Midterm Examination

April 11, 1996

WORK TWO OF THE FOLLOWING PROBLEMS

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.''

END

Wed Apr 10 19:37:51 EDT 1996