April 11, 1996
1. This problem is about the tangent bundle to the 2-sphere 
.
As usual, we write 
, where U is the complement
of the South pole and V is the complement of the North pole.
Show how the tangent bundle admits trivializations over
U and V (i.e. isomorphisms to the product bundles
and 
)
and exhibit the transition function relating those
trivializations over 
.
2. The real projective plane 
can be considered as 
,
where as usual 
, and
the relation 
is 
. The map
defined by
defines a map 
which is invariant under ,
and therefore defines a map 
.
a. Show that v is an immersion, i.e. that v has rank 2 at every
point of .
b. Show that v is one-one.
c. Suppose v is composed with projection onto 
by suppressing
one of the first three coordinates. Is the resulting map still an
immersion? Is it one-one?
3. Let 
be a smooth vectorfield defined in an open set 
, and
suppose S is a smooth, oriented piece of surface lying in U,
with boundary C (C inherits an orientation from S as
usual). The ``Traditional Stokes' Theorem'' states that the
integral over S of the curl of X is
equal to the integral of X along C.
[Recall that the integral of a vectorfield X over a piece
of surface parametrized by 
, 
is
given by 
, and that
the integral of X along a curve parametrized by 
is 
.]
Show that this theorem is equivalent to a special case of our
``Stokes' Theorem.''