+
(22.207)
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