Geography Reference
In-Depth Information
Box 9.4 (Curvature tensors, Gauss's curvature tensors).
Curvature tensor
Curvature tensor
(ellipsoid-of-revolution):
(sphere):
(9.26)
Grad
G
3
grad
g
3
=
−
HG
−
1
G
1
=K
G
1
.
=
−
hg
−
1
g
1
=k
g
1
.
G
2
G
2
g
2
g
2
Gauss's curvature tensor
Gauss's curvature tensor
(ellipsoid-of-revolution):
(sphere):
(9.27)
HG
−
1
hg
−
1
K:=
−
k:=
−
HG
−
1
=
1
/N
,
hg
−
1
=
1
/r
.
0
1
/M
0
1
/r
K:=
−
k:=
−
0
0
A
1
N
:=
(1
−E
2
sin
2
Φ
)
1
/
2
,
A
1
(1
−E
2
)
M
:=
(1
−E
2
sin
2
Φ
)
3
/
2
.
Eigenvalues of the
Eigenvalues of the
curvature tensor:
curvature tensor:
K
1
:=
N
,K
2
:=
M
,
κ
1
=
κ
2
=
r
,
Grad
G
3
=
Grad
g
3
=
=K
G
1
=
K
1
G
1
.
=k
g
1
=
κ
1
g
1
.
(9.28)
G
2
K
2
G
2
g
2
κ
2
g
2
Tangent space:
Tangent space:
G
1
:=
∂
∂Λ
,G
2
:=
∂
∂Φ
.
g
1
:=
∂x
∂λ
,g
2
:=
∂x
∂φ
..
(9.29)
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