Lecture Notes for 120 - UCLA Department of Mathematics

4.2. TANGENT SPACES AND MAPS 81
In particular, the differential is a linear map and is completely determined by the
two partial derivatives @F q
@u
, @F q
@v .
Proof. This follows from the chain rule:
dF q dF (q (t))
(t) =
dt
dt
dF (q (u (t) ,v(t)))
=
dt
= @F q du
@u dt + @F q dv
@v dt
= ⇥ ⇤ apple du
@F q @F q dt
@u @v dv
dt
Example 4.2.11. The Archimedes map satisfies
2 3
sin µ
@ (A q)
= 4 cos µ 5
@ (A q)
, =
@µ
@
0
and the Mercator map
@ (M q)
@µ
2
= 4
sin µ
cos µ
0
3
5 ,
@ (M q)
@
2
4
= 4
Definition 4.2.12. We say that a surface M is orientable if we can select a
smooth normal field. Thus we require a smooth function
N : M ! S 2 (1) ⇢ R 3
such that for all q 2 M the vector N (q) is perpendicular to the tangent space T q M.
The map N : M ! S 2 (1) is called the Gauss map.
0
0
cos
Proposition 4.2.13. A surface which is given as a level set is orientable.
2
0
0
1
3
5
3
5
⇤
Proof. The normal can be given by
if M = {q 2 O | F (q) =c}.
N = rF
|rF |
Example 4.2.14. Aparametrizedsurfaceq (u, v) : U ! R 3
natural map N (u, v) :U ! R 3 defined by
N (u, v) =
@q
@u ⇥ @q
@v
@q
@u ⇥ @q
@v
⇤
always has a
that gives a unit normal vector at each point. However, it is possible (as well shall
see in the exercises) that there are parameter values that give the same points on
the surface without giving the same normal vectors.