Markus Schmidmeier • Mathematical Sciences • Florida Atlantic University

# Modern Analysis

Fall 2013

Hi, here is some information about my course Modern Analysis (CRN: 82421, MAA 4200, 3 credits). We meet Mondays, Wednesdays and Fridays, 10:00am - 10:50am in SC 178.

Is Analysis necessary?

For this course, it is assumed that you have encountered the fundamental ideas of calculus: The system of real numbers, functions, limits, continuity, differentiation, integration. For many purposes, there is no harm at all in an informal approach in which a continuous function is one whose graph has no jumps and a differentiable function is one whose graph has no sharp corners.

Please have a look at the background image of this page. It shows the graph of the function defined by
f(x)=x·sin(1/x) for x≠0 and f(0)=0.

Note that the function cannot be drawn accurately in the vicinity of x=0 since it crosses the x-axis infinitely often. Is the function continuous at x=0? (yes) Is it differentiable? (no)

In this course we ask: What do concepts like continuity and differentiability mean, how can we make them precise? And how can we possibly verify them? Our goal is to discuss how calculus can be built on secure logical foundations. Consider it from a historical perspective: With its foundations established, calculus has lead to amazing developments in applied analysis, in the natural sciences and in engineering.

Prerequisites

Discrete Mathematics (concept of real numbers), Matix Theory (vector spaces), Calculus I and Calculus II.

Course Description

This is one of the "capstone" courses for the mathematics major. The content of the course is a rigorous, proof-oriented treatment of fundamental calculus concepts: sequences, limits, continuity, derivatives, and integrals. In the past, students have found this course to be more challenging than other mathematics courses. This course will most likely take more study time and effort to be successful than other mathematics courses you have taken.

Textbook and Topics

We will use Real Analysis, John M. Howie, Springer Undergraduate Mathematics Series, ISBN 1852333146.

We are going to cover the following chapters:

 Chapter 2 Convergence of Sequences(3 weeks) We review sequences, series, and operations on sequences from Calculus II. Our emphasis is on convergence: We study several definitions of convergence, in particular Cauchy sequences. It turns out that one can speak about convergence even if there is no limit (like for sequences of rational numbers that approach a non-rational number). Chapter 3 Functions and Continuity(4 weeks) Functions, together with addition and scalar multiplication form a vector space. But there are additional operations on the set of functions on the real line, like multiplication and composition. We define continuity and show that this concept is well behaved with respect to the operations mentioned. Chapter 4 Differentiation(2 weeks) You all remember L'Hôpital's rule for critical limits. Here we show why it is true. This involves axioms of real numbers, Rolle's Theorem, and the famous Mean Value Theorem. Returning to vector spaces of functions, we show that differentiation is well behaved with respect to the operations. Midterm Exam The 50 minute exam will be about the material covered so far. It is scheduled for Friday, October 11, during class time (unless we agree on a different date). Chapter 5 Integration(3 weeks) We review the definition of an integral as a limit of a Riemann sum. We study properties of integrals and introduce vector spaces consisting of integrable functions. Of special interest are improper integrals. Chapter 6 log and exp(2 weeks) Your calculus textbook may contain the following line in its title: "early transcendentals". Why is this? Here we do late transcendentals: We discuss how logarithm and exponential functions are introduced rigorously via integrals and inverse functions. This approach lets you derive all the known properties ex and log(x)! (No more memorizing needed.)

Objectives

• Learn about the mathematical foundations of calculus.
• Discuss formal definitions for convergence, continuity, etc.
• Invent and study ``strange'' examples to illustrate the necessity of conditions.
• Apply definitions and results to compose short proofs.
• Practise presentation skills by showing a solution to an exercise to class.

Tutoring

There is free math tutoring available in the Math Learning Center in GS 211.

Credit

Homework:  I will assign homework problems every week. The problems will not be graded, but some may show up on a quiz:   Homework problems.

Quizzes:  We will have a quiz every Friday; the ten best quizzes count for 30 % of the grade. No calculators can be used during the quiz.

Presentation:  For a 5 minute presentation, typically the solution of a non-homework problem from our textbook, you can earn credit which counts for 10 % of the grade.

Midterm Exam:  The midterm exam on Friday, October 11, counts for 20 % of the grade.

Final Exam: The final exam is comprehensive and will count for 40 % of your grade. It has been scheduled for Monday, December 9, 7:45am (sic!) - 10:15am in our classroom. Please bring a picture id (Owl card or drivers licence).

Further Information

For the Disability Policy, the Make-Up Policy, the Code of Academic Integrity, Religious Accommodation, my Grading Scale and Financial Assistance Opportunities please visit Infos for all my courses.

Contact Me

Office hours:  Mondays, Wednesdays and Fridays, 11:00am - 12:00 noon in SE 230.