Math 4320 Introduction to Real Analysis II

Spring 2015 Syllabus

Instructor: Burt SimonCourse dates/times: TR 5:00-6:15

Office: Academic Building, room 4209Course location: AB 4017

Office Hours: MW 2:00-4:00, or by appointmentText: Advanced Calculus, 2nd edition

Phone: 303 315 1710 by Patrick Fitzpatrick

Website: www.math.ucdenver.edu/~bsimon

Email:

Real Analysis II (Math 4320) will start where Real Analysis I (Math 4310) finished off. Roughly speaking, Real Analysis I covered functions of one real variable, and

Real Analysis II will cover functions of more than one real variable, but there are other interesting topics as well. As you already know, analysis is mostly about theorems and proofs, so be prepared!

University Course Catalog Description:

Convergence, uniform convergence; Taylor’s theorem; calculus of several variables including continuity, differentiation and integration; Picard’s theorem in ordinary differential equations and Fourier series.Prereq: MATH 4310.Semester Hours:3 to 3When Offered:SPRING

Course Overview: We will follow the textbook, starting where Math 4310 (Fall 2013) finished off, which is chapter 9. The beginning of the semester will still be on functions of one variable, but we will soon transition to a bit of metric space topology, and then topics on functions of two or more real variables. A lot of the material will (hopefully) strike you as highly analogous to the material from Real Analysis I, but notationallyit will be more complicated due to the additional variables. At the end of the semester we will cover theorems like the Implicit Function Theorem that have no analog in one variable.

Course Goals and Learning Objectives:

  1. Overall Learning Objectives: By successfully completing this course, students will be ready for graduate level mathematics. For many math majors, analysis is the hardest course they take. The experience gives students “mathematical maturity” which is necessary for all advanced mathematics, and other technical fields as well. In particular, students will be able to understand the statements and proofs of the basic theorems of real analysis, and to be able to construct proofs of their own.
  2. Learning Outcomes:
  • Creative Thinking: Students will learn to construct proofs of abstract mathematical statements, and thereby come to understand abstract mathematical concepts at a deep level.
  • Critical Thinking: Students will learn to appreciate mathematical rigor, and will therefore be able to see when an attempt to prove something does not stand up to scrutiny.
  1. Major Topics: Sequences of functions, intro to metric space topology, differentiating functions of several variables, approximating functions of several variables, Inverse Function Theorem, Implicit Function Theorem.
  2. Rationale: Two semesters of Real Analysis are typical requirements for admission to graduate school in mathematics, and come highly recommended in other fields, like economics, financial engineering, and Operations Research.

Course Prerequisites: Math 4310

Course Credits: 3 credit hours

Required Texts and Materials: Advanced Calculus, second edition, by Patrick Fitzpatrick.

Course Schedule:

The following schedule of course materials covered and exams is tentative.

DATES (class times) / TOPICS / SECTIONS FROM TEXT
Jan 21 / Uniform convergence / sections 9.1 – 9.2
Jan 26, 28 / Uniform convergence theorems / Sections 9.3 - 9.4
Feb 2, 4 / Convergence of power series / sections 9.5 – 9.6
Feb 9, 11 / n-dimensional Euclidian Space / sections 10.1 – 10.3
Feb 16, 18 / Continuity and Compactness / sections 11.1 – 11.2
Feb 23, 25 / Metric Spaces and contractions / Sections 12.1 – 12.3
March 2, 4 / Review and Exam #1
March 9, 11 / Differentiating in n-dimensions / sections 13.1 – 13.2
March 16, 18 / Directional derivatives and MVT / sections 13.3, 14.1
March 23 - 27 / SPRING BREAK
March 30, April 1 / Local approximation / Sections 14.2 – 14.3
April 6, 8 / Linear approximation / sections 15.1 – 15.3
April 13, 15 / Review and Exam #2
April 20, 22 / Inverse Function Theorem / sections 16.1 – 16.3
April 27, 29 / Implicit Function Theorem / sections 17.1 – 17.4
May 4, 6
May 11, 13 / Catch-up and Review
Final Exam (precise date to be determined)

Assignments: I will assign homework problems from the textbook (approximately) weekly. Typically the assignment will be posted on the class web page on Monday or Tuesday, and will be due the following Monday. You are expected to turn in every homework assignment on time, since the purpose of the homework each week is to practice the material covered in class that week. Furthermore, homework sets will be discussed in class the day they are due, so late assignments will not receive full credit. Homework sets are expected to be written up neatly so they are easy to read (preferably using Word or Latex).

