Geography Reference
In-Depth Information
Table 22.31
Divergence Theorem, Green's First Identity
“
integration by parts
”
(
Grenn's first identity
)
“
Divergence Theorem
”
h
:=
f
grad
g
(22.208)
div
h
=
div
(
f
grad
g
)=
f
div grad
g
+
grad
f
|
grad
g
(22.209)
↔
grad
f
|
grad
g
=
div
(
f
grad
g>
−
f
div grad
g
(22.210)
“
Directional Derivative
”
∇
t
g
are called
vertical
(
normal
)
directional directive
and
horizontal
(
tangen-
tial
)
directional derivative,
respectively, of the scalar function
g
on the surface
S
.
n
|
grad
g
=:
∇
n
g
and
t
|
grad
g
=:
“
Green's First Identity
”
dS < grad f
gradg >
=
dsf
dSf div grad g
|
∇
n
g
−
(22.211)
Table 22.32
Variational Calculus: x (L, B)
dS
l
2
tr
C
l
G
−
l
= 0, (second part)
“integration by parts:
Green's First Identity
”
1
√
G
11
1
√
G
22
gradf
:=
δGradx
=
Grad δx
=
C
1
δx
L
+
C
2
δx
B
(22.212)
1
√
G
11
1
√
G
22
Grad g
:=
Grad x
=
C
1
x
L
+
C
2
x
B
(22.213)
“The variation
δx
(
S
l
) of a left boundary curve is fixed to zero”:
dS
l
dS
l
δyDivGradx
=0
Grad δy
|
Grad y
=
−
∀
δy
∈
S
l
(22.214)
↔
Div
Grad x
= 0
(22.215)
D
L
1
D
L
x
+
D
B
1
D
B
x
= 0
1
√
G
11
1
√
G
22
√
G
11
√
G
22
Div
Grad x
=
Grad
|
Grad x
=
(22.216)
Outlined in Table
22.34
, first we derive the left surface element
dS
l
in terms of isometric coor-
dinates
by Eqs. (
22.226
)and(
22.227
).
Second
,Eqs.(
22.228
)
and (
22.229
) offer a representation of distortion energy density
tr
C
l
G
−
1
l
as a
Hilbert invari-
ant.
The distortion energy, the integral over the distortion energy density with
fixed boundaries
,
Eq.(
22.230
), is additively decomposed
•
{
longitude
L
, isometric latitude
Q
}
2
,Eq.(
22.231
)'
into an integral
|
proportional to
Grad ξ
2
.
Such a representation is proven according to Eqs.(
22.198
)-(
22.203
). The first variations
•
into an integral
||
proportional to
Grad η
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