Lecture Notes for 120 - UCLA Department of Mathematics

4.5. SPECIAL MAPS AND PARAMETRIZATIONS 95
Proof. Assume we have a different parametrization q (s, t) :V ! M and a
new region T ⇢ V with q (R) =q (T ) and the property that the reparametrization
(u (s, t) ,v(s, t)) : T ! R is a diffeomorphism. Then
ˆ
p
Area (q (R)) = det [I]dudv
=
=
=
=
=
=
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
R
R
R
R
R
R
R
r
det
v
u
t det
v
u
t det
s
det
⇣ ⇥ @q
@u
✓ ⇥ @q
@s
apple @s
@u
@t
@u
apple @s
@u
@t
@u
q ⇥
det @q
@s
q ⇥
det @q
@s
@q
@v
@s
@v
@t
@v
@q
@t
@q
@t
@q
@t
@s
@v
@t
@v
⇤ t ⇥ @q
@u
⇤ apple @s
@u
@t
@u
t ⇥ @q
@s
@q
@v
@s
@v
@t
@v
@q
@t
t q ⇥
det @q
@s
⇤ t ⇥ @q
@s
⇤ t ⇥ @q
@s
@q
@t
@q
@t
⇤ ⌘ dudv
◆ t
⇥ @q
@s
⇤ t ⇥ @q
@s
@q
@t
@q
@t
⇤ t ⇥ @q
@s
⇤
det
apple @s
⇤
dsdt
@u
@t
@u
@q
@t
@s
@v
@t
@v
⇤ apple @s
⇤ apple @s
@q
@t
@u
@t
@u
@u
@t
@u
@s
@v
@t
@v
@s
@v
@t
@v
!
⇤ s det
apple @s
dudv
where the last equality follows from the change of variables formula for integrals.
@u
@t
@u
⇤
!
dudv
dudv
@s
@v
@t
@v
dudv
Exercises.
(1) Check if the parameterization q (t, )=t (cos , sin , 1) for the cone is an
isometry, area preserving, or conformal? Can the surface be reparametrized
to have any of these properties?
(2) Show that he inversion map
F (q) =
q
|q| 2
is a conformal map of R n 0 to it self.
(3) Show that the stereographic projections q ± R n ! R n ⇥ R = R n+1 defined
by
q ± (q) =(q, 0) + 1 |q|2
2
(q, ⌥1)
1+|q|
are conformal parametrizations of the unit sphere.
(4) Show that Enneper’s surface
✓
q (u, v) = u
1
3 u3 + uv 2 ,v
◆
1
3 v3 + vu 2 ,u 2 v 2
defines a conformal parametrization.
(5) Consider a map F : S 2 ! P , where P = {z =1} is the plane tangent
to the North Pole, that takes meridians to radial lines tangent to the
meridian at the North Pole. Sometimes the map might just be defined on
part of the sphere such as the upper hemisphere.