Course MAT201

Calculus III (Multivariable Calculus)

A continuation of MAT103/104, the third semester in the calculus sequence gives a thorough introduction to multivariable calculus and mathematical methods needed to understand real world questions involving quantities changing over time in 3-space.  Topics include limits, continuity and differentiability in several variables, extrema, Lagrange multipliers, Taylor's theorem, multiple integrals, integration on curves and surfaces, Green's theorem, Stokes' s theorem, divergence theorem.  Emphasizes concrete computations over more theoretical considerations.  However, this course demands that students move beyond thinking of mathematics as a set of  rules and algorithms to memorize, and begin to approach problems with greater independence and maturity.  Exams test for thorough conceptual understanding and computational fluency in standard cases.   Although demanding, the exams do not typically involve any proofs, and there is little emphasis on exceptional cases.  Offered both Fall and Spring.  Prerequisite: MAT104 or equivalent.

  • 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.
Description of classes

Classes meet 3 times per week, for 50 minutes.  Sections are generally offered MWF at 10, 11 and 12:30 in both semesters.

The course is organized into small sections 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. The midterm and final count for the bulk of the course grade, typically about 70%.  These exams are graded by all the instructors and graduate student AI's together to ensure uniformity across all sections. Typically there are two to four take-home quizzes, the same for all sections. Homework and quizzes together usually account for about 30% of the course grade.

In order to do well in the course,  we anticipate that most students will need to spend approximately ten hours per week reading the text, reviewing class notes, solving homework problems and working through lots of extra practice problems to prepare for quizzes and exams.  The course will be quite fast-paced and it is essential to work steadily throughout the semester.  Frequent feedback will be given to help students keep up and monitor progress.

  • Usually followed by MAT202 (Linear Algebra). Although MAT201 and MAT202 can be taken in either order, we recommend you take 201 first.
  • The first half of MAT201 generalizes MAT103 to higher dimensions, and the second half generalizes MAT104.
  • Students who have already taken MAT175 should not sign up for MAT201 because there is too much overlap between these two courses.  MAT175 is intended for students who will not take futher mathematics courses at Princeton;  A motivated student who received a grade of at least B could reasonably continue in  MAT202 afterwards, but should expect to find 202 more difficult than 175.
Who Takes This Course
  • Most students in this course are incoming freshmen or sophomores who consider majoring in one of the sciences or engineering.  More mathematically inclined economics majors will take this course along with MAT202 (instead of MAT175).  It gives a solid introduction to multivariable calculus suitable for most students who want to use mathematics as an analytic tool in later studies in other fields.  Although it is not a prerequisite, many students in the course will have had a more basic multivariable calculus course in high school.   
  • Most students in the course are freshmen; in the fall they are students who got a 5 on the BC exam in high school (or its equivalent).  In the spring, they are mostly continuing from MAT104.
  • Students who took AB calculus only should take MAT104 instead.   There is one possible exception to this rule: students with a 5 on the AB exam and a very strong interest in math as a major along with a math SAT score of at least 750 can consider taking MAT215 or MAT214 instead.  This is rather rare, and such students should consult the math placement officer.
  • Students who consider a major in physics or applied math should consider MAT203 or MAT215 instead if they have a 5 on the BC exam (or equivalent) and a math SAT score of at least 750.
  • Future math majors usually learn multivariable analysis (calculus) in MAT218 or in MAT350 instead (after taking MAT215 and MAT217).  Some math majors, especially those who are more interested in applied math opt for MAT203.
  • Non math-track economics majors may take MAT175 instead for a much more basic treatment of multivariable calculus. 
  • AB COS majors are not required to take 201. They need only MAT202 (or MAT204 or MAT217).
Placement and Prerequisites
  • A very solid knowledge of single-variable calculus and precalculus is needed: how to analyze and graph functions, how to compute and interpret derivatives, how to interpret, set up, and calculate definite integrals with speed and accuracy.  An interest in thinking rigorously about problems involving space and time is also needed.
  • If your background is weak or rusty consider MAT104 to get an excellent review of the knowledge assumed in this course.

