Geography Reference
In-Depth Information
∂U
K
=
∂λ/∂Λ
=
a
.
∂u
k
∂λ/∂Φ
0
(9.21)
f
(
Φ
)
∂φ/∂Λ
∂φ/∂Φ
0
If
g
12
=0
,
then
∂Λ
=
g
11
∂λ
2
+
g
22
∂φ
∂Λ
2
2
∂u
k
∂Λ
∂u
l
c
11
=
c
ΛΛ
=
g
kl
,
∂Λ
k,l
=1
∂Φ
=
g
11
∂λ
∂λ
∂Φ
+
g
22
∂φ
∂Λ
∂φ
∂Φ
,
2
∂u
k
∂Λ
∂u
l
c
12
=
c
ΛΦ
=
g
kl
∂Λ
k,l
=1
(9.22)
∂Φ
=
g
11
∂λ
2
+
g
22
∂φ
∂Φ
2
2
∂u
k
∂Φ
∂u
l
c
22
=
c
ΦΦ
=
g
kl
∂Φ
k,l
=1
⇒
c
11
=
c
ΛΛ
=
a
2
r
2
cos
2
φ, c
12
=
c
ΛΦ
=
c
21
=
c
ΦΛ
=0
,
(9.23)
2
(
Φ
)
.
c
22
=
c
ΦΦ
=
r
2
f
The matrix elements
c
KL
of the first Cauchy-Green deformation tensor are summarized according
to (
9.24
). The principal stretches of the first kind amount to (
9.25
).
c
KL
=
a
2
r
2
cos
2
φ
,
0
(9.24)
2
(
Φ
)
0
r
2
f
a
2
r
2
cos
2
φ
A
1
cos
2
Φ
(1
Λ
1
=
c
11
/G
11
=
E
2
sin
2
Φ
)=
ar
cos
φ
−
N
cos
Φ
,
Λ
2
=
c
22
/G
22
=
r
2
f
2
(
Φ
)
E
2
sin
2
Φ
)
3
=
rf
(
Φ
)
M
r
M
d
φ
d
Φ
.
E
2
)
2
(1
−
=
(9.25)
A
1
(1
−
The curvature tensors of the ellipsoid-of-revolution and the sphere, namely the Gauss curvature
scalar and the trace as the alternative curvature scalar, are presented in Box
9.4
.Notethat
N
and
M
are the radii of principal type of the ellipsoid-of-revolution and that
r
is the curvature
radius of the sphere.
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