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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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