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.
Brief Course Summaries [Show]Brief Course Summaries [Hide]
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.
First-hand Account of MAT 215 [Show]
First-hand Account of MAT 215 [Hide]
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.
Recent Changes [Show]Recent Changes [Hide]
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. Logistically, the math department plans to split the two sections like PHY 103 and PHY 105: have the two sections start out together, but then use the first one or two problem sets to differentiate between the two groups.
In addition, many math majors would like to have their introductory math courses done by the end of freshman year. This has resulted in the low popularity of the last intro course, MAT 218. In response to this, the department is now considering switching to a two-semester introductory sequence. Little is know at this point about how or when this will happen.
Links to the Math Department Website [Show]Links to the Math Department Website [Hide]
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”.