Geography Reference
In-Depth Information
Box 1.1 (Left and right Cauchy-Green deformation tensor).
Left CG :
Right CG :
d
s
2
=
d
S
2
=
∂u
μ
∂U
M
∂u
ν
∂U
M
∂u
μ
=
g
μν
{f
λ
(
U
L
)
}
∂U
N
d
U
M
d
U
N
==
G
MN
{F
L
{u
λ
)
}
∂U
N
∂u
ν
d
u
μ
d
u
ν
=
=
c
MN
(
U
L
)d
U
M
d
U
N
,
=
C
μν
(
u
λ
)
du
μ
du
ν
,
(1.16)
c
MN
(
U
L
)=
C
μν
(
u
λ
)=
=
g
μν
(
U
L
)
∂u
μ
∂U
M
(
U
L
)
∂u
ν
=
G
MN
(
U
λ
)
∂U
M
∂u
μ
(
u
λ
)
∂U
N
∂U
N
(
U
L
)
.
∂u
ν
(
u
λ
)
.
2
2
Example 1.3 (Cauchy-Green deformation tensor,
f
:
E
A
1
,A
1
,A
2
→
S
r
).
The embedding of an ellipsoid-of-revolution
M
l
=
E
A
1
,A
1
,A
2
and a sphere
M
r
=
S
r
into a three-
dimensional Euclidean space
{
R
3
,
I
3
}
with respect to a standard Euclidean metric I
3
(where I
3
is
the 3
×
3 unit matrix) is governed by
E
2
)sin
Φ
X
(
Λ, Φ
)=
E
1
A
1
cos
Φ
cos
Λ
+
E
2
A
1
cos
Φ
sin
Λ
+
E
3
A
1
(1
−
1
1
1
=
E
2
sin
2
Φ
E
2
sin
2
Φ
E
2
sin
2
Φ
−
−
−
⎡
⎤
cos
Φ
cos
Λ
cos
Φ
sin
Λ
A
1
⎣
⎦
,
1
=[
E
1
,
E
2
,
E
3
]
(1.17)
E
2
sin
2
Φ
−
E
2
)sin
Φ
(1
−
E
2
:= (
A
1
− A
2
)
/A
1
=1
−
(
A
2
/A
1
)
,
(
A
2
/A
1
)=1
− E
2
,
and by
x
(
λ, φ
)=
e
1
r
cos
φ
cos
λ
+
e
2
r
cos
φ
sin
λ
+
e
3
r
sin
φ
=
⎡
⎤
r
cos
φ
cos
λ
r
cos
ψ
sin
λ
r
sin
φ
⎣
⎦
,
=[
e
1
,e
2
,e
3
]
(1.18)
2
A
1
,A
1
,A
2
respectively. The coordinates (
X,Y,Z
)and(
x, y, z
) of the placement vectors
X
(
Λ, Φ
)
∈
E
2
and
x
(
λ, φ
)
∈
S
r
are expressed in the left and right orthonormal fixed frames
{
E
1
,
E
2
,
E
3
|O}
and
.
Next, we are going to construct the left tangent space
T
{
e
1
,e
2
,e
3
|O}
at their origins
O
and
2
M
l
as well as the right tangent space
2
T
M
r
, respectively. The vector field
X
(
Λ, Φ
) is locally characterized by the
field of tangent vectors
∂
X
∂Λ
,
∂
∂Φ
}
,the
Jacobi map
with respect to the “surface normal ellipsoidal longitude
Λ
”andthe
“surface normal ellipsoidal latitude
Φ
”, namely
{
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