Syllabus for Real
Analysis
Dr. Wildenberg
The following topics
will each take about 3 weeks to investigate.
Texts
1.
“Analysis: with an
Introduction to prooof” by Steven Lay
The
outline below follows Lay’s chapters; sections refer to Lay.
2.
“Exploratory Examples for
Real Analysis” by Snow and Weller
I’ll
use S&W to refer to this.
1.
Logic and Proof (section 1-4)
This
chapter includes the logical connectives with an emphasis on how they are used
in mathematical prooof as well as the universal and existential quantifierr. Negation of complex statements, the role of the contrapositive, are
among the topics.
2.
Sets and Functions (sections
5-8, but NOT section 9)
A
more advanced treatment of this topic than you have previously seen. Included is a discussion of cardinality and
power sets.
3.
The Real Numbers (sections
10-15, with section 15 possible) We’ll have a look at
the properties of the “Real Numbers” that distinguish them from other
mathematical objects. This will include
the properties of bounded sets. And
we’ll investigate in some detail the “topological” properties of the
Reals. Lastly we’ll see that much of
what we know about the Reals is true of other sets as well, in particular
“metric spaces”. Chapters 1 and 2 from S&W will be integrated.
4.
Sequences
(sections 16-19)
together with chapters 3-8 from S&W. Sequences form a powerful tool for
investigating and describing function behavior.
With lots of examples, we can gain an understanding of this.
The
following is a possibility. However, it
is likely that time will not permit doing this chapter.
5.
Limits and Continuity
(sections 20-24) At
long last we arrive at functions, the discussion of which is the goal of the
course. We will have to see how much of
this chapter we can complete. Chapters
9-11 from S&W will be used.
Grading is based on the following:
· Class participation
Attendance counts as does answering and asking questions, showing leadership and/or participation in group or individual class activities. More than two absences is grounds for failure. Emergency situations such as health or family will be handled on an ad hoc basis.
· Homework
Homework serves a dual purpose. Primarily homework is a learning experience. (For me learning is more important than assessment.) Secondarily homework enables me to determine who is working seriously outside of class and how well you are understanding the ideas of this class. Finally homework gives you a chance to present proofs and thus to learn how mathematics is presented. We will accept revised homework for “check credit”. By that I mean that if homework is handed in not fully correct, I will try to give you some direction and an opportunity to have the revised homework checked off . (A corrected homework effectively increases the grade you got on that assignment.) Sometimes I will select only 1 or 2 of the problems for grading. Sometimes homework will be assigned in groups of 2 or 3 or 4 persons. The reason for this is so that you can learn from one another while doing the homeworks. It is essential that homework not merely be copied. After discussing a problem in a group, you should independently write up your results. Each student in a group must hand in their own writeup but should list the names of everyone in the group. Unless specified as a group assignment, I expect your homework to represent independent work.
Please note that I take the admonitions about doing your own writeup very seriously. Group work is an opportunity for discussion and learning – it must not be regarded as an opportunity for copying.
All homework should have your name on each page and be neatly stapled together. You should state each problem. It’s ok to be a bit terser than the text in stating the problems but it should be clear to someone who does not have the text what problem you are answering.
Note that many homework sets will include some historical questions and/or assignments. These may require some research either on the Web or in the library. There will also be a short historical paper required. This should be handed in by the 10th week of the semester. I will give out a more detailed description of this.
· Two in class examinations and a final
These will provide you with an opportunity to consolidate your thinking on the topics and to show your individual work. ALL examinations are “comprehensive with an emphasis on the recent material”. This means that you do need to have learned the material well enough to use it later in the semester if necessary. We will choose the dates the first day of class. For Spring 07, we have chosen the dates Feb 22 and April 3 for exams.
· For many years, students have asked me how I weight the different criteria. There is no answer to that question because I evaluate each student as best I can. I do not just average homeworks and exams. I expect to know each student’s work rather well by the end of the semester and to grade them in accordance with the guidelines below. Please note that passing the midterm and final are necessary to pass the course.
Assessment philosophy
Based on the above inputs, I will give grades according to the following criteria. For each of the following grades a student receiving that grade has demonstrated the level of skills described.
· A – Can solve most homework problems clearly and completely. Can present proofs in good mathematical expository style. Has a good understanding and knowledge of the theorems and definitions.
· B – Can solve most problems. Usually has right idea about proofs but presentation is not always precise. Knows the main theorems and definitions but perhaps not with deep understanding.
· C – Can solve routine problems but seldom gets a proof completely right. Is often unclear about what a theorem or definition means.
· D, F Has little understanding of the topics of the course. Has failed to understand basic definitions and theorems. May have very poor work habits such as failing to hand in homework on time and carefully done.
Advice for learning
mathematics
· I believe that many students have received bad advice on how to make the best possible use of class time. In my opinion (and I admit that this is only an opinion), you should rarely take notes! The reason I say this is that taking notes and thinking about what is being said is virtually impossible. When you are in class, you have an opportunity to get explanations about whatever you find difficult to understand. If you are not thinking about what is being said and instead are just trying to write it down this opportunity to question is largely wasted. In addition, almost everything that is said in class by the professor is in the text. Sometimes it is better said in the text, sometimes not quite as well – usually however, it is all there already written down for you to read and reread.
· One thing you can do to prepare for class is to try to give the next section a cursory reading. Reading mathematics textbooks is not easy and for most people not fun. But by looking over the section before class, you will familiarize yourself with some of the new vocabulary and new ideas that are being discussed – this will make class time much less intimidating.
In
Class Behavior
· Shut off cell phones, pagers, etc.
· Stay “in the class”. I.e. participate in group work, don’t do homework for other classes (or this one), ask questions, look awake, etc.
Contact information
Office: W-113. Office hours are included in my schedule on my web page.
Email: wildenbe@sjfc.edu
Office phone: 585 385 8179 (8179 from campus)
Home phone: 585 461 3495 (I am giving you this number so that if talking to me about a problem can save you a lot of grief you will be able to reach me. However, please respect my privacy by not overusing this number.)
Disabilities
Students with disabilities may be interested in
the college’s policy:
In compliance
with