MAT215 Single Variable Analysis with an Introduction to Proofs

Schedule: (usually) Tuesdays and Thursdays, 11AM-12:20PM and/or 1:30PM-2:50PM, with a Friday precept, in both Fall and Spring semesters, although more students take this course in the Fall. Includes optional evening review sessions run by undergraduate course assistants on a schedule that is set up after classes begin.

Brief Course Description: An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. The course shows how mathematicians think about calculus. Topics include the rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series; continuity, uniform continuity and differentiability of functions; the Heine Borel theorem; the Riemann integral, conditions for integrability of functions and term-by-term differentiation and integration of series of functions; and Taylor’s theorem.

Equal emphasis is given to learning new mathematics and to learning how to construct a mathematical argument by dividing a question into logical steps where each step is explained and justified carefully, giving references if necessary. For most students this will be a completely new and very challenging way to do mathematics, far removed from the process of memorizing algorithms and working through concrete calculations familiar from high school.

Who takes this course? This course is suitable both for prospective mathematics majors and also for non-majors interested in a deeper conceptual understanding of calculus in addition to learning how to understand and construct formal mathematical arguments. It is ideal for students with a strong mathematical background who want to learn more about proofs in preparation for upper-division course work in the mathematics department. It is an especially good choice for students with a serious interest in applied mathematics or theoretical physics.

Prerequisites and Placement Information:  A very strong aptitude for mathematics and genuine mathematical curiosity are essential. Do you like mathematics, especially the part where you try to understand why a method works or why a definition encapsulates a particularly useful point of view? Do you want to make your own conjectures and figure out for yourself whether a mathematical statement is true or false? Would you like to learn how to construct a clear and convincing, even iron-clad, argument to justify a mathematical claim? Would you like to develop an appreciation for the intrinsic value and power of mathematical thinking, separate from considerations of real-world applications or utility?

This course treats all the important definitions and theorems of single-variable calculus from a very theoretical and abstract point of view.  Thus prior knowledge of calculus is not strictly required, although you will need the mathematical maturity and independence that most students acquire by studying calculus seriously in high school. Because the course is more abstract and theoretical, it is quite difficult to judge how much time you will need to adjust to this way of thinking about mathematics. Students should be prepared to invest a lot of time early on.

FAQ: 

• You want to be a math major and you have done some number theory before, so the idea of taking 214 is quite appealing. If you are pretty sure you want to be a math major, then taking 215 in the fall may be a better choice.  There will be time later to look at other branches of mathematics, and the analysis topics covered in 215 and in 216-217 are more broadly useful in the major than the topics in 214. Most math majors will learn more about algebra and number theory in upper-division courses instead. If you decide to start with 214, you will still need to complete 215-217 or 203-204 to prepare for 300-level math courses.

• You are really not sure about a major, but you love math and you are good at it. You have never taken a proof-based class before. Your first semester will give you a lot of information, and the system is quite flexible for the most part. In your first semester at Princeton, you might attend both 203 and 215 for the first few classes to see which one feels more interesting to you at this point in your studies, especially if you are concerned about fulfilling the prerequisites for physics or engineering as quickly as possible. If you plan to take upper-division courses in the math department as part of your work in physics or engineering or finance, then 215 will be very helpful to you, but it won’t teach you vector calculus. Taking 203-204-215 is a reasonable option if you currently believe that you are more likely to do physics or engineering, but if you try 215 and it appeals to you, then taking 215-217-203 is another option that fulfills the prerequisites for all three. And some students decide to take 215 only later, even in the junior or senior year, as preparation for graduate work in other quantitative disciplines like economics or finance.

• How hard should I expect to work in this class?  It can vary quite a bit for individual students, but we expect that you will need to spend substantial time outside of class, reading the textbook and reviewing your notes in order to understand the proofs and the definitions from the textbook well enough to be able to adapt them to create new proofs and construct counterexamples. Following up in office hours or at the undergraduate-led problem sessions or with your study group will likely be important for understanding all the new ideas. The problems require both creativity and knowledge, and so you will need to spend unpredictable amounts of time in active contemplation while you wait for inspiration or the right insight. All in all, you should be ready to spend at least 10 hours per week working outside of class. 

• You have never had a course with rigorous proofs and you are worried it might be too hard. It will probably be a challenge, but the course is designed to be accessible to students who are seeing proofs for the first time. If after a few classes you still find the subject appealing, you should probably persevere, even if you do find the problem sets to be (overly) challenging. Go to office hours and take advantage of the help available in the evening problem sessions where enthusiastic math majors will help you learn how to think like a mathematician. The course is graded on a generous curve to encourage interested students who want to give this kind of thinking a serious try without undue academic risk. Consult your instructor for advice after the first few weeks if you are still worried about how things are going. Placement adjustments can be made even after the drop/add deadline if necessary.

• You have quite a bit of experience with proofs from independent reading or university-level analysis courses you took in high school or from extracurricular activities like math camp. Many of the topics for the course are already familiar to you.  Try the Sample Problems. If you know how to do these problems already, then taking 216 instead might be the right plan. You can discuss this with the math department representatives when you are on campus.