The introductory courses for math majors are MAT 215: Single Variable Analysis, MAT 217: Linear Algebra, and MAT 218: Multivariable Analysis. Like the great majority of math courses at Princeton, these three courses are theoretical and proof-based. The intro courses, especially MAT 215, emphasize this mindset and are geared towards teaching students to read and write proofs.

These three are usually the first math classes that math majors take at Princeton. However, the math department is very flexible in allowing advanced freshmen to skip some or all of these courses. Students who do skip any of these three should make sure they are very comfortable with the corresponding material. Another note is that rarely, people opt to take MAT 214: Numbers, Equations, and Proofs instead of MAT 215. The former is a course which also teaches proofs and rigorous thinking, but in the context of classical number theory.

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#### Brief Course Summaries

The goal of **MAT 215** is to build the theory of analysis from the ground up, teaching students to think rigorously along the way. The course starts by addressing the question: what are real numbers? It then introduces its students to important topological preliminaries such as open and closed sets, compactness, and completeness. The remainder of the course is spent on developing the theory of limits, differentiation, integration, sequences, and series. See below for a first-hand description of MAT 215.

**MAT 217** is a course in linear algebra, a subject at the foundation of almost all branches of pure and applied math. The most basic mathematical object this course deals with the vector spaces, a structure whose elements can be added and multiplied by scalars. One example of this is the set of n-tuples of real numbers. The majority of the course is spent studying linear transformations between vector spaces and their close relatives, matrices.

**MAT 218** is in a sense a continuation of MAT 215: it generalizes the concepts of limits, differentiation, and integration from one to multiple dimensions. Though some of the material at the beginning of MAT 218 might look familiar, fairly soon analysis in several variables takes on a flavor of its own. In particular, linear algebra turns out to play a significant role, especially the space Rn and the determinant. The course briefly touches on the subject of manifolds, i.e., smooth surfaces, which are important in fields such as topology, differential geometry, and Lie theory. MAT 218 concludes with a surprisingly elegant generalization of the fundamental theorem of calculus called Stokes’ Theorem.

#### Recent Changes

The math department has recently seen its enrollment rise, which entails an increasing diversity in levels of preparation. Some incoming math majors will have already seen and done proofs, while others will have not. To accommodate both of these groups, the math department will be splitting up MAT 215 into two sections. One section will introduce students to proofs more fully and gradually, while the other will assume experience with proofs and launch right into the material.

The resulting changes in the course structure are significant. Students start in MAT215 together during the fall, but will be split into two sections after two to three weeks. One section which we will call the fast track will cover the material of 215/217/218 in one year, thus completing all three introductory courses. The other section, which we will call the regular section, will complete 215 in the fall, 217 in the spring, and students are expected to take 218 in the fall of the following academic year.

In short, there will be two sections of math 215/217. The regular track refers to the combination of 215 in the fall and 217 in the spring. The fast track refers to the combination of 215/217/218 integrated in a single year.

The fast track uses Professor Gunning’s notes, which can be found on: https://web.math.princeton.edu/~gunning/

#### First-hand Account of MAT 215

For most students, MAT 215 will be the first course they take in rigorous mathematics. High school math typically involves applying and sometimes slightly modifying a procedure that the teacher has previously demonstrated. In 215, you will never have a problem this routine. There are some recurring techniques that you will learn to apply, but homework and exams will require you to maneuver a little differently in each instance. Lectures follow a common pattern: the professor presents a theorem, supplies a proof or sketch of a proof, and discusses the significance of the result. Rarely will your teacher stop to address the best way to handle a particular class of problems. A few questions will be more computational in nature (e.g. does the following series converge?), but even these will go far beyond basic approaches (e.g. applying the root or ratio test, though both make an appearance in the course). If this seems frightening, relax. A central goal of the course is to build your familiarity with this style of math. The beginning of the semester will be an adjustment, but eventually, you will become comfortable with these sorts of questions.

