Geography Reference
In-Depth Information
S
r
, complete atlas built on six charts
Fig. 3.4.
“Two-sphere”
Φ
1
(
x, y, z
):=
x
=
u
,
(3.17)
y
v
Φ
−
1
(
u,v
)=
u,v,
+
r
2
−
(
u
2
+
v
2
)
∼
x
1
(
u,v
)=
e
1
u
+
e
2
v
+
e
3
r
2
−
(
u
2
+
v
2
)
.
The terms
Φ
−
1
(
u,v
)or
x
1
(
u,v
) determine an open set of the “two-sphere” over the (
x, y
) plane,
namely
{−
r<u<
+
r,
−
r<v<
+
r
}
=:
V
1
.
I
=6:
2
2
Again, the union of the patches (“Umgebungsraume”)
U
∪
U
∪
U
∪
U
∪
U
∪
U
6
=
S
r
is
S
r
,
1
2
3
4
5
completely covered by the six charts
Φ
1
∈
V
1
,...,Φ
6
∈
V
6
,and
V
i
:=
{
]
−
r,
+
r
[
,
]
−
r,
+
r
[
}
(
u,v
)(
i
). An illustration is offered by Fig.
3.4
. In summary, we have generated
the complete atlas of the “two-sphere” constructed by six charts. The choice of the open interval
is motivated by the fact that the functions
Φ
i
(
u,v
)(
i
∈{
1
,
2
,
3
,
4
,
5
,
6
}
)at
u
2
+
v
2
=
r
2
are singular
when differentiated. Indeed, this result is documented by the following expressions:
∈{
1
,
2
,
3
,
4
,
5
,
6
}
⎡
partial deri
vative with res
pect to
u,
1
,
0
,
(
u
2
+
v
2
)
: singular at
u
2
+
v
2
=
r
2
,
partial deri
vative with res
pect to
v,
u/
r
2
⎣
−
−
d
Φ
−
1
(
u,v
)
∼
0
,
1
,
+
v/
r
2
−
(
u
2
+
v
2
)
: singular at
u
2
+
v
2
=
r
2
,
...
(3.18)
⎡
partial
derivative with
respect to
u,
+
u/
r
2
(
u
2
+
v
2
)
,
1
,
0
: singular at
u
2
+
v
2
=
r
2
,
partial
derivative with
respect to
v,
⎣
−
d
Φ
−
6
(
u,v
)
∼
+
v/
r
2
(
u
2
+
v
2
)
,
0
,
1
: singular at
u
2
+
v
2
=
r
2
.
−
End of Example.
Search WWH ::
Custom Search