3-1 Coordinates relating to manifolds 119
Example 3.4 (Sphere S
2
r , complete atlas: I = 6).
By means of an orthogonal projection p = π(P ), we already introduced a first coordinate set of the
“two-sphere” S
2
r in terms of spherical longitude λ and spherical latitude φ. As local coordinates, {λ, φ}
do not cover all points of the “two-sphere”. As a set of exceptional points, we removed the South Pole,
the North Pole, as well as the Greenwich Meridian, the meridian λ = 0. Here, we introduce a special
union of six charts, which covers the “two-sphere” completely. Figure 3.4 illustrates those six charts.
Their generators Φ i = Φ(U i ) (i = 1, 2, 3, 4, 5, 6) are the following:
U
+
z = {[x, y, z] ∈ S
2
r z = +
r 2 − (x 2 + y 2 ) > 0, x
2 + y
2 < r
2
} ,
Φ 1 (x, y, z) :=
x
y
=
u 1
v 1
,
Φ
−1
1 (u 1 , v 1 ) =
u 1 , v 1 , +
r 2 − (u 2
1 + v 2
1 )
∼ x 1 (u, v) = e 1 u 1 + e 2 v 1 + e 3
r 2 − (u 2
1 + v 2
1 ) ,
U
−
z = {[x, y, z] ∈ S
2
r z = −
r 2 − (x 2 + y 2 ) < 0, x
2 + y
2 < r
2
} ,
Φ 2 (x, y, z) :=
x
y
=
u 2
v 2
,
Φ
−1
2 (u 2 , v 2 ) =
u 2 , v 2 , −
r 2 − (u 2
2 + v 2
2 )
∼ x 2 (u, v) = e 1 u 2 + e 2 v 2 − e 3
r 2 − (u 2
2 + v 2
2 ) ,
U
+
y = {[x, y, z] ∈ S
2
r y = +
r 2 − (x 2 + z 2 ) > 0, x
2 + z
2 < r
2
} ,
Φ 3 (x, y, z) :=
x
z
=
u 3
v 3
,
Φ
−1
3 (u 3 , v 3 ) =
u 3 , +
r 2 − (u 2
3 + v 2
3 ), v 3
∼ x 3 (u, v) = e 1 u 3 + e 2
r 2 − (u 2
3 + v 2
3 ) + e 3 v 3 ,
U
−
y = {[x, y, z] ∈ S
2
r y = −
r 2 − (x 2 + z 2 ) < 0, x
2 + z
2 < r
2
} ,
Φ 4 (x, y, z) :=
x
z
=
u 4
v 4
,
Φ
−1
4 (u 4 , v 4 ) =
u 4 , −
r 2 − (u 2
4 + v 2
4 ), v 4
∼ x 4 (u, v) = e 1 u 4 − e 2
r 2 − (u 2
4 + v 2
4 ) + e 3 v 4 ,
U
+
x = {[x, y, z] ∈ S
2
r x = +
r 2 − (y 2 + z 2 ), y
2 + z
2 < r
2
} ,
Φ 5 (x, y, z) :=
y
z
=
u 5
v 5
,
Φ
−1
5 (u 5 , v 5 ) =
+
r 2 − (u 2
5 + v 2
5 ), u 5 , v 5
∼ x 5 (u, v) = +e 1
r 2 − (u 2
5 + v 2
5 ) + e 2 u 5 + e 3 v 5 ,
U
−
x = {[x, y, z] ∈ S
2
r x = −
r 2 − (y 2 + z 2 ), y
2 + z
2 < r
2
} ,
Φ 6 (x, y, z) :=
y
z
=
u 6
v 6
,
Φ
−1
6 (u 6 , v 6 ) =
−
r 2 − (u 2
6 + v 2
6 ), u 6 , v 6
∼ x 6 (u, v) = −e 1
r 2 − (u 2
6 + v 2
6 ) + e 2 u 6 + e 3 v 6 .
