Course Description:

The title of this course is Advanced Calculus, but a good alternate name might be Honors Multivariable Calculus. From a distance, the syllabus will look pretty much like that of Math 121 (Calculus III), but topics will be covered in greater depth and somewhat more attention to mathematical rigor and detail. Consequently, Math 216 will be somewhat more theoretical than Math 121, but applications will also be emphasized and the course is suitable both for math majors and non-majors.

The goal is to study how the ideas of calculus may be applied to functions f : R_m &rarr R_n; indeed, this is precisely what "multivariable calculus" means. Specific topics include: partial derivatives, directional derivatives, graphs of level curves and surfaces, gradients, Taylor approximations, max-min problems, Lagrange multipliers, multiple integrals, vector fields, line integrals, surface integrals, curl, divergence, Green's theorem, Stokes’ theorem, Gauss's theorem, and a brief introduction to differential forms. From time to time, we will also look carefully at questions like convergence, continuity, and the existence of certain limits and integrals. These concepts are fundamental in analysis, the branch of mathematics that follows this course.

There is much to be done! To be honest, the syllabus includes more topics than can be covered thoroughly in one semester, and we will need to rush occasionally. Much of the routine drill will be your responsibility, with the homework as a guide and help sessions as a resource. As in Math 215a, there will be `regular’ and `special’ homework problems. I guarantee that if you do the `regular’ homework conscientiously, you will (a) have a solid working knowledge of the subject matter, and (b) get a satisfactory grade for the course.

Vector Calculus (Fifth edition), by Jerrold Marsden and Anthony Tromba, (W. H. Freeman, 2003).

Three lectures per week (MWF 10:30-11:30). There will be no discussion sections, but I will have two extended office hour periods specifically for this course: Wednesday 4-6PM and one other time to be announced. Please come to these if you have questions about the course.

Weekly homework, two midterm exams (each will consist of an in-class quiz plus a take-home supplement), and a self-scheduled final. These will be weighted as follows in determining the overall course grade:

Each weekly homework assignment will consist of two parts, counted equally in determining the final homework grade.

(1) Standard problems. These are designed to give practice with the routine aspects of material covered each week in lectures. This part of the assignment will normally be posted on Fridays and due the following Friday.

(2) Special problems. Generally one additional "special problem" will be assigned each week. These will be more challenging, often covering material not directly related to the current lectures and drawing on topics from different parts of the course.

Special problems have an extended due date: you should try to do them each week along with the standard problems, but they are officially due four weeks after they are assigned. You may resubmit problems until they are done correctly, within the four-week time limit. I will return them with hints, and record only the grade for the final version. I will accept interim submissions on Fridays (only), and these will be returned the following Monday.

I hope (and expect) that you will find the special problems interesting and perhaps even fun.

Collaboration:

I encourage collaboration on the homework, both standard and special problems. Indeed, I expect you will learn a great deal about this course from each other. It will be to your advantage to form study groups, and many students facilitate this by working together in the Math Question Center (Sunday through Thursday 7-9PM, Hilles 012). This semester I will be in the MQC on Wednesday nights.

Collaboration on homework naturally raises the question, "how much is OK?" I expect that you will share ideas, and perhaps work together at a blackboard, but eventually each student must write up his/her work independently, without reference to another students work or to written work that has been produced jointly. Verbatim copying from another person's paper or blackboard work is definitely "not OK". The safest approach is to write up your final solutions in a different place, and on a fresh sheet of paper.

It's important not to misunderstand these guidelines, so please ask me if you have any questions. You might also want to refer to the department's published guidelines on homework collaboration, which are available on the department website. Collaboration on tests is never permitted. All inquiries about problems on the tests should be directed to me.

Honor code principle:

You must never present others' work as your own. If you have used other students' work in the preparation of homework you must acknowledge it. If you obtain solutions to assigned problems (on homework or take-home tests) from sources other than the textbook or class notes, you must acknowledge such sources. This especially applies to material obtained electronically, e.g., on the web. 

If there are questions about honor code issues, you should seek clarification and guidance from me.

One more comment about collaboration:

Our guidelines about collaboration are more than just about the honor code. If you depend too heavily on help from others, you may be seriously compromised when it comes to the exams, which must be done without collaboration. Please monitor your own level of understanding, and avoid this trap.

Electronic Resources:

(1) Blackboard will be the primary source for course materials. You are responsible for getting the weekly homework assignments from Blackboard. I will also post solutions to homework and tests, handouts, links to course-related websites, sample Mathematica notebooks, and other items that might be quite important (e.g. corrections to and hints to homework assignments, reminders of test dates, etc.). Please check Blackboard regularly.

(2) Mathematica is a powerful software system for doing symbolic calculation and displaying graphics, and we will use it extensively. Since it may also be useful in other courses, it is worthwhile becoming adept with Mathematica at an early stage of your Haverford career. If you are not familiar with Mathematica, please acquaint yourself with it as soon as possible, e.g., by coming to the Math Question Center. This semester you will especially want to acquaint yourself with the Plot, Plot3D, ParametricPlot, ParametricPlot3D, ContourPlot, and ContourPlot3D commands.

This year Haverford has a site license that permits unlimited use by students, as long as they are on the campus network. Complete details on how to download Mathematica and install it on your own computer can be found at http://www.haverford.edu/math/Mathematica/. The installation process requires obtaining a password from the publishers. If you have any difficulty doing this, please come to the MQC and/or contact David Lippel (Math Department lab support person) for assistance.