For the students who choose to enroll in 215, it will likely be their most demanding and time-consuming course that semester. Some will have enough previous experience to complete problem sets in a few hours, such as those who have participated extensively in math competitions, but most should expect to work a minimum of 10-15 hours on assignments; with high probability, there will be at least one problem set that takes 20+ hours. Students should also spend some time each week reviewing class notes so that they can follow along with proofs during lecture- as with any course that builds upon earlier material, falling behind is a bad idea. Basically, you should be ready to work really hard. But there is good news: 215 is famous for its large study groups. I rarely worked with fewer than 6 other people- all of us struggling, but struggling together. Office hours usually turn into seminars, with the majority of the class attending for homework help. 215 is the closest that you will come to team math in your time at Princeton. As an incoming freshman, you will have a ready-made group of friends who will share your interest in math. These study groups became some of my favorite memories from school.

Here are suggestions provided by 4 students who took 215/217 in the academic year of 2012-2013. The first three students were in the fast track section, and the fourth student was in the regular section.

##### Student 1 (Fast track)

*1. Given the structure of the new 215/217/218 track (or the regular 215/217 track), what did you like (or not like) about it(e.g. pace of course, readings, lectures, pset loads), and what advice would you give to incoming freshmen?*

The pace of the rapid section is pretty fast, yet manageable. It did help a lot to have to have some background in linear algebra though; Gunning doesn’t waste much time there. Gunning’s lectures were quite helpful for the most part. Eighty minutes could start feeling long towards the end, but Gunning was pretty good about keeping me engaged. Problem sets typically took quite a bit of time. I probably averaged well over fifteen hours on each one, but I believe that was longer than my typical classmate.

*2. Would you recommend 215/217/218 fast track or the slow track?*

Having familiarity with basic linear algebra and/or having experience writing more or less rigorous proofs would make the decision for me. Being pretty comfortable with linear algebra and the idea of proofs going in, the fast track was reasonable, but I think I would have been in over my head had I never been exposed to one or either of those.

3. Any suggestions about what to prepare before the start of the semester?

If you don’t have any experience writing mathematical proofs, at least do a little reading on induction, contraposition, etc. If you have taken a course in linear algebra or have some exposure otherwise, consider doing a little reading there to refresh yourself.

##### Student 2 (Fast track)

*1. Given the structure of the new 215/217/218 track (or the regular 215/217 track), what did you like (or not like) about it(e.g. pace of course, readings, lectures, pset loads), and what advice would you give to incoming freshmen?*

215 started out quite gently but the pace rapidly increased after half a semester, reaching a climax in the second half of 217.

Reading notes is important because the lectures basically follow the notes.

Gunning’s lectures were very enjoyable.

Some of the pset problems were actually theorems; since Gunning didn’t have time to squeeze all the theorems/lemmas/useful results in the lectures, some of those were assigned as pset problems for us to prove ourselves instead.

Overall, the flavor of the content was very abstract and different concepts were often investigated in their most general form, occasionally leading to a lack of specific (numerical?) examples to aid understanding. Extra reading at a “lower” level may be helpful for one to learn how to apply the abstract knowledge in solving down-to-earth problems on the psets.

But the exams were mostly easy. Nothing much tricky. You will definitely be able to demonstrate what you have learned and feel a sense of accomplishment.

*2. Would you recommend 215/217/218 fast track or the slow track?*

If you plan to be a math major, I’d recommend the fast track unless you have serious difficulty or haven’t tasted proofs before at all (anyway, you can always switch to the slow track if necessary, so no harm starting out on the fast track for a while). Not only is the fast track a deeper introduction to mathematical concepts that will be useful in further studies, it also allows you to only take two courses and complete the math department prerequisite (without having to take 218), thus saving more course capacity for higher-level endeavors. Even if you can’t understand all of the materials in the fast track, you will be able to digest them gradually, possibly as you re-encounter them in future learning. So it is fine.

But if you are not a math major, don’t want/need too deep mathematical knowledge and worry about the difficulty of the fast track, then of course the slow track is a good alternative as well.

3. Any suggestions about what to prepare before the start of the semester?

Get familiar with some basic multivariable calculus and algebra, because they will arise as special cases of the overarching theories visited in 215/217. So some understanding in these areas will enhance your understanding of their generalized “parents.”

##### Student 3 (Fast track)

*1. Given the structure of the new 215/217/218 track (or the regular 215/217 track), what did you like (or not like) about it(e.g. pace of course, readings, lectures, pset loads), and what advice would you give to incoming freshmen?*

The fast track’s pacing is, unsurprisingly, challenging. It would be extremely difficult to fall behind and try to catch up a couple weeks later in this class.

