Course MAT215 in Fall 2011

Analysis in a Single Variable

Strongly recommended for future math majors, this course covers calculus more thoroughly and more theoretically, giving an introduction to important mathematical techniques and results that give a foundation for further work in analysis. It serves as an introduction to the rigorous proofs and formal mathematical arguments needed in all upper division math courses.

Equal emphasis is given to learning new mathematics and to learning how to construct and write down a correct mathematical argument by dividing the question into logical steps where each step is explained and justified carefully, giving references if necessary. For most students this will be a completely new and very challenging way of doing mathematics, very far removed from the process of memorizing algorithms and working through concrete calculations typical of high school math.

Textbook(s)
Topics

The course covers most of the first seven chapters of the textbook, Principles of Mathematical Analysis by Walter Rudin (3rd edition).

Topics include the rigorous ε-δ treatment of limits, convergence, and uniform convergence of sequences and series, continuity, uniform continuity, and differentiability of functions, the Heine-Borel Theorem, the Rieman integral, conditions for integrability of a function and term-by-term differentiation and integration of series of functions, Taylor's Theorem.

Description of Classes

Classes are usually taught in a two sections in the fall semester and a single section in the spring semester.  The two fall sections are closely coordinated, with the same problem sets and exams.

Problem sets count for approximately 15% of the course grade.  Collaboration is allowed on these, but each student must write up and submit his/her own solutions and take care to understand thoroughly any results that are obtained jointly.  Late homeworks are not accepted.  There will be an in-class midterm (35%) and an in-class final exam (50%) which count for the remainder of the course grade.

We expect that most of you will need to spend several hours outside of class each week reading the textbook and reviewing class notes as you puzzle over the homework problems.  After you finally understand the questions, and you start to get an idea how to solve the problems, then you go on to write up your solutions clearly, carefully justifying each step, thereby exposing the gaps and errors in your thinking, which you then correct, and finally, having beaten the problems into submission, you go take a quick nap before class!

Notes

Alternatively, prospective math majors can take MAT214 for a more algebraic introduction to rigorous proofs and formal mathematical arguments in the context of classical number theory.

This course is usually followed by MAT217 (linear algebra) and then MAT218 (analysis in several variables).

Who Takes This Course
  • Strongly recommended for incoming students who seriously consider majoring in math.
  • Students who are choosing between math and physics as a major should might consider MAT203 instead, but many physics majors take the MAT215/217/218 sequence, especially those who are interested in theoretical physics or mathematical physics.
  • Students who are choosing between math and computer science might consider MAT214 instead.
  • Students who wish to skip MAT215 and begin in MAT217 or higher should consult the placement officer (to be talked out of this plan).
Placement and Prerequisites:

A very strong aptitude for mathematics and real mathematical curiosity is essential. Do you want to

  • make your own conjectures and figure out for yourself whether a mathematical statement is true or false?
  • be able to construct a clear and convincing, even iron-clad, argument to justify a mathematical claim?
  • develop an appreciation for the intrinsic value and power of mathematical argument, separate from considerations of real-world applications or utility?

Typically students have a 5 on the BC calculus exam together with a math SAT score of at least 750. A very solid knowledge of one-variable calculus is assumed, and this course will build on that knowledge to give you a much deeper understanding of the concepts and theorems you first saw in high school.

Students who learned only AB calculus can also take this course provided they have sufficient mathematical aptitude along with a very serious interest in being a math major.   This is very rare, and students in this situation shouldconsult the placement officer.  Most students who learned only AB calculus should start in MAT104.   

Sample Material

A math major who took this course in the Fall of 2010 has prepared this list of Sample Problems designed to help you understand for yourself what MAT215 will be like.  If these questions seem intriguing to you, then take the course to find the solutions!

FAQ
  1. How hard should I expect to work in this course?
    •     Pretty hard.  Most math courses require a steady time commitment.  We expect that the weekly problem sets will try to take over your life, especially at the beginning.  The time you will need to invest can vary quite a lot depending on your background and goals.   It is quite difficult to judge how much time you will need to master the more abstract parts of the course, but for most future math majors, this will be the most demanding (and rewarding) course you take this semester.
    • For many students this course requires a very big adjustment both in effort expended and in your definition of success.  Be prepared to invest quite a lot of time early on, learning how to think about proofs and counterexamples and adapting old techniques to new situations, rather than chugging through computational problems just like the textbook and lecture examples.  
    • This is a course for people who are more afraid of being bored than they are of being lost sometimes.  If you quickly find yourself longing for the good old days back in high school when you always knew exactly what you were doing, then you should think about switching to MAT203 or MAT201 for a more concrete approach to calculus.
  2. I already have a good background with mathematical proofs -- do I need to take this course?
    • It is very difficult to judge, but working through the sample problems listed above should help.  This is the right course for most future math majors, even those who have had some prior experience with proof-based calculus courses, but each year a few students with an extremely strong mathematical background do skip this course and start in MAT217 or higher.   If you can solve the sample problems, then consult the placement officer at freshman registration or at the "Academic Expo" during orientation for permission to start in a more advanced course.
  3. I have never had a course with rigorous proofs -- will this course be too hard for me?
    • If you have a serious interest in being a math major you should sign up for this course and give it a try.  The first few classes will tell you whether you find this way of doing mathematics appealing or not.  Many students who excelled at math in high school (and loved it) discover in the first few weeks of MAT215 that they do not actually enjoy thinking about math in this more abstract way.  No problem!  Consider switching to MAT203 or MAT201 instead where you can continue your serious calculus studies with a more concrete approach.
    • Students who have had previous experience constructing mathematical proofs will have an advantage at the beginning, and some of these students can be very intimidating to the other students in the class.  Nonetheless, the course is designed to be accessible to students who are seeing proofs for the first time.   If after a few classes you still find this approach to calculus appealing, you should persevere even if you do find the problem sets to be (overly) challenging!  Go to your instructor's office hours and take advantage of the help available at the McGraw Study Halls, where enthusiastic math majors will help you learn to think like a mathematician. 
    • This course is graded on a generous curve to encourage interested students who want to give this kind of thinking a serious try without undue academic risk.   Consult your instructor for advice after the first few weeks if you are worried about the decision.
  4. I took only AB calculus but I really want to be a math major!  Am I qualified for this class?  Should I take this course?
    • You should know enough calculus if you got a 5 on the AB exam and you have a good math SAT score (at least 750).   It is more a question of how much you like this kind of thinking and how quickly you can learn to construct rigorous proofs.   We recommend that you sign up for both MAT104 and for MAT215 and use the drop/add period to make a decision.    Consultation with the placement officer at freshman registration is strongly recommended.
  5. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
    • Probably not.  If you want to be a math major you should take this course (or MAT214) here at Princeton.  This course (or MAT214)  will show you what it will mean to be a math major at Princeton.  The first rigorous proof course, followed by MAT217 and MAT218, sets the foundation for all the more advanced courses for math majors and you really need to be sure that this foundation is as solid as possible.  Credit for this course at another university will be granted only under very exceptional circumstances and we strongly prefer that you take this course here at Princeton.
  6. My question is not listed above. Where can I find an answer?
    •Try the undergraduate home page.   You will find links there to more information for future math majors and contact information for the various people who can advise you.  Representatives from the math department will be available at the academic expo during orientation and at freshman registration.