Course MAT218 in Fall 2011

Analysis in Several Variables

This is a rigorous course in multivariable analysis. Continuation of MAT215/217 (or MAT203/204, with instructor's permission). It covers differentiation and integration in several variables, including both theoretical results (with fairly detailed proofs) as well as explicit calculations, to ensure both an understanding of the theory and the ability to apply it to concrete problems.

The primary reading comes from the Lecture Notes prepared by Professor R. C. Gunning.  In addition the following supplementary texts will be on reserve in the math library:

  • An alternate treatment of the more challenging ideas: Calculus on Manifolds by M. Spivak (Benjamin, 1965)
  • The usual textbook for our intermediate vector calculus course (MAT203) to help you brush up on/master the basic calculations: Vector Calculus by J. Marsden and A. Tromba (Freeman, 1996)
  • Similar to Marsden and Tromba: Advanced Calculus by G. Folland (Prentice Hall, 2002)

The treatment in the course follows most closely that in Spivak's book. The treatments in Folland and in Marsden and Tromba follow the more traditional paths, but these books have a number of useful illustrative problems and more detailed discussions of techniques for solving the problems.


Metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini's theorem, change of variables theorem, and the theorems of Green, Gauss, and Stokes.

Description of Classes

The class is generally taught in a single section. 

There are three hours of lecture each week (traditionally scheduled for MWF from 10 to 10:50AM) and an hour of precept on Thursday evenings, from 7:30 to 8:20PM.

It is expected that all students will attend all four hours of the course each week. In addition to weekly problem sets, there is an in-class midterm exam, a take-home midterm problem set, a final in-class exam and a final take-home problem set. There are two kinds of homework problems in this course: some consist of straightforward problems and basic calculations (similar to the problems in MAT203) and others are much more theoretical.

Who Takes This Course

Mostly students who seriously consider majoring in math, or possibly physics, especially theoretical physics.

Placement and Prerequisites:

A very strong aptitude for mathematics and real mathematical curiosity is essential.

A very sound knowledge of calculus in one variable is also required, preferably from a fairly rigorous and abstract point of view, as it is taught in MAT215. A good source to review this background material is Rudin's Principles of Mathematical Analysis, particularly its first few chapters. If that material is unfamiliar to you, you should take MAT215 instead.

Good knowledge of the basic properties of vector spaces, linear transformations and determinants from MAT204 or MAT217 or the equivalent is needed.  Prospective majors in a hurry to complete the prerequisites for the major sometimes take both MAT217 and MAT218 in the same semester.  This makes for a very demanding semester so choose your other courses wisely!

If you are an incoming freshman and you believe that you are ready for MAT218:

  • Check out the course web pages for MAT215 and MAT217. Be sure you can work the sample problems there. If possible, take a careful look at the texts for these courses as well, to be sure you really know this material already.
  • Check out the sample MAT218 problems below.

After doing this research, if you still think that MAT218 is the best choice for you, contact the placement officer at  the Academic Expo during Orientation or at freshman registration. (If possible, bring your graded work from the MAT215 and MAT217 equivalents you took elsewhere.)

Sample Material

The math department is grateful to Daniel Shenfeld who, based on his experience as the 218 preceptor in the spring of 2011, prepared this list of Sample 218 Problems to help potential students decide if 218 is the right course for them. If you have the prerequisite knowledge for the course, you should be able to begin thinking about these questions, even if you cannot produce a complete solution. These problems should give you a good sense of how 218 will extend and deepen your knowledge of calculus and abstract mathematical argument. If you find these questions intriguing, that is an excellent indication that you have the mathematical curiosity and ability that this course demands. Otherwise, consider taking MAT201 or MAT203 instead to acquire a less abstract, but nonetheless solid, knowledge of vector calculus.

  1. How hard should I expect to work in this course?
    Pretty hard.  If you already took MAT214 or MAT215 then you should have a pretty good idea of what is involved.  It requires a steady time commitment, but the time you will need to invest can vary quite a lot depending on your background and goals.   It is quite difficult to judge how much time you will need to master the more abstract parts of the course.  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.
  2. I have never had a course with rigorous proofs -- will this course be too hard for me?
    • If you have time, consider starting in MAT215 instead.  If that is not an option (are you sure?) but nonetheless you have a serious interest in being a math major, then you should give it a try.  The first few classes will tell you whether you find this way of doing mathematics appealing or not.  Students who have had the expected previous experience constructing mathematical proofs will have an advantage, but it is not impossible to catch up.   You will certainly need to work much harder for the first few weeks since you will be learning both new mathematics and a new way of thinking about mathematics.  Go to your instructor's office hours and take advantage of the help available at the McGraw Study Halls, where enthusiastic math majors will help you learn to think like a mathematician.  
    • It would be nice if you could attending both this course and MAT203 for the first few weeks of the semester until you can tell which course will work better for you.  This is not usually an option because they are not often offered in the same semester.  You may need to talk to one of the math major advisors or the departmental representative.  Contact information can be found on the undergraduate home page
    • This course (like MAT203) 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 worried about your preparation and prospects.
  3. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
    • Probably not.  If you want to be a math major you should take this course here at Princeton.  Courses like MAT217 and MAT218, along with MAT215 or MAT214, set the foundation for all the more advanced courses for math majors and you really need to be sure that this foundation is as solid as possible.  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.
  4. My question is not listed above. Where can I find an answer?
    • Try the undergraduate home page.   You will find links there to more information for future math majors and contact information for the various people who can advise you.  Representatives from the math department will be available at the academic expo during orientation and at freshman registration.We strongly prefer that you take this course here at Princeton. This course sets the foundation for all the more advanced courses in analysis for math majors, and you really need to be sure that this foundation is as strong as possible.