Gunning’s notes were generally well organized and a great reference, although he was actively writing them as the course progressed and there were often mistakes and many revisions, but this should not be a problem in the near future.

It is important to realize that much of the learning in a hard math class such as this happens not in class, but in the problem sets, because you’re expected to learn so much that they can’t possibly cover all the details in lecture.

*2. Would you recommend 215/217/218 fast track or the slow track?*

If you, for example, have never studied the epsilon-delta treatment of limits before, don’t take the fast track. If you haven’t already had some exposure to writing proving mathematical theorems before, don’t take the fast track. There are some places, especially in the introduction of differentiability, where you will be lost if you take the fast track and haven’t already developed some understanding of the subject in calculus in a single variable. It would also help if you’ve done some multivariable calculus before so you have a clue about multiple integrals and differentiation of vector functions, because Gunning doesn’t stop to provide all the geometric interpretation and the intuition and spatial reasoning skills. One of the defining moments in the class for me is when Gunning went right into defining differentiability of a function from n dimensions to m dimensions, without first introducing differentiability in functions of a single variable. Gunning’s style is to start by writing out formally the most abstract and general definition you will use, and then working back to the particular properties and principles that motivate the math we’re studying. The introduction and exploration of of differential forms was also carried out in this manner. I found it extremely rewarding. Don’t take the fast track if you are very attached to always understanding fully the motivation behind why a class is exploring some topic: Gunning requires you to follow him in a few leaps of faith that he eventually gets back to filling in the motivational basis for, sometimes days or weeks later. But if you’re willing to follow and put in the work to absorb the groundwork he lays out, his class is very rewarding. Gunning is a great professor.

Each semester of the regular track can be taken independently. If you quit math after taking the regular MAT 215, you will still have learned analysis in a single variable. However, Gunning’s fast track is more of a whole-year commitment to make sense of. The first semester of the fast track spends more time covering bits of algebraic fundamentals, vector and metric spaces, and topology than topics that students would recognize as being like calculus, which a student reading the official MAT 215 course description might expect more of. The fall semester will end around the introduction of differentiability and derivatives, and the spring semester picks up from there.

3. Any suggestions about what to prepare before the start of the semester?

Try to have some idea of how mathematical proof works if you don’t already have that. Know your single variable calculus, including how limits and differentiability work. Study multivariable calculus to get an intuitive sense of what these differential operations and multidimensional integrals mean, because you are expected to develop intuition on your own, and it is not “explained” in class, which is taught in rigorous mathematical language.

##### Student 4 (Regular track)

*1. Given the structure of the new 215/217/218 track (or the regular 215/217 track), what did you like (or not like) about it (e.g. pace of course, readings, lectures, pset loads), and what advice would you give to incoming freshmen?*

I took the regular track and found that despite it not covering the material of 218, it was still challenging because it went much deeper into the 215/217 content than I anticipated. Many of our problem set problems were very intricate and involved. When I took it, I had Professor Damron for both classes, and he was very good at explaining the material and building our intuition. The pset loads are relatively intense (I probably spent the most work on 215/217 of all my classes, except maybe Writing Sem) and a lot of important concepts are also presented in the homeworks.

*2. Would you recommend 215/217/218 fast track or the slow track?*

As someone who has had a little bit of proof exposure before, I kind of wish I took the fast track, but took the slow track to be extra careful. However, the slow track builds a very strong foundation, so I can pick up the 218 material with relative ease.

*3. Any suggestions about what to prepare before the start of the semester?*

Professor Damron’s course notes on his website are very clear and helpful. I tried reading Rudin to prep for 215, but it was too dense and I gave up after about one chapter. We used Hoffman and Kunze for 217, and I really liked the way that was written. That text is much more accessible to someone who hasn’t taken either of these topics before.

#### Links to the Math Department Website

The above descriptions for the introductory courses are fairly short, due to the fact that the math department has done a good job of providing information about these courses on its website. Please see here for more details on course content, sample material, and other information. The FAQ sections of the pages for the intro courses provide answers to questions such as “how hard is this course” and “is this the right course for me”.

#### Contacts

Chris McConnell ’15 (cmcconne[at]princeton[dot]edu)

Gene Katsevich ’14 (ekatsevi[at]princeton[dot]edu)

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