MATH 4111. INTRODUCTION TO ANALYSIS

News: You can download test 1 , test 2  and the FINAL TEST
with my comments/solutions.
           You are welcome
to contact me if you want me to elaborate on those.
          
Here is the graph of the function in problem 4, hw 10.
           Here is the proof of the
Weierstrass approximation theorem.


This course is a first part of what used to be an undegraduate analysis sequence 411-412 and has been modified to become 4111-4112. It presents the theory of one and several variable calculus followed by the introduction to Lebesgue theory.  This sequence is usually required for those wishing to complete the so-called "traditional" math major, a group normally limited to students contemplating going on to do graduate work in mathematics, physics, mathematical economics, or some other highly quantitative discipline in which theory plays a large role.  Students who are not traditional math majors are welcome to take the course provided they have an interest in understanding the theoretical basis for calculus. Those interested in building up their prowess with techniques for doing calculus problems and uninterested in understanding why the techniques work are STRONGLY ADVISED NOT to take 4111-12.  While it's true that occasionally techniques will be introduced in 4111-412 which aren't covered in other calculus courses, these will be few and far between; also, most of the lecture time will be devoted to the theory of why these methods work rather than illustrations of the calculations.  In brief, there is virtually nothing in common between the style of earlier math courses (nothing proven, hand-waving justifications, reliance on physical intuition, working out numerical exercises, developing computational skills) and the style of 4111-12 (nearly everything proven with a fairly high degree of rigor, a large part of the homework and exam questions theoretical with only a few numerical problems tossed in for good measure, emphasis on building up skills at proving theorems and developing mathematical intuition).
Many 4111-12 students may have gotten a taste of theory in Math 310; others may never have gone past the statements of unproved theorems in Calculus 1-3.  In any event, it will not be presumed that students have any prior experience in proving theorems or working out well-reasoned arguments responsive to theoretical homework questions.  To the contrary, one of the main goals of 4111-12 is to provide this kind of experience.

Location:   Cupples I 111, -T-T---, 10:00AM - 11:30AM

Instructor: Ilya Krishtal

             Office:  Cupples I, Room 202 (between the floors)
             Office Hours:  Tu,Th 11:40 - 12:40 or by appointment
             Office Tel:    935-6785 (has voice-mail)
             E-mail:        krishtal @ math.wustl.edu

Textbook:  Rosenlicht, Maxwell. Introduction to analysis.
           Dover Publications, Inc., New York, 1986. viii+254 pp. ISBN 0-486-65038-3

Topics: The real number system and the least upper bound property; metric spaces (completeness, compactness, and connectedness); continuous functions (in R^n; on compact spaces); pointwise and uniform convergence; Weierstrass approximation theorem; differentiation (mean value theorem; Taylor's theorem); the contraction mapping theorem; the inverse and implicit function theorems. Other topics will be tossed in if time permits. Not all of the topics are covered in the textbook. For those that are not other references or a handout will be provided.

Exams:  There will be two in-class mid-term exams, on Oct 4 and OCT 27. Each of these exams will ask for the proof of one or more theorems discussed in class. There will also be questions asking for definions, examples, and counter-examples for the material covered by the exam. Finally, there will be some problems not previously covered in class but often analagous to homework problems. The final exam will likely be a take-home exam given out in early December and due back in roughly a week.  We will settle on the exact dates when everyone has a firm schedule for exam week. The final exam will consist entirely of problems not covered in class.

Homework:  There will be weekly homework assignments to write up and hand in.  Usually the homework will be due on Tuesday and the assignment handed out the previous Tuesday.  The homework will consist of a mixture of selected exercises from the textbook and supplementary exercises made up by the instructor. Click on current homework to get it.

Grading:   Each of the two mid-term exams will count 20% toward the final grade, the final exam will count 30%, and the final homework average the last 30%.

Academic Integrity: As with all Washington University courses, cheating on exams will be taken very seriously with evidence supporting a cheating allegation forwarded to the Arts and Sciences Integrity Committee for adjudication.  When the Committee concludes that a student cheated on an exam, it normally directs the instructor to assign the student a failing grade for the course.
      Cheating on homework consists of either blindly copying off someone else's solutions or not acknowledging the receipt of assistance from others in completing the assignment.  It's anticipated that students will make a genuine effort to solve the homework problems themselves.
However, if the effort has lead nowhere, discussing problems with others is a way to avoid frustration and gain useful insight. All students are expected to write up their own assignments and to indicate in a short note at the top of the first page the names of any people (other than the instructor) with whom they discussed the problems or from whom they received some hints.  Violation of these requests will result in an instructor-imposed penalty (e.g., something like half credit for the assignment) but won't be treated as a "hanging" offense--in particular, won't be brought to the attention of the Arts and Sciences Integrity Committee.