(3.16)
Example 3.4 (Sphere S
2
r , complete atlas: I = 6).
By means of an orthogonal projection p = π(P ), we already introduced a first coordinate set of the
“two-sphere” S
2
r in terms of spherical longitude λ and spherical latitude φ. As local coordinates, {λ, φ}
do not cover all points of the “two-sphere”. As a set of exceptional points, we removed the South Pole,
the North Pole, as well as the Greenwich Meridian, the meridian λ = 0. Here, we introduce a special
union of six charts, which covers the “two-sphere” completely. Figure 3.4 illustrates those six charts.
Their generators Φ i = Φ(U i ) (i = 1, 2, 3, 4, 5, 6) are the following:
U
+
z = {[x, y, z] ∈ S
2
r z = +
r 2 − (x 2 + y 2 ) > 0, x
2 + y
2 < r
2
} ,
Φ 1 (x, y, z) :=
x
y
=
u 1
v 1
,
Φ
−1
1 (u 1 , v 1 ) =
u 1 , v 1 , +
r 2 − (u 2
1 + v 2
1 )
∼ x 1 (u, v) = e 1 u 1 + e 2 v 1 + e 3
r 2 − (u 2
1 + v 2
1 ) ,
U
−
z = {[x, y, z] ∈ S
2
r z = −
r 2 − (x 2 + y 2 ) < 0, x
2 + y
2 < r
2
} ,
Φ 2 (x, y, z) :=
x
y
=
u 2
v 2
,
Φ
−1
2 (u 2 , v 2 ) =
u 2 , v 2 , −
r 2 − (u 2
2 + v 2
2 )
∼ x 2 (u, v) = e 1 u 2 + e 2 v 2 − e 3
r 2 − (u 2
2 + v 2
2 ) ,
U
+
y = {[x, y, z] ∈ S
2
r y = +
r 2 − (x 2 + z 2 ) > 0, x
2 + z
2 < r
2
} ,
Φ 3 (x, y, z) :=
x
z
=
u 3
v 3
,
Φ
−1
3 (u 3 , v 3 ) =
u 3 , +
r 2 − (u 2
3 + v 2
3 ), v 3
∼ x 3 (u, v) = e 1 u 3 + e 2
r 2 − (u 2
3 + v 2
3 ) + e 3 v 3 ,
U
−
y = {[x, y, z] ∈ S
2
r y = −
r 2 − (x 2 + z 2 ) < 0, x
2 + z
2 < r
2
} ,
Φ 4 (x, y, z) :=
x
z
=
u 4
v 4
,
Φ
−1
4 (u 4 , v 4 ) =
u 4 , −
r 2 − (u 2
4 + v 2
4 ), v 4
∼ x 4 (u, v) = e 1 u 4 − e 2
r 2 − (u 2
4 + v 2
4 ) + e 3 v 4 ,
U
+
x = {[x, y, z] ∈ S
2
r x = +
r 2 − (y 2 + z 2 ), y
2 + z
2 < r
2
} ,
Φ 5 (x, y, z) :=
y
z
=
u 5
v 5
,
Φ
−1
5 (u 5 , v 5 ) =
+
r 2 − (u 2
5 + v 2
5 ), u 5 , v 5
∼ x 5 (u, v) = +e 1
r 2 − (u 2
5 + v 2
5 ) + e 2 u 5 + e 3 v 5 ,
U
−
x = {[x, y, z] ∈ S
2
r x = −
r 2 − (y 2 + z 2 ), y
2 + z
2 < r
2
} ,
Φ 6 (x, y, z) :=
y
z
=
u 6
v 6
,
Φ
−1
6 (u 6 , v 6 ) =
−
r 2 − (u 2
6 + v 2
6 ), u 6 , v 6
∼ x 6 (u, v) = −e 1
r 2 − (u 2
6 + v 2
6 ) + e 2 u 6 + e 3 v 6 .
(3.16)