26
1 From Riemann manifolds to Riemann manifolds
Box 1.14 (Orthogonal projection S
2
R + onto P
2
O , polar coordinates, the first problem).
x = r cos α , y = r sin α ,
Λ(x, y) = arctan
y
x
= α ,
Φ(x, y) = arccos
p
x 2 + y 2
R
= arccos
r
R
.
(1.95)
Right Jacobi matrix:
J r :=
"
D α Λ D r Λ
D α Φ D r Φ
#
=
2
4
1
0
0 −
1
√
R 2 − r 2
3
5 ,
D α Λ = 1 , D r Λ = 0 ,
D α Φ = 0 , D r Φ = −
1
p
1 − r 2 /R 2
1
R
= −
1
√
R 2 − r 2
.
(1.96)
Right Cauchy–Green matrix:
C r := J
∗
r G l J r =
2
6
4
r
2
0
0
R
2
R 2 − r 2
3
7
5 ,
G l = R
2
"
cos
2 Φ 0
0
1
#
=
"
r
2
0
0
R
2
#
.
(1.97)
Right Cauchy–Green tensor:
x(α, r) = e 1 r cos α + e 2 r sin α ,
g 1 := D α x = −e 1 r sin α + e 2 r cos α , g 2 := D r x = +e 1 cos α + e 2 sin α ,
g 11 :=
˙
g 1 g1
¸
= r
2 , g 12 :=
˙
g 1 g2
¸
= 0 , g 22 :=
˙
g 2 g2
¸
= 1 ,
G r =
"
r
2 0
0 1
#
,
(ds)
2 = r
2 (dα)
2 + (dr)
2 ,
g
µ =
2
X
ν=1
g
µν g ν , g
1 =
1
g 11
g 1 =
1
r 2 g 1 , g
2 =
1
g 22
g 2 = g 2 ,
C r =
2
X
µ,ν=1
g
µ ⊗ g
ν C µν =
= g
1 ⊗ g
1 r
2 + g
2 ⊗ g
2
R
2
R 2 − r 2 = g 1 ⊗ g 1 1 + g 2 ⊗ g 2
R
2
R 2 − r 2 .
(1.98)