Working collectively on the homework assignments is encouraged! There is obviously no penalty, but please list the people you worked with on the papers you turn in.

Basis for Final Grade: Your final grade will be based on your exam grades (two midterm exams and a final exam), weekly homework sets, and intangibles such as class participation. The intangibles can only increase your grade. Exam and homework grades will be letter grades (A through F). I grade proofs the same way I would grade an “essay”, so letter grades are more appropriate.

In the absence of “intangibles”, your final grade will be a weighted average of the (letter)grades on your exams and homeworks. The weightings will be as follows: Homework (20%), Midterm Exam #1 (25%), Midterm Exam #2 (25%), Final Exam (30%).

Grade Dissemination: I will try to grade homework sets within a week of when they are due, and exams by the next class after they are taken. They will be returned in class with grades on them. If you sense a mistake in my grading, please send me an email, or come to my office hours to discuss.

Course Policies

  1. Attendance: I will not take attendance, but students are expected to attend every class. You will be responsible for material I cover in class, whether or not it is in the textbook. Class participation is one of the important “intangibles” that can impact your grade. Please try to turn in homeworks sets when they are due (by email if necessary). If there is a problem with the timing of an exam, let me know as soon as possible so arrangements can be made.
  2. Grades of “Incomplete”: I will follow university procedures on “incompletes”, i.e., they are only given in situations where unexpected emergencies prevent students from completing the course and the remaining work can be easily finished the following semester. Incomplete work must be finished the next semester or the grade automatically turns into an F.
  3. Group Work Policy: Students are encouraged to collaborate on homework sets, as long as they acknowledge their collaborators. There is no penalty for working together. Of course, no collaboration is allowed on exams, as that is considered cheating.
  4. Announcements: I will use the class web page for all communication that is meant for the whole class. Please check the page regularly. Private communication is best done by email. I will typically respond within a day.
  5. Laptops, Cell Phones, etc.: You are free to use your devices as you see fit during class. (No phone calls or texting, of course.) The rules during exams will be announced prior to the tests. Usually my exams are open-book, open-notes, but electronic devices (calculators, computers, etc.) are not allowed. If your textbook is on your computer, let me know so there is no disruption during the test.
  6. Civility: Students are expected to be quiet and attentive during class, although raising your hand to ask a question or make a comment is welcome and encouraged.
  • Dishonesty: Students are expected to understand intuitively what proper ethical conduct means in the context of a college mathematics course. If you are caught cheating you could fail the class or (at least) have your grade lowered, so don’t even try it.

