Rigorous theoretical introduction to the foundations of analysis in one and several variables, including basic linear algebra, for students who already have substantial experience with formal mathematical proofs and a good knowledge of one-variable calculus. Covers basic set theory, vector spaces, metric and topological spaces, continuous and differential mappings between n-dimensional real vector spaces. Offered Fall Only. Normally followed by MAT218. (New in Fall 2014.)

# Course MAT216

## Accelerated Honors Analysis I

- sets and numbers, groups, rings and fields
- finite dimensional vector spaces, bases, homomorphisms, determinants
- normed vector spaces
- metric spaces
- topological spaces, open, closed, compact sets, Hausdorff spaces, complete spaces
- continuous mappings between topological spaces, homeomorphisms, limits
- differentiable mappings between n-dimensional spaces, mean value theorem, chain rule

Classes usually meet on Tuesday and Thursday for 90 minutes.

Problem sets count for approximately 30% 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. There is typically an in-class midterm (30%) and a final exam (40%).

The class usually includes popular, but optional, weekly problem sessions with undergraduate and graduate course assistants.

- MAT216/218 is an accelerated alternative to the standard honors sequence, MAT215/217, for students with a very strong interest and aptitude for mathematics and with substantial background in university-level proof-based mathematics.
- It is very easy to switch between MAT215 and MAT216 during the first few weeks of the semester, and the course instructors will help students decide between these two courses after the first problem set is returned.
- More detailed information about the content and level of MAT216/218 can be gleaned from the Course Notes, available from the Professor Gunning's departmental web page.
- The sample problems listed for MAT215 also give useful information about the level of MAT216.

- This course is intended for future math majors with
**substantial**background with formal proofs and a very solid knowledge of one variable calculus. - The standard recommended sequence for prospective math majors in the freshman year is the honors sequence MAT215/217. MAT216/218 is an accelerated version of the honors sequence.
- For students with the necessary background with proofs, the accelerated sequence MAT216/218 gives an integrated treatment of linear algebra and analysis in several variables, sufficient for any of our 300 level course offerings.
- MAT215/217 gives a very substantial introduction to formal mathematical argument and a thorough knowledge of analysis in a single variable in MAT215 and Linear Algebra in MAT217. Prospective majors should continue with MAT350, an introduction to manifolds (and analysis in several variables). (New in Fall 2014, replaces the old MAT218 (Analysis in Several Variables).
- Prospective physics majors or students with a strong interest in applied math may choose to take MAT203/204 instead of MAT216/218 or MAT215/217.
- Students who are choosing between math and computer science might consider MAT214 instead.

- A very strong aptitude for mathematics and deep mathematical curiosity is essential for MAT216/218, as for MAT215/217. However 216/218 assumes substantial previous experience in analysis and with formal proofs. Students without this prior experience with proof-based mathematics will likely prefer MAT215. Instructors for MAT215 and 216 will assist with placement adjustments during the drop/add period. Additional information will be available at the Academic Expo, part of freshman orientation in September.
- The sample problems listed for MAT215 can also help prospective majors choose between these two introductory sequences.
- A draft of the MAT216/218 Course Notes is available from the Professor Gunning's departmental web page.

**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 in your first semester.

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• 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. The undergraduate course assistants and the problem sessions can make a big difference in getting off to a good start.

• 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 MAT215 for a somewhat slower pace or to MAT203 or MAT201 for a more concrete approach to (multivariable) calculus.**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 for MAT215 should help. If you can solve the sample 215 problems, you should sign up for 216, and then consult with your 216 instructor or the junior advisor (usually one of the 215 instructors) early on. About half of the math majors start in this course, and about half start in MAT215. You can also consult the placement officer or go to information session at the Academic Expo during orientation, when you arrive in September.**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 MAT215, and be sure to go to the problem sessions for extra help if you need it.**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?**

• This course assumes you have experience with University level proof-based math courses. Check out the MAT215 info and discuss your situation with the placement officer at freshman registration or the Academic Expo.**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 MAT215) here at Princeton. This course will show you what it will mean to be a math major at Princeton. Credit for this course at another university will be granted only under very exceptional circumstances and we strongly prefer that you take this course (or MAT215) here.**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.