J.D. Mireles James
Mathematical Sciences
Florida Atlantic University

# Modern Analysis

Fall 2017

What, where, and when?
(MAA 4200 001, 3 credits). We meet on Mondays, Wednesdays, and Fridays from 12:00pm-12:50p.m. in the College of Education - Room 114.

Contact Me

Office hours:  Mondays and Wednesdays, 4pm-5pm in SE 262.

Phone:  561-297-2490

E-mail:  jmirelesjames@fau.edu.

(Description) Two of the most important aspects of mathematics are argument and explaination. For a mathematician, it is never enough to say that this or that formula is true. It is critical that we give an argument for why it is true. It is important to explain why it should be true in the first place. Knowing how to compute a derivative or an integral is great. But from a mathematical point of view it is even more important to understand what a derivative is and what are its properties. Why does it act this way and not that way?

That being said, learning to construct correct mathematical arguments is one of the most dificult parts of the subject. It takes years of hard work and practice. Building correct arguments is especially challenging in Calculus, becuase calculus is built on the real number system -- and most of us never think very seriously about what a real number is before a course like Modern Analysis. We tend to take the real numbers for granted. But if we want to build correct arguments -- if we want to understand why Calculus works -- then we have to understand exactly what the real numbers are. What properties do they have? What can we do with them? How should we think about them?

The goal of this course is to re-examine Calculus from the point of view of "Why?" rather than "How?" We will spend most of our time studying in great detail parts of the subject that were glossed over very rapidly when you took high school or Engineering Calculus. We will spend almost all our time asking why? rather than how? In fact I assume that you are already expert in using Calculus: computing limits, derivatives, integrals, and infinite sums. In this course we will delve into why these calculations work and ask what they tell us. Developing a deep understanding of the real numbers and of Calculus is necessary for teaching the subject, and lays the foundation for all further work in mathematical analysis.

Prerequisite

MAC 2313 (Calculus III) and MAD 2104 (Discrete mathematics) both with a minimum grade of C.

Textbook and Topics

Textbook: Understandin Analysis-- by Stephen Abbot

We will cover at least the following material:

 Chapter 1 The Real Numbers After reviewing more basice number systems like the Natural Numbers, the Integers, and The Rational Numbers, we will discuss the Real Numbers. We'll be especially interested in basic properties such as completeness and uncountability. Chapter 2 Sequences and Series The most important concept is the notion of the limit of a sequence or real numbers and we will study this idea in great detail. Then we will use it to define infinite series and study their properties. Chapter 3 Basic Topology of the Real Numbers Once we understand sequences of real numbers we use them to study deep properties of the real numbers, such as compactness and connectedness. Chapter 4 Functions: limits and continunity This is the most important chapter in the course, where we commence the serious study limits of functions and continunity.

Time permitting we may discuss some material form Chapters Five, Six, and Seven.

Objectives

• Know the definitions of objects like the real numbers, greatest upper bounds, limits, continunity, convergence of infinite series. Know their basic properties.
• Be able to use the basic definitions to construct correct mathematical arguments about the Real Numbers, Sequences/Series of Real Numbers, and functions defined on the real numbers.
• Be able to give examples and counter examples of important concepts.
• Generall improve our ability to formulate mathematical arguments and give mathematical explainations.

Tutoring

Tutoring is available at the Math Learning Center (MLC), located at GS 211. For tutoring resources, visit MLC

FREE MATH TUTORING for FAU students: The MLC provides the following FREE academic support services for FAU students:

1. Drop-in tutoring in the SAM LAB (Succeed At Methods) in GS207 during all hours of operation
a. ALL METHODS TUTORING is done in the SAM Lab except on Sundays. On Sundays, please visit the MLC as the SAM Lab is closed.
b. Monday – Thursday: 9am – 5pm and Friday: 9am – 4pm

2. Drop-in tutoring in the MLC GS211 during all hours of operation
a. Monday – Thursday: 9am – 5pm, Friday: 9am – 4pm, and Sunday: 1pm – 5pm

