Math 5070 – Applied Analysis, Spring 2006

Jan Mandel

Department of Mathematics
University of Colorado at Denver

This web page http://www-math.cudenver.edu/~jmandel/classes/5070s06 serves as the class syllabus and source of further information. Homeworks, material covered, and handouts are in the homeworks file. Photographs and scans are in the photos directory.


Final: Tuesday, May 9


Prerequisites: MATH 4320

Textbook: Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill, 3rd edition, 1976
Time and location:  Tuesday and Thursday 4-5:15 WC 144
Office hours: UCD building 640, Tuesday and Thursday 2-3:30. I am also available at other times, ask.
Contact: Phone 303-556-4475; email jmandel@math.cudenver.edu. I am usually pretty good at answering email even at late hours.


 

The objectives of the class are to learn how to

  • Formulate and prove statements from Analysis
  • Tell if a statement is true, false, or meaningless because of incorrect formulation
  • Solve problems
  • Write mathematically correctly and clearly so that the results can be verified

This is the minimum of Analysis needed for applications and to prepare (in part) for the PhD Preliminary Examination in Applied Analysis.

 

Material to be covered

Selected sections from the following, depending on the progress of the class:

  • Basic logic and notation (handout)
  • Real and Complex number system, infinum, supremum (Rudin 1.1-1.20, 1.23-1.38)
  • Real Line and Metric Space Topology: Rudin 2.1-2.42, 2.45-2.47
  • Numerical Sequences and Series: Rudin 3.1-3.55, Buck 5.2
  • Continuous Functions: Rudin 4.1-4.19, 4.25-4.33
  • Differentiation: Rudin 5.1-5.19
  • Riemann Integration: Rudin 6.1-6.9, 6.12-6.18, 6.20-6.27
  • Sequences and Series of Functions: Rudin 7.1-7.26
  • Power Series: Rudin 8.1-8.5
  • Functions of Several Variables: Rudin 9.1-9.29, 9.39-9.42

Further required material is in the following recommended exercises. You may need to consult other books to solve some of the exercises.

  • Rudin Ch. 1 exercises 1-5, 8-19
  • Rudin Ch. 2 exercises 1-16, 19-27, 29
  • Rudin Ch. 3 exercises 1-18, 20, 21, 23-25
  • Rudin Ch. 4 exercises 1-18, 20-25
  • Rudin Ch. 5 exercises 1-7, 9-14, 19, 20, 22-24
  • Rudin Ch. 6 exercises 1-5
  • Rudin Ch. 7 exercises 1-13,15-20,24
  • Rudin Ch. 8 exercises 1-5,7-9
  • Rudin Ch. 9 exercises 1-8,9 (for convex set), 11-16,20,27,30,31

The class will cover about 50-70% of the above material. Students who plan to take the prelim will need to study the rest on their own.

 

Class approach

This class is unusual. It does not teach "material" as much as the way of careful and brutally honest thinking - no fog can be tolerated. Do not trust anything until you have verified it - this includes what I say and what is in the book. This will take great time and effort. Please budget your time and energies accordingly.

Sometimes students say "What was the class for? I had to learn everything by myself."  But this is exactly how it should be. No amount of  lecturing and explanations in the class can replace the process of understanding and internalizing the knowlege that you must go through by yourself. So, the class time will focus on an overview of the material and some discussion and problem solving, but you are expected to study in detail the complete sections in the book that are covered and solve or at least attempt seriously all recommended exercises. The good thing about exercises is that only the first few from any topic are hard; then, once the first exercises make you go back to the text many times and you get the general idea, the rest are easy. I strongly recommend that you study in groups, work on the exercises in groups, and bring to class or office hours the exercises where you need help - but work on them seriously first. The homeworks can cover only a very small part of the exercises; again, you should solve all the exercises yourself even if you do not turn them in.

The progress of the class will be recorded and all assigments available in the class notes file.

 

Assignments and tests

There will be homeworks given most Tuesdays and due always the following Tuesday, mostly consisting of exercises from the textbook. Homeworks will be considered on time if they are turned in on the due date at the beginning of the class or left at my office before the class. Late homeworks will not be accepted except when arranged in advance, and then at 50% if the rest of the homeworks was already graded.  Please do not send homeworks by email. There will be a midterm and final exams in class. You cannot use any books, computers, or notes at the exams, except for one 3" by 5" card where you can write anything you want. The midterm will be on March 16. The final will be during the finals week as determined by the Auraria finals schedule. The final will be from material since the midterm. Because the grading system allows for skipping one of the exams or all homeworks and still getting the full score, there will be no make-up exams or work for extra credit.
 

