SAMPLE FINAL EXAM PROBLEMS
Problem 1. Prove that the inverse map
GL(n)
T Inv (T ) = T 1 GL(n)
is dierentiable, and that
DInv (X )(H ) = X 1 HX 1 ,
for every X GL(n), and every H1 , H2 M (n).
Problem 2. Prove that the inverse map
GL(n)
T Inv (T ) = T 1 GL(
Math 509 - Advanced Analysis
Herman Gluck
Thursday April 16, 2015
4. MULTILINEAR ALGEBRA
Calculus + Linear Algebra Vector Calculus
Calculus + Multilinear Algebra Differential Forms
We will begin here with Multilinear Algebra,
and then move to the study of
Math 509 - Advanced Analysis
Herman Gluck
Tuesday February 17, 2015
Incomplete
2. REVIEW OF ADVANCED CALCULUS
INTEGRATION (following Spivak's "Calculus on Manifolds")
Basic Definitions
The definition of the integral of a real-valued function
f: A R define
Math 509 - Advanced Analysis
Herman Gluck
Tuesday April 21, 2015
5. DIFFERENTIAL FORMS
We will study differential forms defined on open subsets of
Euclidean space Rn .
Elements of Rn may be regarded as points p or vectors v .
Fix a point p Rn . Then the s
MATH 509 Advanced Analysis
SAMPLE MIDTERM EXAM
Name :
Student I.D. # :
Date : March 12, 2013
Problem 1.
4points
Problem 2.
4points
Problem 3.
4points
Problem 4.
4points
Problem 5.
4points
Problem 6.
4points
Problem 7.
4points
Problem 8.
4points
TOTAL:
32p
MATH 509 HOMEWORK 12.
due on Thursday, April 18.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 8 Some Special Functions.
Power Series
The Exponential and Logarithmic Functions
The Trigonometric Functions
Fou
MATH 509 HOMEWORK 11.
due on Thursday, April 11.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 8 Some Special Functions.
Power Series
The Exponential and Logarithmic Functions
The Trigonometric Functions
Fou
MATH 509 HOMEWORK 9.
due on Thursday, March 28.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 8 Some Special Functions.
Power Series
The Exponential and Logarithmic Functions
The Trigonometric Functions
Four
MATH 509 HOMEWORK 7.
due on Thursday, February 28.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 7 Sequences and Series of Functions
The Stone-Weierstrass Theorem
Chapter 8 Some Special Functions.
Power Serie
MATH 509 HOMEWORK 6.
due on Thursday, February 21.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 7 Sequences and Series of Functions
The Stone-Weierstrass Theorem
Chapter 8 Some Special Functions.
Power Serie
MATH 509 HOMEWORK 5.
due on Thursday, February 14.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 7 Sequences and Series of Functions
Discussion of Main Problem
Uniform Convergence
Uniform Convergence and Cont
MATH 509 HOMEWORK 4.
due on Thursday, February 7.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 7 Sequences and Series of Functions
Discussion of Main Problem
Uniform Convergence
Uniform Convergence and Conti
MATH 509 HOMEWORK 1.
due on Thursday, January 17.
Text: Principles of Mathematical Analysis by Walter Rudin, third edition
Topics:
Chapter 7 Sequences and Series of Functions
Discussion of Main Problem
Uniform Convergence
First Homework Assignment.
Rea
SAMPLE FINAL EXAM PROBLEMS
Denition. Let fi : i Rmi , i = 1, 2 , be functions dened on the open subsets
i Rni . We denote by
f1 f2 : 1 2 Rm1 Rm2
the map f1 f2 (v1 , v2 ) = (f1 (v1 ), f2 (v2 ).
Problem 1. Let i L(Rni , Rmi ), i = 1, 2. Prove that 1 2 L(Rn1
Math 509 - Advanced Analysis
Herman Gluck
Tuesday March 3, 2015
2a. MORE ABOUT INTEGRATION
Here is the basic existence theorem for Riemann integrals,
stated but not proved in the preceding chapter, which we
now prove here.
Theorem. Let A be a closed recta