Geography Reference
In-Depth Information
9-17 Deformation of the Second Kind
We then consider the deformation of the second kind: the deformation tensor of the second
kind is based upon the second fundamental form of differential geometry. We shall compute
its representation. First, in the coordinate system
u
1
,u
2
{
u,v
}
=
{
}
. Second, in the transformed
coordinate system
u
k
U
K
=
U
K
(
u
k
). Or from the spherical coordinate system to the ellipsoidal
→
coordinate system.
2
2
2
∂u
k
∂U
K
∂u
l
∂U
L
.
h
kl
d
u
k
d
u
l
=
d
KL
d
U
K
d
U
L
,
KL
:=
II :=
h
kl
(9.30)
k,l
=1
K,L
=1
k,l
=1
Box 9.5 (The matrix
d
KL
).
∂u
1
∂u
1
∂U
1
=
∂λ
∂U
2
=
∂λ
∂Λ
=
a
and
∂Φ
=0
,
∂U
1
=
∂φ
∂u
2
∂U
2
=
∂φ
∂u
2
∂Φ
=
f
(
Φ
)
∂Λ
=0 and
(9.31)
⇔
∂U
K
=
∂λ/∂Λ ∂λ/∂Φ
=
a
.
∂u
k
0
(9.32)
f
(
Φ
)
0
∂φ/∂Λ ∂φ/∂Φ
If
h
12
=0
,
then
∂Λ
=
h
11
∂λ
2
+
h
22
∂φ
∂Λ
2
2
∂u
k
∂Λ
∂u
l
d
11
=
d
ΛΛ
=
h
kl
,
∂Λ
k,l
=1
∂Φ
=
h
11
∂λ
∂λ
∂Φ
2
∂u
k
∂Λ
∂u
l
d
12
=
d
ΛΦ
=
h
kl
∂Λ
k,l
=1
+
h
22
∂φ
∂φ
∂Φ
,
(9.33)
∂Λ
∂Φ
=
h
11
∂λ
2
+
h
22
∂φ
∂Φ
2
2
∂u
k
∂Φ
∂u
l
d
22
=
d
ΦΦ
=
h
kl
∂Φ
k,l
=1
⇒
r
2
f
2
(
Φ
)
.
a
2
r
cos
2
φ, d
12
=
d
ΛΦ
=0
,
22
=
d
ΦΦ
=
d
11
=
d
ΛΛ
=
−
−
(9.34)
In summary, let us here present the diverse coordinates of the
second deformation tensor
,namely
its eigenvalues.
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