Math 531 Midterm
Please abide by the following rules:
Time Limit: 3 hours
You are allowed 1 untimed break
This test is open notes and open Spivak. You may not consult any additional sources (or
people)
You are to answer 2 of the rst 3 questions and 1
Math 531 Midterm Solutions
1. In multivariable calculus, given a function f : Rn R, you dene the gradient vector
f by
f
.
xi xi
f=
i
f
In other words, it is vector eld with component functions
.
xi
Consider the more general situation where M is a smooth
Math 531 Midterm Review
The midterm will be a timed, take-home exam (taken on the honor system). You will have 3 hours to complete the exam. You may use your notes
and Spivak, but you may not obtain outside help from anyone. The problems
will be a mix of
Math 531 Midterm Review Solutions
1. Let f : R2 R be dened by
f (x, y ) = x3 + xy + y 3 + 1.
1
For which points p = (0, 0), p = ( 3 , 1 ), p = ( 1 , 1 ), is f 1 (f (p) an embed3
3
3
ded submanifold in R2 ?
The answer here is quite subtle. The points p = (
Math 531 Final
May 15, 2008
Solve any 2 of the following 3 problems:
1. Show directly that RP n is a smooth manifold by giving local coordinate patches and calculating
the transition functions (Hint: use homogeneous coordinates).
2. Let M = R3 , and consi