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(the
Frobenius matrix maps a two leg of tangent vectors to a two leg which is orthonormal and is also
called
Cartan frame
of reference); (iii) the matrix G = J
∗
J of the metric of
namely I
3
= diag [1, 1, 1]); (ii) the Frobenius matrix F whose elements are called
{
a, b, c, d
}
2
S
R
whose elements
are called
(the letter
G
has been chosen in honour of C. F. Gauss, the matrix G builds
the first fundamental form
I
:d
s
2
=[d
Λ,
d
Φ
]G[d
Λ,
d
Φ
]
∗
); (iv) the matrix H of second derivatives,
whichisdefinedby[
{
e, f, g
}
∂
2
X
/∂U
K
∂U
L
] with respect to the surface normal vector
G
3
and
the embedding function
X
=
X
(
U
1
,U
2
)or
X
=
X
(
Λ, Φ
) (the letter
H
has been chosen in
honour of L. O. Hesse, the matrix H builds the second fundamental form
II
:[d
Λ,
d
Φ
]H[d
Λ,
d
Φ
]
∗
,
the elements of the Hesse matrix are denoted by
{l,m,n}
); (v) the Jacobi matrix J of the first
derivatives of the embedding function, precisely [
D
Λ
X,D
Λ
Y,D
Λ
Z,
]and[
D
Φ
X, D
Φ
Y, D
Φ
Z,
] (the
letter
J
has been chosen in honour of C. G. J. Jacobi); the curvature matrix K =
−
HG
−
1
,its
negative trace taken half (denoted by the letter
h
), also called
mean curvature
, and its determinant
(denoted by the letter
k
), also called
Gaussian curvature
;(vi)the
Christoffel symbols
of the second
kind, which are named after E. B. Christoffel (
G
3
|
∗
10 November 1829, Monschau;
†
15 March 1900,
Strassburg), which are used to compute
geodesics
, and which are defined by
M
KL
=
1
2
G
MN
(
D
K
G
NL
+
D
L
G
KN
− D
N
G
KL
)
.
(5.1)
Fig. 5.3.
The special azimuthal projective maps
2
R
is summarized. According to the above considerations, F
(with elements
a, b, c, d
) is the Frobenius matrix, G (with elements
e, f, g
) is the Gauss matrix, H
(with elements
l,m,n
) is the Hesse matrix, J is the Jacobi matrix, and K is the curvature matrix,
leading to the mean curvature
h
and to the Gaussian curvature
k
,and
G=J
∗
J
,
H=
G
3
In Box
5.1
, the ID card of the sphere
S
∂U
K
∂U
L
=[
G
3
|
X
KL
]
,
J=
∂X
J
∂U
K
,
∂
2
X
(5.2)
1
K=
−
HG
−
1
, h
=
−
2
tr[K]
, k
= det[K]
.
Box 5.1 (ID card of the sphere
S
2
R
).
Spherical coordinates (1st chart:
Λ, Φ
):
{Λ, Φ, R}→{X,Y,Z}
:
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