Course MAT203 in Fall 2010

Advanced Multivariable Calculus

Covers largely the same mathematical topics as MAT201 (namely, limits, continuity, differentiation and integration for functions of several variables), but aimed at future physicists or students interested in applied math.  Just like MAT201, this course introduces mathematical methods needed to understand real world questions involving quantities changing over time in 3-space but requires greater independence and maturity. Exams test for thorough conceptual understanding as well as computational fluency. In this course we assume that students will not need much help to master standard, straightforward calculations, allowing the instructor to emphasize more subtle aspects of the definitions and important exceptional cases that come up in applications. Offered Fall semester only.


The course covers most of the material in the textbook Vector Calculus by J. Marsden and A. Tromba.

The first half of the course quickly reviews vectors, lines and planes in space and standard algebraic tools like matrices, cross-products, dot products and determinants. The notions of limit, continuity and differentiability are generalized for vector-valued functions with rigorous definitions and proofs. We study the graphs and level sets of these functions and their extrema (maxima/minima) using Taylor's theorem and Lagrange multipliers.

Integration in 2- and 3-dimensional space as well as line and surface integrals, arc length, surface area, volume and the integration theorems (of Gauss, Green and Stokes) generalizing the Fundamental Theorem of Calculus are the focus of the second half of the course. We work extensively with curves, surfaces and vector fields in space, with special attention to problems (like those governed by Maxwell's celebrated equations) of fundamental interest in physics and engineering. Although historically the underlying mathematics here was developed for physics, this course provides essential tools for engineering, economics and other sciences as well.

Description of Classes

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

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 assistants 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 (before the corresponding lectures), 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.

  • Most students will continue with MAT204 (Advanced Linear Algebra), offered in the Spring semester only. Although MAT203 and MAT204 can be taken in either order, we recommend you take MAT203 first.
  • MAT201 gives a very solid treatment of the same material, sufficient for most purposes.  This course is intended for students with a very strong interest in mathematics, especially as applied to other fields such as physics or engineering and it requires a greater commitment of time and energy.  If you do not expect to take any 300-level courses in the math department, then MAT201 is a good choice.
  • MAT203 is graded with a more generous curve than is MAT201 to remove considerations of grades as a deciding factor in the choice between these two courses.
Who Takes This Course
  • The typical student in this course is strongly interested in physics as a major, but the course is suitable for anyone with a strong background in one-variable calculus (e.g. 5 on the BC calculus exam and 750 or more on math SAT) and a serious interest in how mathematics can be applied to other fields such as economics, science or engineering.
  • Naturally, since math and physics have much in common, many of the students in MAT203 are also considering math as a major. If you think you are more likely to major in math rather than physics, we recommend that you take MAT215/217/350 instead.  Don't worry if you cannot decide for sure, or if you change your mind. You can still decide to be a math major if you take MAT203.
  • Students who took AB calculus only should take MAT104 instead.  Such students are not prepared for this course.  (In rare cases, for students with an AB score of 5, a math SAT above 750 and a very strong interest in math as a major, we recommend MAT215. If that describes you, please talk to the placement officer at freshman registration.)
  • Most mathematically inclined economics majors (or others interested in a finance certificate) take MAT201/202 instead.  If you cannot imagine yourself as a math or physics major, then this course is probably not for you.
  • AB COS majors are not required to take multivariable calculus. They need only linear algebra (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. Even more than in MAT201, it will be assumed that routine single-variable calculus is second-nature to you, and the course will focus mostly on developing a deeper, more conceptually sophisticated grasp of the main ideas of calculus as you generalize them to higher dimensions.
  • If you got a 5 on the BC calculus exam and you have a math SAT score of at least 750, you can sign up for MAT203 instead of MAT201. But be warned! You will likely need to work very hard. Be ready to get extra help and plan to switch down to MAT201 if need be. If you plan to be a physics major, MAT203 is the recommended course; so start here, but be realistic! You will learn a lot of multivariable calculus in MAT201 if you decide to take that instead.
  • You absolutely must have a serious interest in thinking rigorously about problems involving space and time. If you were happy with the way calculus was taught in high school, you may well find that you prefer the style of MAT201 instead. For most engineering majors, MAT201 is also an excellent choice which will give you the background you need for your upper division courses.
  • MAT203 covers the same mathematics as MAT201, but more deeply and with more abstraction; most students find MAT203 to be more challenging. For most students who are not planning to major in math or physics, it is mostly a matter of taste whether this extra challenge is worthwhile. It may take a few days of going to both courses to figure out which version appeals to you more. To encourage students who find the challenge of MAT203 appealing, we grade this course on a somewhat more generous curve than MAT201 with the intention that a qualified student who chooses MAT203 will be able to get a grade comparable to what he/she would have received by playing it safe and taking MAT201 instead.
  • If your background is weak or rusty, you should not take this course.  Consider MAT201 instead or take MAT104 to get an excellent review of the knowledge assumed in this course.

