Geography Reference
In-Depth Information
Table 22.33
Variational calculus:
y
(
L,B
)
δ
dS
l
2
tr
C
l
G
−
l
= 0 (second part)
“integration by parts:
Green's First Identity
”
1
√
G
11
1
√
G
22
Grad f
:=
δGrady
=
Grad δy
=
C
1
δy
L
+
C
2
δy
B
(22.217)
1
√
G
11
1
√
G
22
Grad g
:=
Grad y
=
C
1
y
L
+
C
2
y
B
(22.218)
“The variation
δy
(
S
l
) of a left boundary curve is fixed to zero”:
dS
l
dS
l
δyDivGrady
=0
δy
Grad δy
|
Grad y
=
−
∈
S
l
(22.219)
↔
Div
Grad y
= 0
(22.220)
D
L
1
D
L
y
+
D
B
1
D
B
y
= 0
1
√
G
11
1
√
G
22
Div
Grad y
=
Grad
|
Grad y
=
√
G
11
√
G
22
(22.221)
“The variation
δy
(
S
l
) of a left boundary curve is fixed to zero”:
dS
l
dS
l
δyDivGrady
=0
δy
Grad δy
|
Grad y
=
−
∈
S
l
(22.222)
↔
(22.223)
Div
Grad y
= 0
(22.224)
D
L
1
D
L
y
+
D
B
1
D
B
y
= 0
1
√
G
11
1
√
G
22
√
G
11
√
G
22
Div
Grad y
=
Grad
|
Grad y
=
(22.225)
•
δ
|
=0
,
Eq.(
22.233
),
•
=0
,
Eq.(
22.234
)
Lead to the variational functions Eqs.(
22.234
)and(
22.235
).
These first order variational functions are transformed into their standard form Eqs.(
22.238
)
and (
22.240
) if integration by
parts
or
Green' first identity
is applied. Tables
22.35
and
22.36
present us with the result by means of Eqs.(
22.237
)and(
22.239
) where we have assumed
fixed
boundary data
which make the first integral of the right side of Eq.(
22.211
) vanish. Obviously,
the
Laplace-Beltrami equations
in isometric coordinates
{L
,
B}
of type Eqs.(
22.238
)and(
22.240
)
are much simpler than
Laplace-Beltrami equations
in Gauss surface normal coordinates
δ
||
{
L
,
B
}
of type Eqs.(
22.216
)and(
22.225
).
The topic of Chap.
22
is based on the contribution of
Grafarend
(
2005
), We thank
F. Krumm
,
Department of Geodesy and GeoInfomatics, Stuttgart University (
Germany
) he programmed this
new
Harmonic Map
and prepared the graphics.
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