Analysis in Several Variables

Description of Course Goals and Curriculum

MAT 218 is the second part of the MAT216-218 introductory math sequence. It continues as an introduction to the mathematical discipline of analysis, and as preparation for other higher-level math classes. When compared with MAT216, 218 focuses more on linear algebra topics and multi-variable analysis. Topics include real analytic mappings and series, the Stone-Weierstrass theorem, endomorphisms, inverse mapping theorem, implicit functions, integration over Jordan domains, and Stoke’s Theorem. The course is intended as a continuation of MAT216. Together the courses aim to provide students with a basic experience in a rigorous math class, in a style that will be similar to the style that students will encounter in upper level math classes. By the time students are in MAT 218, they have already had a semester of proof-based mathematics and thus MAT 218 does tend to focus more on foundational concepts in math than MAT 216 does. The class mainly focuses on linear algebra and multivariable calculus. As such, having some experience in both is helpful to contextualize the rigor and the proofs in the class, although no background is assumed (beyond what was covered in MAT 216).

Learning From Classroom Instruction

Professor Gunning usually covers material quite quickly during class. Especially later in the course, Professor Gunning will often skip small steps within proofs to present the big-picture idea of how a proof is moving along. As such, it is important to follow both Professor Gunning’s speech and his written work, since sometimes he will say important things but not write them down or vice-versa. Professor Gunning does provide lots of examples of the material that he covers, and usually has plenty of time to answer any questions from the class. While Professor Gunning covers the majority of topics deeply in class, there are topics that he skips over during class that are still important for a good understanding of the material presented in class. A solid resource to rely in these cases (or in general if lectures are confusing), is Professor Gunning’s compiled course notes which is essentially the textbook for the class. If something is every confusing in lecture, reading the textbook is usually a helpful and solid start since the lectures and the textbook generally cover identical material (even including notation!). In order to make the best use of Professor Gunning’s deep, but fast lectures, it is important that students have a good understanding of the material already covered. Since lectures usually don’t spend too much (if any) time on review and Gunning does not often connect theorems directly to each other, not understanding what happened previously can lead to a scattered understanding of the material. Understanding methods of proof is also essential, since once a method of proof has been encountered once or twice, the class will generally go over it only briefly, skipping many of the small details. As such, it is important for students to have their own understanding of the proofs and the way they were proved. After all, while MAT 218 does present material that is new for some students, the general purpose of the class is still to provide an introduction to mathematical thinking rather than teach a specific set of material. To this end, students should always remember to try and connect the various ideas that are presented in the class and figure out how they relate to each other. While the material presented is useful, the ability to link and understand mathematical concepts is far more important in a long-term pursuit of mathematics.

Learning For and From Assignments

Weekly assignments in MAT 218 are similar to what could be found in MAT 216. The assignments in general are challenging, but teach students how to engage with increasingly theoretical material. In general, students should expect to spend a lot of time on the problem sets, but with the assurance that the problem they encounter are very much within their abilities, and do not require any outside knowledge. Problem sets are almost entire proof-based. When approaching a problem set, it is most helpful to first consider the proofs that have been shown in class and seeing what approaches worked, and why. Then, it is also important, particularly in analysis, to go over definitions and basic concepts to try and get a better grasp of what the problem is saying. Finally, it is usually instructive to try and solve a simpler version of the problem, or to do some numerical examples to see why a particular statement would be true. Exams are generally similar to the problem sets, although shorter and easier due to the three-hour time limit. Although they are open-note, students should have a solid understanding of the material before the exam. Since the exam, like the problem sets, are mostly proofs, looking up definitions or theorems could detract from time spent thinking about the proofs.

External Resources

Since MAT 218 is still introductory content, the internet can serve as a valuable resource. Wikipedia in particular stands out. The math pages are generally quite well written and seeing statements and proofs from another angle can be illuminating. The math department also offers problem sessions that are taught by upperclassmen. Even when the problem sessions don’t cover a specific question that a student might have, it is usually instructive to listen to classmates and tutors explain how to think about different problems and proofs. Professor Gunning’s office hours are also great. He is very open to questions and generally responds with hints that do not fully explain the problem, but give the student good insight and intuition into it.

What Students Should Know About This Course For Purposes Of Course Selection

MAT218 has a very theoretical bend and requires MAT216 as a pre-requisite. As such, it is mainly intended for those students who are seriously considering studying rigorous math in the future. Although it is usually taken by freshmen looking to concentrate in math, there are also computer science and physics students who take the class to gain a better grasp of rigorous thinking and of underlying mathematical concepts. MAT218, along with MAT216, is definitely one of the most time-intensive and demanding pre-requisite courses at Princeton. It is however, very well-taught and rewarding, and a memorable experience for any student considering the deeper pursuit of mathematics.