# Math 215

### Analysis in a Single Variable

Aimed primarily at future math majors, this course covers calculus more thoroughly and more theoretically, giving an introduction to important mathematical techniques and results that give a foundation for further work in analysis. It serves as an introduction to the rigorous proofs and formal mathematical arguments needed in all upper division math courses.

Equal emphasis is given to learning new mathematics and to learning how to construct and write down a correct mathematical argument by dividing the 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 of doing mathematics, very far removed from the process of memorizing algorithms and working through concrete calculations typical of high school math.

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 Rieman integral, conditions for integrability of a function and term by term differentiation and integration of series of functions, Taylor's Theorem.

Classes: are usually taught in a two sections in the fall semester and a single section in the spring semester. The two fall sections are closely coordinated, with the same problem sets and exams.

Textbook: Principles of Mathematical Analysis by Rudin, McGraw-Hill Science/Engineering/Math, 3rd edition.

Notes:

• Alternatively, prospective math majors can take 214 for a more algebraic introduction to rigorous proofs and formal mathematical arguments in the context of classical number theory.
• This course is usually followed by 217 (linear algebra) and then 218 (analysis in several variables).

### Who Takes This Course?

• Primarily incoming students who seriously consider majoring in math.
• Students who are choosing between math and physics as a major should perhaps consider 203 instead.
• Students who are choosing between math and computer science might consider 214 instead.

### Placement and Prerequisites

• A very strong aptitude for mathematics and real mathematical curiosity is essential. Do you want to
• make your own conjectures and figure out for yourself whether a mathematical statement is true or false?
• be able to construct a clear and convincing, even iron-clad, argument to justify a mathematical claim?
• develop an appreciation for the intrinsic value and power of mathematical argument, separate from considerations of real-world applications or utility?
• Typically students have a 5 on the BC calculus exam together with a math SAT score of at least 760. A very solid knowledge of one-variable calculus is assumed, and this course will build on that knowledge to give you a much deeper understanding of the concepts and theorems you first saw in high school.

### Sample Problems

A math major who took this course in the Fall of 2010 has prepared this list of Sample Problems designed to help you understand for yourself what 215 will be like. If these questions seem intriguing to you, then take the course to find the solutions!

### FAQ

1. How hard should I expect to work in this course?
• It requires a steady time commitment. We expect that the weekly problem sets will take at least 3 hours to complete. To do well on exams, you need to spend a lot of time digesting the course material, learning the proofs well enough to adapt them to new situations and combine various standard ideas in new ways on an exam. All in all, this course may easily take 10 hours/week outside of class, on average. It is quite difficult to judge how much time you will need to master the more abstract parts of the course. For many students this is a big adjustment, unlike the math courses you have taken before, so be prepared to invest quite a lot of time early on, learning how to think about proofs and counterexamples and adapting old techniques to complicated new situations, rather than chugging through computational problems just like the textbook and lecture examples.
2. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
• Credit for this course at another university will be granted only under very exceptional circumstances and we strongly prefer that you take this course here at Princeton. This course (or 214) shows you what it will mean to be a math major at Princeton. It sets the foundation for all the more advanced courses for math majors, and you really need to be sure that this foundation is as strong as possible.
3. My question is not listed above. Where can I find an answer?
• Try the general math FAQ page. It should contain a section just for prospective math majors (eventually).