For those deciding between MAT104 and MAT201:

  • Keep in mind that a score of 5 on the BC calculus exam is minimally equivalent to MAT104, probably equivalent to a grade of C.  As a result, many students who scored a 5 on the BC calculus exam or took a similar course in high school (and did well) opt to start in MAT104 and find it to be quite challenging.  Although many of the topics in MAT104 are somewhat familiar after a BC calculus course, the depth of coverage and mastery of the subject required here is much greater, with more emphasis on independent thinking.  Additional topics that are not covered in most BC calculus courses are also included. 
  • Note that a score of 7 on the IB MathHL exam (not SL!) or an A on the British A-levels exam is treated as equivalent to a 5 on the BC exam here and at many other universities.  These scores indicate that MAT201 is a reasonable starting point for you.
  • If in doubt, sign up for MAT201 and be prepared to re-evaluate during the two weeks of classes.

For those deciding between MAT201 and MAT175:

  • You should probably take MAT201 and switch down after a couple of weeks, if necessary.  Taking MAT175 limits your options and not every program accepts it as a substitute for MAT201; you should not take it if you may later need further math courses at Princeton.

General information about courses and useful contacts can be found on the undergraduate program home page.

Sample Material

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.

Additional Sample Finals (without solutions):

  1. I already took multivariable in high school, do I have to take this course?
    • Most students in MAT201 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 MAT201 even though some material will be review. 
    • Not convinced?  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 MAT201 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.
    • You might consider MAT203 instead (or even MAT215/217/350 if you want to be a math major) but these courses are not for everyone -- they require an intense commitment and interest in math for its own sake, not just fulfilling a requirement.  
  2. Can I take MAT201 and MAT202 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?  
    • Most math courses require a steady time commitment.  We expect that the weekly problem sets will take at least three hours to complete, although this can vary quite a lot depending on your background and goals.  To do well on math exams, you need to work through a lot of extra problems.  All in all, you should be ready to spend up to ten hours per week working outside of class.
  4. If I think MAT201 is too hard, what should I do? 
    • You may have a couple of options, depending on which courses you have already taken here at Princeton and depending on your major.   You may consider switching into MAT175 but only if you are completely sure that you don't need to take any further math courses and that your program will allow you to substitute MAT175 for MAT201.  If you want to major in engineering, then your only option is to consider dropping back to MAT104 to get a thorough review of all the material assumed in MAT201.  As a future BSE major, starting in MAT104 will not throw you off-track, and taking the time to strengthen your foundations can really pay off in the long run.
    • Try an old final exam in MAT104. Can you do at least half of the problems correctly?  (Try the problems -- don't just read the questions!)  If you do need to switch down to MAT104 because your knowledge of one-variable calculus is insufficient, then you should decide this as quickly as possible because there is very little overlap between the first half of MAT201 and MAT104.  If you wait too long, it will be very difficult to catch up and do well in MAT104.
    • Peer tutoring can be arranged through your residence college, and there is also help available at the McGraw Study Halls.  Talk to your instructor!
  5. If I think MAT201 is too easy, what should I do?
    • Have you had a quiz yet? You may be in for a surprise. Homework and the first couple of weeks of class can be misleading since the first few topics are not too difficult and the homework problems are quite routine compared to the exam questions.  Try a sample quiz or midterm for this course.  Remember -- don't just read the question.  See if you can produce correct solutions to most of the problems in the allotted time.
    • 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 MAT203 or MAT215 instead.
  6. I need both MAT104 and MAT201 for my major. After checking the math placement information, I think MAT201 is probably the right course for me, but I don't qualify for AP credit for MAT104. Do I have to take MAT104 or can I sign up for MAT201 instead?
    • Be cautious. Students often underestimate the difficulty of MAT104 and of MAT201 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 MAT201, you can sign up for it. If you pass MAT201 in your freshman year, you will automatically receive AP credit for MAT104. Just be ready to re-consider and switch down to MAT104 early on if necessary since there is very little overlap with the first half of MAT201 and the material in MAT104.
  7. If I want to switch courses, what should I do?

  8. I would like to switch sections within MAT201, what should I do?
    • If you have a time conflict, there is usually 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. Consult the departmental drop/add/swap procedures.  But don't delay -- If you are going to switch, do it soon!
  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.   Check out our summer course approval procedures.
  10. I have more questions that are not answered here. What should I do?
    • First, check the undergraduate home page for more information about how our courses work in general and about who to contact if you need to discuss your situation with someone from the math department.  Also: representatives from the math department will be available at freshman registration and at the academic expo in September.