Lecture Notes for 120 - UCLA Department of Mathematics

4.2. TANGENT SPACES AND MAPS 82
Definition 4.2.15. The parameters u, v on a parameterized surface q (u, v)
define two differentials du and dv. These are not mysterious infinitesimals, but
linear functions on tangent vectors to the surface that compute the coefficients of
the vector with respect to the basis @q
@u , @q
@v . Thus
✓ @q
du (v) = du
@u vu + @q ◆
@v vv = v u
✓ @q
dv (v) = dv
@u vu + @q ◆
@v vv = v v
and
v = ⇥ @q
@u
@q
@v
⇤ apple du
dv
(v) = ⇥ @q
@u
@q
@v
⇤ apple v u
From the chain rule we obtain the very natural transformation laws for differentials
or
Exercises.
apple du
dv
du = @u @u
ds +
@s @t dt
dv = @v @v
ds +
@s @t dt
=
apple @u
@s
@v
@s
@u
@t
@v
@t
apple ds
dt
(1) Show that the ruled surface
◆
q (t, ) = (cos , sin , 0) + t
✓sin cos , sin sin , cos 2 2 2
defines a parametrized surface. It is called the Möbius band. Showthatit
is not orientable by showing that when t =0and = ±⇡ we obtain the
same point and tangent space on the surface, but the normals
N (t, )=
@q
@t ⇥ @q
@
@q
@t ⇥ @q
@
are not the same.
(2) Show that q (t, )=t (cos , sin , 1) defines a parametrization for (t, ) 2
(0, 1) ⇥ R. Showthatthecorrespondingsurfaceisx 2 + y 2 z 2 =0,z>
0. Show that this parametrization is not one-to-one. Find a different
parametrization of the entire surface that is one-to-one.
(3) The inversion in the unit sphere is defined as
F (q) =
q
|q| 2
(a) Show that this is a diffeomorphism of R n 0 to it self with the
property that q · F (q) =1.
(b) Show that F preserves the unit sphere, but reverses the unit normal
directions.
v v