Mathematics 481-2, 681-2
Real Analysis 
(3 credit hours)

This page: [ Prerequisites | Comments |  Catalog Description | Grade | Presentations | Objectives | Text ]
Teacher:
 Dr. Caldwell, office 429 Holt Humanities, office phone 7336.  Department office phone 7360.  E-mail: caldwell@utm.edu.   E-mail is the standard method of letting me know if you are going to miss class, and for us to communicate if there is a campus closure (I will use your ut.utm.edu e-mail address).  If that fails, phone the office; and if that fails, find me on FaceBook by searching for "Caldwell, PrimeMogul".)
Comments:
 This course and the course text are intentionally difficult. Mathematics 461-2, 471-2 and 481-2 are the capstone courses for our undergraduate majors and are important transition courses for those students continuing on to graduate school.  Plan to work hard, and in return, to mature mathematically.

Suggestions:  (1) Do the homework!  (2) Study with a friend (make a new friend if necessary).  (3) Use other real analysis books as references.  (4) Stay ahead of the class in the text (we will move through it sequentially, skipping some of the optional sections but we will not "hop around").  (5) Come by my office (make appointments as necessary).  I want you to succeed and will gladly help you.  (6) For the first chapter your discrete text should be very helpful.

To help you learn (and to motivate you to keep up) we will try to do a great deal of board work (student proofs in class on the board, often in teams of two).  These will be done in a supportive way, using the class as the audience for your proof. If you had me in 241/314, then you will remember my slogans: "learn to learn," and "write to be read."  They will be emphasized here as well.
Prerequisites:
 Discrete Mathematics (Math 314 = 241) and Multivariate Calculus (Math 320)  (The need for a firm basis in the first of these courses will be obvious the very first day.  The second is basically a maturity requirement: if you can not pass Math 320, you can not as this course.)
Catalog Description:
 Sets and countability. The real number systems. Sequences, limits, infinite series, metric spaces, continuous functions, uniform continuity, and convergence.  Riemann and Lebesgue integration. Students are required to submit written work and make an oral presentation.
Grade:
 The course grade will be a weighted average of the homework (17%), tests (53%), final (17%), board-work (13%).  For 681-682 students, the above will be 71% of the grade and a research project (paper and presentation) the other 29%.

Bonus points: (should these exams be offered on campus)
    2%  for each correct solution on the Virginia Tech. Competition
    1%  for attempting each 3 hour half of the Putnam Competition
    5%  for each apparently correct solution on the Putnam Competition
Homework:
 Homework will be assigned and collected daily at the beginning of the following class period (unless otherwise announced).  No late homework will be accepted.  Much of the homework will be from the text, but we will augment the text wherever necessary.  If you will miss a day, ask ahead of time what the homework will be.
Presentations:
 This semester I will emphasize constant board work, so I will not also require formal presentations of those taking 481-482.  (See note above for those taking 681-682.)
Departmental
Objectives:
 The student will: 
  1. Define the real numbers, least upper bounds, and the triangle inequality.
  2. Define functions between sets; equivalent sets; finite, countable and uncountable sets. Recognize convergent, divergent, bounded, Cauchy and monotone sequences. 
  3. Calculate the limit superior, limit inferior, and the limit of a sequence. 
  4. Recognize alternating, convergent, conditionally and absolutely convergent series. 
  5. Apply the ratio, root, limit and limit comparison tests. 
  6. Define metric and metric space. 
  7. Determine if subsets of a metric space are open, closed, connected, bounded, totally bounded and/or compact. 
  8. Determine if a function on a metric space is discontinuous, continuous, or uniformly continuous. 
Text:
 Principle of Real Analysis ["baby Rudin"] 3ed McGraw Hill.  ISBN: 978-0070542358 (0-07-054235-X) hardback or 978-0070856134 (0070856133) paperback.