Geography Reference
In-Depth Information
9-15 The Curvature Tensor of the Sphere, the Second Differential
Form
Fourth, we compute the second differential form of the surface of type
sphere
. The second dif-
ferential form is related to the
determinantal form
of the sphere. We compute first the surface
normal vector and second the surface derivatives of the tangent vectors. In summary, we refer to
the coordinates of the curvature tensor of the sphere.
1
∂
2
x
/∂u
k
∂u
l
∂
2
g
k
/∂u
l
det[
g
kl
]
[
g
K,L
,
g
1
,
g
2
]
,
h
kl
=
g
3
|
=
g
3
|
=
g
3
=cos
φ
cos
λe
1
+cos
φ
sin
λe
2
+sin
φe
3
,
(9.17)
det[
g
kl
]=
r
2
cos
φ,
g
2
||
g
1
×
g
2
||
g
1
×
g
3
=
,
∂
g
1
/∂u
1
r
cos
2
φ,
l
(Gauss) :=
g
3
|
:=
h
λλ
=
h
11
=
−
∂
g
1
/∂u
2
m
(Gauss) :=
g
3
|
:=
h
λφ
=
h
12
=
h
21
=
h
φλ
=0
,
(9.18)
∂
g
2
/∂u
2
n
(Gauss) :=
g
3
|
:=
h
φφ
=
h
22
=
−
r.
Λ
0
)and
φ
=
f
(
Φ
), let us here compute
the deformation tensor of the first kind and the deformation tensor of the second kind.
Based upon the general mapping equations
λ
=
λ
0
+
a
(
Λ
−
9-16 Deformation of the First Kind
We first consider the deformation of the first kind: the deformation tensor of the first kind is
based upon the first fundamental form of differential geometry.
2
2
2
∂u
k
∂U
K
∂u
l
∂U
L
.
I:d
s
2
=
g
kl
d
u
k
d
u
l
=
c
KL
d
U
K
d
U
L
,
KL
:=
g
kl
(9.19)
k,l
=1
K,L
=1
k,l
=1
The first invariant differential form I := d
s
2
=
2
K,L
=1
c
KL
d
U
K
dU
L
is to be computed next. In
Box
9.3
, the various steps of computing the matrix
c
KL
are outlined.
Box 9.3 (The matrix
c
KL
).
∂u
1
∂u
1
∂U
1
=
∂λ
∂U
2
=
∂λ
∂Λ
=
a
∂Φ
=0
,
and
(9.20)
∂U
1
=
∂φ
∂u
2
∂U
2
=
∂φ
∂u
2
∂Φ
=
f
(
Φ
)
∂Λ
=
a
and
⇔
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