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

Sample Material

Working problems from these sample quizzes from Spring 2011 can give you a good general 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!

Solutions to these quizzes are not available. Take the course to find the answers!

  1. 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.  Instructors will assume that students in this course have a strong commitment to becoming real experts in the long run, and so are willing to invest more time and energy in the course.
    • 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 already took multivariable in high school. Do I really have to take this course?
    • Many students in MAT201 and MAT203 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 multivariable calculus even though some material will be review.  The sample quizzes and exams posted for this course should give you a pretty clear idea of whether you already know the material taught in this class. 
    • If you can't solve the problems on the sample exams for this course, try the sample final for MAT201. 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?  If not, you really should take a multivariable course here at Princeton.  It is then a matter of taste and interest which multivariable course you should choose.
    • In rare cases, the placement officer will decide that your prior work is indeed equivalent to MAT201 (or maybe even MAT203) 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.
  3. Can I take MAT203 and MAT204 in the same semester?
    • No, they are never offered in the same semester. You can however take them in either order, and you can substitute MAT201 for MAT203 or MAT202 for MAT204 to satisfy most programs or departments that require these courses. Both MAT201 and MAT202 are serious courses and, for most students, it really doesn't matter which version you choose.
  4. If I think MAT203 is too hard, what should I do?
    • Of course you can consider switching to MAT201
    • The BC calculus exam does not test your knowledge of single variable calculus very thoroughly, so it is even possible that you would be better off in MAT104 where you can get a thorough review of all the math assumed in multivariable calculus.  Try an old final exam  from MAT104 to see if you can do most of the problems with confidence.  If not, starting in MAT104 might really pay off in the long run and it won't throw you off-track if you are a BSE student.
  5. If I think MAT203 is too easy, what should I do?
    • Have you had a quiz yet? You may be in for a surprise. 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 in the meantime.
    • If you are also taking PHY103 or PHY105, you might consider just enjoying the fact that MAT203 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 MAT215 of maybe even MAT216, especially if you are interested in math or physics as a major.  Talk to your instructor or the placement officer for advice. 
  6. If I want to switch courses, what should I do?
    • Talk with your MAT203 teacher. You may also want to talk with the teacher in charge of the other course. Or consult Jennifer Johnson, the person who supervises most of the lower division courses for the math department.  During the drop/add period it is quite simple to switch.  It can also be arranged after the official drop/add period ends, but the sooner you can decide the better!
  7. I would like to switch sections within MAT203. What should I do?
    • If you have a time conflict, there should be 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 department's drop/add procedures.
  8. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university over the summer?
    • Yes, but it may be difficult to find an equivalent course. Many multivariable courses at other universities cover only about half of 201 and the equivalent of 203 will likely be an upper division course for math majors at many universities.  Upper division courses are rarely offered in summer sessions.   If you find a possible candidate course, you need to apply for summer course approval from the math department.
  9. 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 departmental open houses during registration week.