Math 396 (Winter 2011). Final exam solutions
F.1. This is false. As we discussed in class, there are many connected manifolds whose
fundamental group is noncommutative. For instance, for any nite group one can construct a smooth manifold with that fundame
Math 396 (Winter 2011). Solutions for Problem Set 4
4.1. Let e1 = (1, 0, 0, . . . , 0) Rn denote the rst standard basis vector. If M Rn and
X is the vector eld x1 , then the denition of an integral curve in this case amounts to
the following: a smooth map
Math 396 (Winter 2011). Solutions for Problem Set 1
1.1. Consider the function f : R2 R dened by
x if x > 0 and 0 < y x2 ;
f (x, y ) =
0
otherwise.
Note that f (x, y ) 0 as (x, y ) (0, 0), so f is continuous at p = (0, 0). It is also easy
to check that th
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Math 396 (Winter 2011). Take-home midterm solutions
M.1. The usual projection : S n Pn (where S n is the n-dimensional sphere, Pn is
the real n-dimensional projective space, and (p) is the line spanned by p for any p S n )
has the property that its dieren
Math 396 (Winter 2011). Solutions for Problem Set 6
6.1. (a) Using our standard notation for dierential forms, write
fI dxI ,
=
|I |= k
where x1 , . . . , xm are the standard coordinates on Hm and fI : U R is a C function
for each I = (i1 , . . . , ik ),