Math 301, Real Analysis, Spring 2006

Harriet Pollatsek, 400 Clapp, X 2341, hpollats@mtholyoke.edu

This information and weekly assignments also appear on the course web page

(see www.mtholyoke.edu/courses/).

 

Text:  Introduction to Analysis by Arthur Mattuck, Prentice Hall, 1999.

 

Homework will be due on Monday and (normally) returned on Wednesday.  Rewrites are due the following Monday (attach the original).  The higher of the two “grades” will be recorded.  There will be 10 homework assignments, including some exercises to be read by me (HP) and some by a student assistant (SA).   Each assignment will be graded “check plus”, “check” or “check minus”.  Papers graded “check minus” should definitely be rewritten.  Graded homework counts for up to 220 points total out of 1000 (HP 150, SA 70).  Working in groups is strongly encouraged, but you should write up the homework for me on your own, so I can give you feedback on your writing.  Some exercises will be suggested but not assigned to hand in; these are valuable practice and helpful to your learning.  They can be used for oral presentations (see below).  In addition, there will be almost weekly “challenge problems,” read by me.  These are optional, but some success with challenge problems is expected for course grades of A or A-.  You have two weeks to work on each challenge problem; they too can be rewritten, and rewrites are due the next week, same rules as for homework for me.

 

Note that the text includes three levels of tasks: “questions” at the end of each section (for which answers are provided at the end of the chapter), “exercises” keyed to sections (from which homework assignments are usually chosen), and “problems” (which are more challenging).  You should answer all the questions when you do the reading.

 

Exams:  There will be two take-home midterm exams worth 170 points each.  The first, on chapters 1-6, will be handed out 2/27 and due 3/6; the second, on chapters 7-15, will be handed out 4/10 and due 4/17.  Note: The Science Symposium is 4/10, so the 4/10 class (exam review) will be scheduled earlier so we can go to the Symposium---TBA.  Rewrites of the midterms are possible, and the two scores will be averaged.  The final, worth 220 points, will be cumulative.  It will be a take-home exam due at the end of the exam period (earlier for seniors).  No rewrites of the final will be possible (obviously).

 

Quizzes on definitions and statements of theorems will be on Wednesdays.  You’ll always know exactly what material each quiz will be drawn from.  There will be eleven quizzes at 20 points each, and I’ll drop the lowest two, so the maximum quiz total will be 180 points.  Precise language is especially crucial in analysis, and grading of quizzes will be picky.

 

Oral presentations:  Everyone will be graded on two oral presentations, each of one exercise, one before spring break and one after.  The two presentations are worth a total of 16 points.  Consultation on presentations, with me and/or with other students, is encouraged.  Reserve presentation problems ahead of time with me.

 

Explorations on dynamical systems:  During the weeks when you are working on exams, we will use class time for three modules from Discovering Dynamical Systems by Johnson, Madden and Sahin.  You will use Maple for guided explorations, looking for patterns, and you will work in small groups to prove conjectures.  Active participation in the three explorations will be worth a total of 24 points.

 

 

Schedule (Do reading before class -- except 1/30; Fridays will be regular classes.)

2/1: ch 1.1-1.4; 2/3: app A.0, A.1; 2/3: ch 1.5-1.5, App A.3, A.4 p. 411

2/6:  ch 2 and app A, A.2;  2/8: ch 3

2/13: ch 4; 2/15: ch 5

2/20 ch 6.1-6.3; 2/22: ch 6.4-6.5; 2/24: review

2/27 & 3/1 & 3/3: Module 1, Introduction to dynamical systems (no reading)

3/6: ch 7; 3/8: ch 8.1

3/13: (ch 9), ch10; 3/15: ch 11

3/27: ch 12; 3/29: ch 13

4/3: ch 14; 4/5: ch 15; TBA: review

4/10 & 4/12 & 4/14: Module 2, Classifying fixed points (no reading)

4/17: (ch 16), 17; 4/19: ch 18

4/24 ch 19; 4/26: ch 20

5/1 & 4/3 & 5/5: Module 3, Introduction to symbolic dynamical systems

5/8: metric spaces, review

 

Foundations:  We will assume that we know everything about the ordinary algebra of real numbers, including the order properties.  We will place special emphasis on the completeness property of the real numbers.  These properties taken together (algebraic, order, completeness) completely characterize the real numbers as a unique system and provide the working basis we will need in order to create an analytic superstructure of grand proportions.  A warning:  don’t be too impatient to arrive at novel theorems.  We first need to go over old ground and strengthen our grasp of fundamental ideas.  (Sometimes we’ll also assume facts from calculus to illustrate the ideas, but eventually our point of view will shift, and we will carefully prove theorems from first year calculus.  These shifting ground rules will be made explicit.) 

 

Apology:  This term is used in the same sense as the title A Mathematician’s Apology by G. H. Hardy, the greatest English mathematician of the first half of the twentieth century.  The passage below, from his book Pure Mathematics,  refers to the “limit of the sum is the sum of the limits” theorem for convergent sequences.

… the argument may possibly appear to the reader to be merely a piece of useless pedantry, or an attempt to manufacture difficulties out of what is really obvious.  We do not assert that such an opinion is, in this case, entirely groundless.  The result really is very obvious: nor would any mathematician think it worthwhile as a rule to state arguments for what is so obvious at such length.

 

But the reader must remember that the theorem, obvious though it may be, is one of the most fundamental and important theorems in all mathematics.  It is one which every mathematician uses, consciously or unconsciously, twenty times a day.  The proof of such a theorem must be made absolutely clear, explicit, and rigorous: no room must be left for any possible apprehension or confusion.  And this is not all.  The great majority of theorems concerning limits are, as the reader will discover before long, far from being so simple and so obvious as this one.  In this case the result obviously indicated by common sense was true.  In more difficult cases common sense as often indicates an untrue result as a true one:  sometimes it fails to give any indication at all.  In such cases, vague general arguments are worse than useless:  they lead to mistakes not only gross in themselves but entirely confusing in their consequences.  And unless the reader is prepared to take the trouble to try and understand the way in which rigorous methods apply to simple and obvious cases, where their application is easy, [s]he will find when [s]he comes to difficult questions which cannot be settled without them, [s]he has not the capacity to use them.