Math 147, Homework 1 Solutions
Due: April 10, 2012
1. For what values of a is the set:
Ma = (x, y, z ) : x2 + y 2 z 2 = a
a smooth manifold? Give explicit parametrizations for open sets covering Ma
in the cases where Ma is a smooth manifold.
Solution.
Con
Math 147, Homework 4 Solutions
Due: May 1, 2012
1. On Homework 3 you constructed a smooth function f : R1 R1 with a dense
set of critical values. Can you construct a smooth map f : S 1 S 1 whose
critical values are dense?
Solution.
No, there is no such ma
Math 147, Homework 3 Solutions
Due: April 24, 2012
1. Let Y R3 be the surface of revolution dieomorphic to S 1 S 1 you studied
in Homework 1 and let f and g be the smooth functions:
f, g : Y R1
dened by f (x, y, z ) = x and g (x, y, z ) = z . What are the
Math 147, Homework 2 Solutions
Due: April 17, 2012
1. Show that 0 is the only critical value of the map f : R3 R1 dened by:
f (x, y, z ) = x2 + y 2 z 2 .
Prove that if a and b are either both positive or both negative, then f 1 (a) and
f 1 (b) are dieomor
Math 147, Homework 5 Solutions
Due: May 15, 2012
1. Let f : R3 R6 and : R3 R3 be the smooth maps dened by:
f (x, y, z ) = (x2 , y 2 , z 2 , xy, xz, yz ) and (x, y, z ) = (x, y, z ).
(a) Show that f is proper and that f is an immersion except at the origin
Math 147, Homework 6 Solutions
Due: May 22, 2012
1. Let T = S 1 S 1 be the torus. Is it possible to nd a nite set S = cfw_P1 , . . . , Pn
of points in T and an embedding of the complement T \ S into R2 ? [Hint: You
may nd the Jordan-Brouwer separation th
Page 1
Math 147 Midterm
Your name:
Instructions: You may only use pen and paper. No reference material
(notes, books) or electronic device (calculator, laptop) is allowed.
As a courtesy to others, please turn o your cell phone at the beginning of
the exam
Math 147, Homework 8 Solutions
Due: June 5, 2012
You are encouraged to collaborate on the homework problems. Each student must
understand and write up his or her own clear and legible solutions.
1. In class we showed that if v : M Rk has a non-degenerate