# MATH 365C: REAL ANALYSIS 1

### General Information

```      Instructor: Dave Rusin (rusin@math.utexas.edu)
Office hrs: I will be in my office (RLM 9.140) at these times for you:
Tuesdays 11-2 pm
Wednesdays noon-2:30 pm
Thursdays 11-2 pm
Other times available by appointment.

Text: Principles of Mathematical Analysis, Walter E Rudin.
We will cover the first six chapters and selected topics from
the remainder, as time allows.

Class meets Tuesdays and Thursdays in PMA/RLM 5.122, 2-3:30 pm

YOUR FINAL EXAM. The Registrar's Office has accurate information
about the time and place of all final exams.

```

Course webpage: http://www.ma.utexas.edu/~rusin/365C-19b/

BONUS MATERIAL:

After discussions with the class, I have set the date of the second exam to be THURSDAY, NOV 7. It will cover the topics we have discussed since the previous mid-term (continuity and differentiablity).

Here is an answer key for exam 2.

I promised that I would share with you something I wrote up to amuse/annoy my Calculus students. It is an example of what Rudin calls "Summation By Parts", which I illustrate with a specific example: how can one decide whether or not the series Sum( sin(n)/n ) converges? Here is the handout.

Here is the test itselfi. It was too long. I'm sorry about that; I will arrange it so that grades will be better than some of you fear. Here is a Test 1 answer key

### Catalogue Description

Course description: This course is an introduction to Analysis. Analysis, together with Algebra and Topology, form the central core of modern mathematics. Beginning with the notion of limit from calculus and continuing with ideas about convergence and the concept of function that arose with the description of heat flow using Fourier series, analysis is primarily concerned with infinite processes, the study of spaces and their geometry where these processes act and the application of differential and integral to problems that arise in geometry, PDE, physics and probability.

A rigorous treatment of the real number system, Euclidean spaces, metric spaces, continuity of functions in metric spaces, differentiation and Riemann integration of real-valued functions of one real variable, and uniform convergence of sequences and series of functions.

### Pre-requisites

Prerequisite and degree relevance: Either consent of the Undergraduate Mathematics Faculty Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who receive a grade of C- in one of the prerequisite courses are advised to take Mathematics 361K before attempting 365C. Students planning to take Mathematics 365C and 373K concurrently should consult a mathematics adviser.

Homeworks: These will be assigned approximately weekly. The due dates will be marked. Links will appear here when they are available:

There will also be two mid-terms and a final exam.

Each component will be converted to a letter grade according to the following scale:
 97.0-100 A+ 93.0-96.9 A 90.0-92.9 A- 87.0-89.9 B+ 83.0-86.9 B 80.0-82.9 B- 77.0-79.9 C+ 73.0-76.9 C 70.0-72.9 C- 67.0-69.9 D+ 63.0-66.9 D 60.0-62.9 D- 0-59.9 F

I will also use an alternative method and then whichever method gives you the better letter grade will be the one I record for you for that exam. Here's the alternative method: After I compute the mean and the standard deviation of the class grades, I will determine how many standard deviations above or below the mean your grade is. If your score is greater than the mean by less than one standard deviation, you will get a B (or B+ or B-, as appropriate); higher scores get A's, lower scores get C's, D's, and F's.

After each component is converted to a letter grade, I will compute your semester grade as a weighted average of these: The homeworks count 20%, the exams 25% each, and the final 30%

Textbooks, notes, and electronic devices (including phones and calculators) are not permitted during exams.

### Policies

Classroom activity: Our meeting times together are very short so we must make the most of them. Come to class daily and ask questions. Please silence your cell phones.

Make-ups: it is in general not possible to make up missing exams or homework assignments after the due date. If you believe you will have to miss a graded event, please notify me in advance; I will try to arrange for you to complete the work early.

Students with disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.

Religious holidays: If you are unable to participate in a required class activity (such as an exam) because it conflicts with your religious traditions, please notify me IN ADVANCE and I will make accommodations for you. Typically I will ask you to complete the required work before the religious observance begins.

Academic Integrity. Please read the message about Academic Integrity from the Dean of Students Office. I very much prefer to treat you as professionals whose honesty is beyond question; but if my trust is violated I will follow the procedures available to me to see that dishonesty is exposed and punished.

Campus safety: Please familiarize yourself with the Emergency Preparedness instructions provided by the university's Campus Safety and Security office. In the event of severe weather or a security threat, we will immediately suspend class and follow the instructions given. You may wish to sign up with the campus alert programs.

Counseling: Students often encounter non-academic difficulties during the semester, including stresses from family, health issues, and lifestyle choices. I am not trained to help you with these but do encourage you to take advantage of the Counselling and Mental Health Center, Student Services Bldg (SSB), 5th Floor, open M-F 8am-5pm. (512 471 3515, or www.cmhc.utexas.edu

Add dates: If you enroll within the first four class days of the semester, and have missed any graded material, I will adjust the weighting of your graded sections accordingly so that you are not penalized. No such accommodation is made for students who enroll on the 5th day or later. (Such students must enroll through the MPAA advising center in RLM, and ordinarily I do not admit students who ask to enroll then if they have missed any graded activities).

Drop dates: Sep 3 is the last day to drop without approval of the department chair; Sep 13 is the last day to drop the course for a possible refund; Oct 31 is the last day an undergraduate student may, with the dean's approval, withdraw from the University or drop a class except for urgent and substantiated, nonacademic reasons. For more information about deadlines for adding and dropping the course under different circumstances, please consult the Registrar's web page, http://registrar.utexas.edu/calendars/19-20/