3. Small group tutoring by appointment. Appointments can be made in TutorTrac. Go to www.fau.edu/tutoring and click on ‘Find a Tutor,’ then ‘Click Here to Make an Appointment.’ Login with your FAU ID and password and click on ‘Search for Availabilities.’ For Center, choose SAM Lab for Methods of Calculus and Math Learning Center for everything else. Choose your Section (Class) and click ‘Search.’ Choose your time and then click ‘Save.’ If there are no appointments listed for your course, please email bferoz@fau.edu and request an appointment.

Credit

Homework:  I will assign homework problems every day, or at least once a week. I will collect all the homework. . Quizzes:  I will give some short pop-quizes during the semester. These may not be announced ahead of time. The quiz questions will be similar to assigned exercises, and will help you understand how ready you are for an exam situation.

Final Exam:  Date, time and location of the final exam to be announced.

Item Homework/Quizzes Date Percentage Assigned/administered in class 30% (Range) Mid/Late September & Late November 20% (each) Date and time to TBA 30%

 % letter grade 93-100 90-92 87-89 84-86 80-83 75-79 70-74 60-69 59 and below A A- B+ B B- C+ C D F

The instructor reserves the right to adjust the grading scale in the students favor. The grade of I (incomplete) will only be given for reasons specified on page 67 of the FAU Undergraduate Catalog.

Missed Exams and Quizzes

A missed midterm or final exam may be made up; however, it is the student’s responsibility to establish with documentation that the exam was missed for an allowable reason. The student cannot make up a missed midterm or final exam without such documentation. A missed quizz cannot be made up, but the two worst quiz grades will be dropped.

Attendance

Attending lectures is an important part of making progress in this course. You will find that if you don't come to class you will get behind very quickly. Any announcement made in class (quizz warnings, exam dates, and so on) are considered official course information. Also, I will assign homework in class. These assignemnts will not be posted online or anywhere else (if you must miss class you should try to check with a classmate about the assignment or any other important announcments). The instructor reserves the right to take attendance on any given course meeting. The following is quoted from FAU university wide policy: Students are expected to attend all of their scheduled University classes and to satisfy all academic objectives as outlined by the instructor. The effect of absences upon grades is determined by the instructor, and the University reserves the right to deal at any time with individual cases of non-attendance. Students are responsible for arranging to make up work missed because of legitimate class absence, such as illness, family emergencies, military obligation, court-imposed legal obligations or participation in Universityapproved activities. Examples of University-approved reasons for absences include participating on an athletic or scholastic team, musical and theatrical performances and debate activities. It is the student’s responsibility to give the instructor notice prior to any anticipated absences and within a reasonable amount of time after an unanticipated absence, ordinarily by the next scheduled class meeting. The instructor will allow each student who is absent for a University-approved reason the opportunity to make up work missed without any reduction in the student’s final course grade as a direct result of such absence.

Code of Academic Integrity policy statement

Students at Florida Atlantic University are expected to maintain the highest ethical standards. Academic dishonesty is considered a serious breach of these ethical standards, because it interferes with the university mission to provide a high quality education in which no student enjoys an unfair advantage over any other. Academic dishonesty is also destructive of the university community, which is grounded in a system of mutual trust and places high value on personal integrity and individual responsibility. Harsh penalties are associated with academic dishonesty. For more information, see University Regulation 4.001 at http://www.fau.edu/regulations/chapter4/4.001_Code_of_Academic_Integrity.pdf

Students with Disabilities

In compliance with the Americans with Disabilities Act (ADA), students who require special accommodation due to a disability to properly execute coursework must register with Student Accessibility Services (SAS) and follow all SAS procedures. SAS has offices across three of FAU’s campuses – Boca Raton, Davie and Jupiter – however disability services are available for students on all campuses.

Contact: Office for Students with Disabilities (OSD) -- in Boca Raton, SU 133 (561-297-3880).