Spring 2015 CLAS Academic Policies
The following policies, procedures and deadlines pertain to all degree-seeking students in the College of Liberal Arts and Sciences (CLAS) and are aligned with the Official University Academic Calendar. Please also see you advisor if you have questions or concerns.
  • Schedule verification: It is each student’s responsibility to verify online that his/her official registration and schedule of classes is correct before classes begin and prior to the drop/add deadline. Failure to verify schedule accuracy is not sufficient reason to justify a late adds or drops.
  • E-mail: Students must activate and regularly check their official CU Denver e-mail account for university related messages. Those who forward email should check CU Denver e-mail regularly for messages not automatically forwarded.
  • Waitlists:
  • Students are automatically notified if they are enrolled in a class from a waitlist via their official CU Denver email account.
  • Students are not automatically dropped from a class if they never attended, stopped attending, or do not make tuition payments, though instructors may request administrative drops.
  • Waitlists are purged after the 1st week of classes.
  • Late adds (after 4 February) will be approved only when circumstances surrounding the late add are beyond the student’s control. This will require a written petition and verifiable documentation. Petition forms are available for undergraduate students in the CLAS advising office and from the Graduate School for graduate students. The signature of a faculty member on a SAF does not guarantee that a late add petition will be approved.
  • Late drops/withdrawals (after 20 April) will be approved only when circumstances surrounding the late drop have arisen after the published drop deadline and are beyond the student’s control. This will require a written petition and verifiable documentation. The signature of a faculty member does not guarantee that a late drop/withdrawal petition will be approved after April 20. After April 20th, students should meet with their CLAS advisor or Graduate School Coordinator to learn more about how to petition to drop or withdraw from a course late.
  • Tuition: Students are responsible for completing arrangements with financial aid, family, scholarships, etc. to pay their tuition prior to Census Date (4 February). Students who drop after that date are (1) financially responsible for tuition and fees, (2) academically responsible and will receive a "W" grade, and (3) are ineligible for a refund of COF hours or tuition.
  • Graduation:
  • Undergraduate students wishing to graduate in Spring 2015 should (1) first meet with their CLAS advisor; (2) meet with their major and minor advisor(s), who will complete the electronic form required to verify eligibility to graduate; and (3) apply for graduation online through UCDAccess. These steps must be completed by no later than 5PM on February 4, 2015, which is an absolute deadline without exception.
  • Graduate students wishing to graduate in Spring 2015 must apply for graduation online through UCDAccess and have a Request for Admissions to Candidacy on file with the CU Denver Graduate School (LSC 1251) no later than 5PM on February 4, 2015, which is an absolute deadline without exception.

Important Dates and Deadlines
  • January 19, 2015: Martin Luther King Holiday. Last day to withdraw from all classes via UCDAccess and receive a refund of all tuition that has been paid.
  • January 20, 2015: First day of classes.
  • January 25, 2015: Last day to add or waitlist classes using UCDAccess without permission. After this date, a CLAS Instructor Permission to Enroll form is required for any late add requests.
  • January 26, 2015: Last day to drop without a $100 drop charge. No adds permitted on this day.
  • January 27 – February 4, 2015:
  • To add a course from January 27-February 4, it is the student's responsibility to get a CLAS Instructor Permission to Enroll form at http://www.ucdenver.edu/academics/colleges/CLAS/clas-advising/Pages/CLASAdvising.aspx, have it signed in person and submit it or route it electronically, according to the instructions on the form. Students and faculty will receive emails with directions for confirming registration via official university email and students must verify their schedules before census, February 4th, by 5p.m. After February 4th, students should meet with their CLAS advisor or Graduate School Coordinator to learn more about how to petition to add a course late.
  • February 4, 2015: Census date.
  • 2/4/115, 5 PM: Last day to add structured courses without a written petition for a late add. This is an absolute deadline and is treated as such. This does not apply to independent studies, internships, project hours, thesis hours, dissertation hours, and modular courses, though it is in the student’s best interest to have requested all late adds by this date.
  • 2/4/15, 5 PM: Last day to drop a Spring 2015 courseorcompletely withdraw from all courses using the UCD Access Portal and still receive a tuition refund, minus the drop fee(s). After this date, tuition is forfeited and a "W" will appear on the transcript. This includes section changes and is an absolute deadline.
  • 2/4/15, 5 PM: Last day to request Pass/Fail or No-Credit option for a course using a Schedule Adjustment Form.
  • 2/4/15, 5 PM: Last day for a graduate student to register for a Candidate for Degree and last day for a Ph.D. student to petition for a reduction in hours.
  • 2/4/15, 5 PM: Last day to apply for Spring 2015 graduation.
  • February 16-25, 2015: Early Alert open to faculty
  • March 23-29, 2015: Spring Break-(no classes; campus open).
  • April 6, 2015, 5 PM: Last day for non-CLAS students to drop or withdraw without a petition and special approval from the academic dean. After this date, a dean’s signature is required.
  • April 20, 2015, 5 PM: Last day for CLAS students to drop or withdraw with signatures from the faculty and dean but without a full petition. After this date, all schedule changes require a full petition.
  • May 11-16, 2015: Finals Week. No schedule changes will be granted once finals week has started--there are no exceptions to this policy. Commencement is May 16.
  • May 21, 2015, 5 PM: Due date for faculty submission of grades.
  • May 24, 2015: Spring final grades available on UCD Access (tentative).