Honors Analysis (Single Variable)

Description of Course Goals and Curriculum

MAT 215 covers topics that you may have seen in prior calculus classes in a more rigorous fashion. Coming out of the course, you will not only have a deeper understanding of calculus but also greater intuition for how to do proofs. The class follows the first 8 (more or less, depending on the semester you take it) chapters of Rudin’s textbook “Principles of mathematical analysis” pretty closely. Those are: 1. How to do a basic proof 2. Counting and Basic Topology (Open/closed/compact sets etc.) 3. Limits 4. Continuity 5. Derivatives 6. Integration 7. Uniform Convergence 8. Other special topics

Learning From Classroom Instruction

Class time in a nutshell:

• MAT 215 meets for class (lecture-style) twice a week for 80 minutes.
• Professors hold 2+ weekly office hours.
• There are optional problem sessions at night (think 7:30-9 pm) 2-4 times a week depending on how many students are enrolled in the course.

In lecture, which was 11am-12:20pm Tuesdays and Thursdays when I took it, the instructor goes over a section of the Rudin textbook. The expectation is that students follow along with the proofs in class and ask questions if any step of the proof is unclear. Approximately every Thursday, there is a problem set due at the beginning of class. We also had biweekly 15 minute quizzes at the beginning of every other Tuesday class, but I’ve heard that they have removed this component of the class. Instructor office hours and TA problem sessions are optional but help with the homework and conceptual questions for the class.

Learning For and From Assignments

Outside-of-class work in a nutshell:

• There is a weekly homework that takes ~20 hours due at the beginning of class every Thursday.
• There is a midterm and final exam. Whether these exams are take-home or in-class depends on the instructor and semester, but usually one is in-class and one is take-home.

The homework and exams prepare you well in developing your mathematical maturity. They also help with your understanding because you have to apply the concepts and building blocks learned in class. The questions are similar in difficulty between the homeworks and exams (if not the exam questions being slightly easier) and will primarily consist of asking you to prove new things using theorems learned in class. In doing the homeworks, if you run into any difficulty, you can go to the instructor’s office hours and TA problem sessions for hints. If you want more practice doing proofs before you attempt the homework problems, try to recreate proofs from Rudin or from class. My first two pset grades were the lowest, but I improved over the course of the class. If you aren’t doing well right now, don’t get discouraged! The course staff genuinely wants to help you succeed in the class. In preparing for exams, there are past homeworks, quizzes (if you have them this semester), your midterm, the textbook, Stack Exchange and other math forums online, past exams from other universities posted online, and so much more. Start by definitely being able to regurgitate definitions in the rigorous way Rudin defines them. Once you have those building blocks, you can start preparing by proving the things you proved in class using those definitions, then continue developing intuition doing more practice with past homework and textbook problems, and lastly if you still have time you can practice with exams from other universities (for example MIT 18.100C has assignments and exams posted online).

External Resources

Before moving to outside resources, I think that it’s important to consider the internal resources for the class first. Every week, I would read the relevant sections of Rudin (ask your instructor for which sections are relevant) and go to office hours with questions about the proofs from class/Rudin that I didn’t understand. I would read a proof once, then write it out with help from the book/class notes, then cover up the proof and try to recreate it by myself. This really helped me learn how to do proofs, which in turn helped with the homework. Beyond this, there are probably over 10 hours total per week of help available outside of class in the form of the TA hours and instructor office hours. There are also additional exercises in Rudin not assigned for homework that can be done. If you are still lacking in practice problems for the class, the equivalent course of MAT 215 on MIT OCW is 18.100C, and there are plenty of additional practice problems there if necessary. You can also do an online search for other universities using the Rudin textbook and see if they have homeworks and exams posted on their websites. Struggling in this class is common, even if students do have proof experience, due to the pace and rigor of the course. However, it really does wonders in developing your mathematical maturity, and there are plenty of resources available to help you succeed.

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

What background knowledge and skills are needed for MAT 215, and how will it work with my schedule? When the course description says that no background knowledge in proofs is necessary for MAT 215, they really mean it! The majority of the course comes in with no proof experience and only a passion for learning. That being said, MAT215 is really time consuming, especially for people who have never had exposure to proof based math before, so you want classes that aren’t that time consuming to balance it out. For some people, this is additional STEM classes, where they can finish psets really fast to focus their time on 215. For others, it’s reading-heavy classes that allow them to good breaks from 215. How could MAT215 fit into my course of study at Princeton and long-term goals? MAT 215 is a gem of a course. MAT 215 is extremely valuable because it develops introductory logic and proof skills which carry over well even if you decide not to be a math major.   If you do decide to become a math major, MAT 215 is a great stepping stone into the department’s courses. You may never have to take a class more rigorous than MAT 215 if you choose all of the applied courses in the department for your requirements. That’s not to say MAT 215 doesn’t prepare you well for the more rigorous versions of those requirements—many of my classmates who started off in MAT 215 have done incredibly well in upper level MAT departmental courses considered to be “hard”.   At the same time, if you end up in the more applied-math realm, the course will help you succeed in other “more proofy” courses in ECO, ORF, and COS, as well as logic courses such as PHI 201 and 312. Examples that come to my mind that became easier after taking MAT 215 (or would have been easier had I taken MAT 215 earlier) include ECO 310, ORF 309, COS 340, and COS 445.   Many PhD programs in areas of applied mathematics (such as in economics, statistics, and more theoretical applications of computer science) require you to have exposure to some analysis, and MAT 215 is sufficient for an analysis course as a prereq for applying to graduate school.   Math is hard, yet extremely rewarding. My last piece of advice would be to not give up and to enjoy the process of learning.