Office hours

 

I enjoy discussing technical topics related to the class with my students during my office hours or any other time you walk in. Please do your homeworks ahead of time, and do not come to my office on the day the homework is due and expect me to do it for you.   Please do not try to negotiate grades (except in the case of clear error on my part), and especially not partial credit.  I usually need few minutes before the class to get organized, so please respect that and do not show up right before the class.

 

Grading

In all homeworks and exams, please show clearly all your work, but do not write irrelevant things just to fill more paper - this will not help, just the opposite because more writing means more opportunity for errors. Do not write incorrect statements. Points will be taken off for incorrect statements even if they are not relevant to the problem being solved. It is not enough that you understand the mathematics correctly – so that we can understand your solution, you must also follow the rules for mathematical writing. The point of most problems is the justification rather than a formal solution, so formal solutions count for little even if correct.

All work will be graded on the scale 0-4: 0=nothing relevant, 1=understands definitions and the problem, or a correct formal solution 2=good progress or a correct partial solution, 3=some minor glitch, 4=OK

The final score will be determined by the sum of the best 2 out out of the following, each scaled separately from 0 to 50: 1. homeworks 2. midterm 3. final. The letter grades will be A >= 90,  A- >= 85, B+ >= 80, B >= 75, B- >=70,  C+ >= 65, C >= 60, C- >= 55, D+ >= 50, D >= 45, D- >= 40.

 

Class Policies

 

Students are encouraged to study in groups, so certain similarity in the solutions (and errors) in homeworks is OK. Solutions clearly taken as a whole from other sources, such as another book or internet, in an obviously different setting than as done in the course, with mature elegant formulation, etc., will receive zero credit unless student can explain the solution well during office hours. Students found plagiarizing solution during in-class exams will receive zero credit for the exam and will be reported to the department and the college for disciplinary action.

 

Please do not eat during the class. Drinks are OK. All cell phones must be turned off before the class. Cell phone calls are very disruptive. If your cell phone rings, you will be asked to take the call outside and you cannot return to the class.

 

If you need any accommodation because of religion or disability, please let me know.

 

 

Incomplete grades

 

            Incomplete grades (IW or IF) are NOT granted for low academic performance.  To be eligible for an incomplete grade, students MUST (1) successfully complete a minimum of 75% of the course,   (2) have special circumstances beyond their control that preclude them from attending class and      completing graded assignments, and (3) make arrangements to complete missing assignments with the original instructor via completion of a CLAS Course Completion Agreement.  Verification of special circumstances is required.   The CLAS Course Completion Agreement is available from the CLAS Advising Office, NC 2024 or from the Department of Mathematics.

 

 

 

Spring 2006 Registration and Academic Deadlines

 

·        CLAS students must always have an accurate mailing and e-mail address:  http:/www.cudenver.edu/registrar

·        Students are responsible for completing financial arrangements with financial aid, family, scholarships, etc.

·        January 12, 2006 (5:00 pm)   Payment plan deadline for students registering by December 16, 2005.   Students who have not applied for financial aid are administratively dis-enrolled for non-payment on January 13, 2006. 

·        January 19, 2005 (midnight) Last day to be added to the wait-list for a closed course.

·        January 17 – January 27, 2006   Students are responsible for verifying an accurate Spring 2006 registration via SMART.

·        January 26, 2006 (midnight)   Last day to add courses via the web SMART system.

·        February 1, 2006 (5:00 pm)   Last day to drop a Spring 2006 course for tuition refund and no transcript notation.

·        February 1, 2006 (5:00 pm)   Last day for undergraduates and graduates to apply for May 2006 graduation.

·        February 10, 2006 (5:00 pm)   Last day for CLAS students to add a Spring 2006 course.  Treated as an absolute deadline.

·        April 3, 2006 (5:00 pm)   Last day for students to drop a Spring 2006 course without college approval.

·        April 14, 2006 (5:00 pm)   Last day for CLAS students to drop a Spring 2006 course.  Treated as an absolute deadline.

·        May 1, 2006 (5:00 pm)   Last day to withdraw (drop all courses) without a written petition.

 

Last updated 1/17/06