Geography Reference
In-Depth Information
Table 22.27
Simultaneous diagonalization of the matrix pair (
G
l
,
G
r
):
left eigenvectors
Λ
i
G
l
)
F
il
=0
(
C
l
−
∀
i
∈{
1
,
2
}
(22.161)
“1st
left eigencolumn:
Λ
1
”
F
1
l
:=
F
11
F
12
=
c
22
−
1
Λ
1
G
22
(
c
22
−
(22.162)
−
c
12
G
22
)
2
+
c
12
Λ
1
G
22
“2nd
left eigencolumn:
A
2
”
F
2
l
:=
F
21
F
22
=
1
(
c
11
−
2
G
11
)
G
22
+
c
12
G
11
c
12
c
11
−
2
−
(22.163)
G
11
due to
G
12
=0
“
special form of
C
l
,
G
l
”
x
2
B
+
y
B
−
1
M
2
1
(
x
2
B
+
y
B
−
1
N
2
cos
2
B
)
2
+(
x
L
x
B
+
y
L
y
B
)
2
F
1
l
=
(22.164)
−
(
x
L
x
B
+
y
L
y
B
)
1
(
x
L
x
B
+
y
L
y
B
)
x
L
+
y
L
−
2
N
2
cos
2
B
−
F
2
l
=
(
x
L
+
y
L
−
2
(22.165)
N
2
cos
2
B
)
2
M
2
+(
x
L
x
B
+
y
L
y
B
)
2
N
2
cos
2
B
“
Cartan 2-leg
”
G
11
=
X
L
/
(
N
coscos
B
)
C
1
:=
X
L
÷
X
L
=
X
L
÷
(22.166)
G
22
=
X
B
/M
C
2
:=
X
B
÷
X
B
=
X
B
÷
(22.167)
“
left eigenvectors in the Cartan 2-leg
”
F
1
=
C
1
F
11
+
C
2
F
12
=
C
1
C
2
x
2
B
+
y
B
−
Λ
1
M
2
(22.168)
−
(
x
L
x
B
+
y
L
y
B
)
F
2
=
C
1
F
21
+
C
2
F
22
=
C
1
C
2
(
x
L
x
B
+
y
L
y
B
)
x
L
+
y
L
−
−
(22.169)
Λ
2
N
2
cos
2
B
lead to the variational function Eqs.(
22.206
)and(
22.207
).
Those first order variational functions are transformed into their standard form Eqs.(
22.216
)
and (
22.225
)if
integration by parts
or
Green's first identity is applied
.Table
22.31
presents us
with the auxiliary “
Divergence Theorem
”, Eqs.(
22.208
)-(
22.210
), for two scalar functions
f
and
g
which define the third function
h
:
=fgradg
. As soon as we define the
vertical directional
derivative
t
g,
also
called
tangential derivative
, we are able to formulate by means of Eq.(
22.211
)
Green's first
identity.
On its right-hand side we are led to the
boundary integral
along a closed curved of
f
∇
n
g,
also called normal derivative, or the
horizontal directional derivative
∇
∇
n
g
as well as to the
surface integral
“over”
fdivgradg
.
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