Exam 2: Math 7350 Spring 2013
Professor William Ott
Problem 1. (Cartan lemma) Let M be a smooth n-manifold and let ( 1 , . . . , k ) be an ordered k -tuple of
smooth 1-forms such that ( 1 |p , . . . , k |p ) is linearly independent for every p M . Given s
Assignment 1: Math 7350 Spring 2013
Professor William Ott
Problem 1. Let X = (x, y ) R2 : y = 1 or y = 1 and let M be the quotient space obtained from X
by identifying (x, 1) with (x, 1) for all x = 0. Show that M is locally Euclidean and second countable
Assignment 2: Math 7350 Spring 2013
Professor William Ott
Problem 1. Dene F : Rn RPn by F (x1 , . . . , xn ) = [x1 , . . . , xn , 1]. Show that F is a dieomorphism
onto an open, dense subset of RPn .
Problem 2. Suppose A and B are disjoint closed subsets
Assignment 3: Math 7350 Spring 2013
Professor William Ott
Problem 1. Consider the map : R4 R2 dened by
(x, y, s, t) = (x2 + y, x2 + y 2 + s2 + t2 + y ).
(a) Show that (0, 1) is a regular value of .
(b) Show that 1 (0, 1) is dieomorphic to S2 .
Problem 2.
Assignment 4: Math 7350 Spring 2013
Professor William Ott
Problem 1. Let M be a connected smooth manifold. Prove that for every p, q M , there exists a
dieomorphism F : M M such that F (p) = q . (This problem shows that the group of dieomorphisms
acts tra
Exam 1: Math 7350 Spring 2013
Professor William Ott
Problem 1. Let G be a Lie group. Prove that the multiplication map m : G G G is a smooth
submersion. Hint: use local sections.
Problem 2. Let G be a Lie group.
(a) Let m : G G G denote multiplication. Sh
Justin Le
Professor Sara Rolater
English 1304
6 November 2016
Fluorides Consequences
Looking at one of Americas most controversial topics is the addition of fluoride into
Americas water supply. I wanted to research the different reasons as to why the topi