Math 201

Calculus III (Multivariable Calculus)

Third semester of the 3-semester calculus sequence 103/104/201. Introduces mathematical methods needed to understand real world questions involving quantities changing over time in 3-space. Students must move beyond thinking about mathematics as a set of rules and algorithms to memorize. Instead they learn to approach problems with greater independence and maturity. Exams test for thorough conceptual understanding and computational fluency in standard cases. In other words, although demanding, the exams do not typically involve proofs, and there is little emphasis on exceptional cases.

The first part of the course introduces basic objects in space: lines, planes, curves, (quadric) surfaces, and basic properties like arc length, surface area and volume.

The second part moves on to quantities that change with position in 3-space, like temperature or population density, which can be described by functions of several variables. We study their graphs, level sets, rates of change (partial derivatives) and their extreme points (maxima/minima) using Lagrange multipliers and Taylor's theorem.

The third part is devoted to vector fields, quantities that change with both position in space and with time. Think of how storms move, how radio signals make cell phones work or how blood flows. Although we do not explicitly study such applications, we lay the mathematical foundation for more advanced courses in science, engineering, and economics. The course ends with the theorems of Green, Gauss, and Stokes.

Classes: meet 3 times per week, for 50 minutes. Generally offered MWF at 10, 11 and 12:30 in both Fall and Spring semesters. The course is organized into small precepts of 20 to 30 students. There is one course head who coordinates with all the instructors to write the exams. All students have the same homework assignments and take the same midterm and final exam. These exams count for most of your course grade. They are graded by all the instructors together to ensure uniformity across sections. In addition there are two to four take-home quizzes, the same for all sections.


Textbook: Thomas' Calculus Customized for Princeton University by Thomas, Weir, and Hass, Vol 2, (12th edition) Pearson: Addison Wesley.

Who Takes This Course?

Placement and Prerequisites

General information about placement and contact information for the placement officer can be found on the math placement page.

Sample Problems

Working problems from these sample quizzes and exams can give you a good idea of the expectations and content in this course as you think about which course is right for you. Just reading the questions or the solutions can be very misleading however. Try the problems yourself!

Once you have tried the problems, you can check your answers:

Quiz One Solutions, Midterm Exam Solutions, Quiz Two Solutions, Final Exam Solutions.


  1. I already took multivariable in high school, do I have to take this course?
    • Most students in 201 have had some multivariable calculus and/or linear algebra before, but rarely with the same depth and thoroughness. If you need the course for upper division courses in your major, then you are probably better off to take 201 even though some material will be review.
    • Take the sample final. Can you do any of the problems? For most students, the answer will be no. Review your old notes and try again. Can you do at least 60% of the exam?
    • In rare cases, the placement officer will decide that your prior work is indeed equivalent to 201 at Princeton. It will be helpful if you can bring your graded exams from the course you took to show the placement officer. He/she may also require you to take an exam to demonstrate your knowledge. Contact info for the placement officer can be found at the adept rep home page.
  2. Can I take 201 and 202 in the same semester?
    • It is not impossible, but we do not recommend it. It makes midterm week particularly unpleasant, but if you have a very good reason for it and you are a very strong student, it can be done. It will likely mean that you will get a lower grade in one of them that you would otherwise have done.
  3. How much work is 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 work a lot of extra problems from old exams. So all in all we expect this course to take a minimum of 10 hours/week outside of class, on average.
  4. If I think 201 is too hard, what should I do?
    • Try an old final exam in 104. If you can do most of the problems with confidence, you should stay in 201. Otherwise, you may want to switch to 104 to get a thorough review of all the math assumed in 201. Starting in 104 won't throw you off-track if you are a BSE student and taking the time to strengthen your foundations can really pay off in the long run.
  5. If I think 201 is too easy, what should I do?
    • Have you had a quiz yet? You may be in for a surprise. Homework will be much easier than exams in this course. The general math FAQ page has a whole section on how Princeton's calculus courses work which may give you a more realistic view of the situation.
    • If you are also taking PHY103, you might consider just enjoying the fact that MAT201 is too easy. The combination of a demanding physics class and a demanding math class has been responsible for quite a few academic distress stories at Princeton.
    • Consider taking 203 or 215 or 218 instead.
  6. I need both 104 and 201 for my major. After checking the math placement information, I think 201 is probably the right course for me, but I don't qualify for AP credit for 104. Do I have to take 104 or can I sign up for 201 instead?
    • Be cautious. Students often underestimate the difficulty of 104 and of 201 because many have seen some of the techniques taught in the first few weeks of these courses. Consider the information in the previous two questions as you think about your decision.
    • If you are really sure that you belong in 201, you can sign up for it. If you pass 201 in your freshman year, you will automatically receive AP credit for 104. Just be ready to re-consider and switch down to 104 early on if necessary.
  7. If I want to switch courses, what should I do?
    • Talk with your 201 teacher. You may also want to talk with the teacher in charge of the other course. If you are going to switch, do it soon!
  8. I would like to switch sections within 201, what should I do?
    • If you have a time conflict, there is no problem. If you want to switch to another section at the same time, be prepared for a possible refusal --- the instructor may not have room for another student. Course and section changes must be made through SCORE.
  9. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
    • Yes, but it may be difficult to find an equivalent course. Many multivariable courses at other universities cover only about half of 201. More info at the adept rep home page.
  10. I have more questions that are not answered here. What should I do?
    • First, check the general math FAQ page for more information. (There is a whole section there on how Princeton's calculus courses work.) If you still have questions, representatives from the math department will be available at freshman registration.