Fall 2014 MTH 435 Mathematical Analysis and Topology I
MW 3-4:15, F 3-3:50, Lippitt 204

Textbooks:

• Russell A. Gordon Real Analysis - A First Course (Second Edition), ISBN: 0-201-43727-9
• Theodore W. Gamelin, Robert Everist Greene Introduction to Topology (Second Edition), ISBN 0-486-40680-6

Instructor:

Barbara Kaskosz, Lippit Hall 202J, 874.4445, bkaskosz@math.uri.edu.
Office hours: W 1-2, F 4-6.

MTH 435 and its second part MTH 436 provide an introduction to the wonderful world of real numbers, real analysis and topology. Real analysis is the precise mathematical theory behind Calculus. The emphasis will be on mathematical precision and developing strong skills in proving theorems. You will see complete proofs done in class and learn how to write your own proofs. At the same time, you will gain a deep understading of the fundamental concepts of Calculus that you already know from the intuitive point of view. Those include limits, continuity, sequences of numbers and differentiation.

Many theorems in analysis that we will prove are particular cases of more general theorems in metric spaces. Therefore, we will study metric spaces which belong to the area of mathematics called topology.

Current Downloads

You will find here current downloads: solutions to homework problems, additional class notes etc.

Homework 1 Solutions   Homework 2 Solutions   Homework 3 Solutions   Homework 4 Solutions  

Homework 5 Solutions   Homework 6 Solutions   Homework 7 Solutions   Homework 8 Solutions   Homework 9 Solutions

Exams and Evaluation

There will be two exams during the semester, Friday Oct 17, 3-5 and Friday Nov 21, 3-5, Lippitt 204. Our Final is scheduled (by the Office of Enrollment Services) on Fri, Dec 12, 3-6pm, Lippitt 204. Final Exam Schedule.

Your grade will be based upon a possible total of 550 points, as follows:

• Exam 1 -- 100 points,
• Exam 2 -- 100 points,
• Homework -- 150 points,
• Final exam -- 200 points,

Weekly homework will be assigned in class.

Topics

INTRODUCTION

Mathematical Logic
Mathematical Induction

REAL NUMBERS

Axioms, Ordering
The Completeness Axiom
Density of Rationals and Irrationals
Countable and Uncountable Sets

SEQUENCES OF REAL NUMBERS

Intervals, Neighborhoods, the Triangle Inequality
Convergence, Divergence, Limits
Cauchy Sequences
Subsequences
The Bolzano-Weierstrass Theorem

REAL FUNCTIONS

Limits
Continuity
Boundedness Theorem
Min-Max Theorem
Intermediate Value Theorem
Uniform Continuity
The Derivative
Differentiability
Proofs of the Product, Quotient, Chain Rules
Local Extrema
Mean Value Theorem
Monotonicity and the Derivative

METRIC SPACES

Definition
Open and Closed Sets
The Real Line as a Metric Space
Sequences in a Metric Space
Completeness
Compactness
Continuous Functions