Prerequisites: Before beginning this class you are expected to have had the equivalent of a second semester course in calculus. You should have experience with the fundamental theorem of calculus, as well as integration techniques. If some time has passed since you've worked with these topics, then it is up to you to review the appropriate material - there will be very little time to do much review in class during the semester as there are a number of new topics to cover. Although it is expected that you will take the initiative to review any material you need to on your own, you should always feel free to talk to your section leader or course assistant to ask for help.

Sectioning: There are a several variants to the basic Math 21a course. There are essentially three types of sections, known as Regular, Physics and BioChem (if a section doesn't have a special label, then it is automatically a Regular section). All sections cover functions of several variables, differentiation and integration of such functions, parametric curves and surfaces, optimization, vector fields, linear approximations and topics in partial differential equations.

Towards the end of the semester there is a fork in the road. At this point the Regular and Physics sections will study line and surface integrals, and multivariable generalizations of the Fundamental Theorem of Calculus (Green's theorem, the divergence theorem and Stokes' theorem).

Instead of covering these topics, the BioChem section will cover various introductory topics in statistics and probability. Note that the BioChem section does not require any Biology or Chemistry background, this section simply covers a different set of math topics from the Regular sections, but it is accessible to anyone who is interested. If you are planning on concentrating in BioChemical Sciences, then you are strongly encouraged to enroll in the BioChem section. This section would also be especially useful for those concentrating in economics or social sciences.

The Physics section will use examples and some homework problems drawn from Physics to illustrate topics covered in the Regular section. The Physics section is designed for potential Physics concentrators. If you are planning on concentrating in Physics, however, you will still cover the exact same mathematics in either a Regular or a Physics section, so if you are unable to sign up for the Physics section because of scheduling conflicts, then you should just sign up for a Regular section.

Note, you can't go wrong in a Regular section, so if you aren't sure of your concentration, then take a Regular section.

To section: if you have an email account, log on to the Harvard computer system, then type "telnet ulam.fas.harvard.edu" instead of "pine," then when prompted just type "section." Or if you'd prefer, you can always go to the Math Department's home page, http://www.math.harvard.edu and click on the sectioning link in the upper right corner of the page. If you are having problems sectioning then please contact Susan Milano via email at milano@math.harvard.edu by noon on Thursday.

Classes, Problem Sessions and Course Assistants: Math 21a is taught in sections which meet three hours per week. The philosophy behind the sections is that it is far better to work on math in smaller groups than in one huge, impersonal lecture setting. This gives you a better opportunity to ask questions in class, and interact with the teacher. Make sure you take advantage of this arrangement, and try to get the most out of being in these smaller groups. Any questions you ask in class will likely be ones that other students will be having as well, so get over any hesitation you might have, and ask questions as the material is presented. Remember, the class is being held for you to learn the material, not just for you to copy notes off of the blackboard, so be sure to take advantage of the class setting.

You will also be attending a problem session led once per week by a Course Assistant (CA). Course Assistants grade homework and hold weekly problem sessions. They also attend classes alongside you, so you will get to know them well during the semester. The problem sessions are an integral part of the course and will be devoted mainly to working problems and reviewing material. You are strongly urged to attend these problem sessions. Even if you find you're not having difficulty doing the homework problems, you should still make a habit of going to these sessions. A schedule of all of the problem sessions will be posted in the Science Center and on our 21a course web page, so that if you have a scheduling conflict with your particular class section's problem session, then you can still attend another problem session.

Question Center: In addition to class, problem sessions, and office hours, the math department operates a Math Question Center in Loker on Sunday through Thursday evenings from 8pm to 10pm. The Question Center is staffed by Course Assistants as well as by graduate students and other teaching staff. This is a good place to meet with other students in your class to do homework. You should feel free to drop in any time you want a bit of help, or if you just want to solidify your basic math understanding by doing some review problems.

Homework: There is no question but that the best way to learn math is by doing math. Homework exercises are an essential part of any math course. I know from personal experience that if you just go to a math class and watch the teacher work problems, but don't actually try doing any problems on your own, then there is very little chance you will really learn what is going on. It is also very unlikely that you will do well on math exams without working through homework problems ahead of time! While doing homework, don't just crank through computations and write down answers - think about the problems posed, your strategies, the meaning of your computations, and the answers you get. The main point is not to come up with specific answers to the specific problems you're working on, but to develop an understanding of what you're doing so that you can apply your reasoning to a wide range of similar situations. It is very unlikely that later on in life you will see the exact same math problems you're working on now - so learn the material in such a way that you are prepared to use your general math knowledge in the future, not just how to apply particular formulas for very specific problems.

We encourage you to form study groups with other students in the class so that you can discuss your work with each other. Your Section Leader or CA will provide names and contact information for everyone in your section to help you out with this. Although we encourage you to work together with your classmates, all work submitted must be written up individually. Make sure that even if you do work in groups, that you come away with the ability to explain everything you end up writing up in your homework. You're cheating yourself of learning if you just copy down someone else's answers.

There will generally be two problem sets each week. MWF classes will typically have a problem set due on Wednesday, and another shorter one due Friday. T/Th classes will typically have a problem set due each class period. Assignments will be graded by your Course Assistant and will typically be returned at the following class meeting. We will then post solutions to the homework on the math 21a website. Check the solutions so that you can learn from your work. To make it possible to post solutions as soon as possible, and in light of the fact that getting behind in a math class is one of the most uncomfortable things you can do to yourself, homework must be turned in on time. We will make a general policy for the course of dropping your 3 lowest homework grades. Because of this, please do not try to harass your CA into accepting a late homework - the homework policy is a course-wide policy, and it would be unfair if certain CAs were lenient when others weren't. If you want to gripe about homework, please come see me instead! Although it may seem as though you're doing homework all the time at first, most students have been thankful in the end that the course was designed to keep you working throughout the semester - it means less last minute catching-up for you when you're getting ready for the tests.

There will be times when problems for homework will look a bit different from problems discussed in class. This is not the result of your teacher not covering all the material in class. The point is that we want you to get used to thinking about the math and learning how to apply it in unfamiliar settings. If you don't get used to this now, then there will be little chance that you will feel confident about applying your math understanding to general situations later on in life, and the class will not have been as useful to you as it could have been. Test and exam questions will be similar to the problems which you have worked on in class and on homework, but only up to a point - they will not simply be copies of problems you have already seen. Be prepared to spend some time thinking during tests, not just spending time busily write down formulas.

Textbook - Our main course book this semester will be Stewart's Multivariable Calculus: Concepts and Contexts, available at the Harvard Coop. For the BioChem sections, there is also a required text on probability and statistics by Rosner, Fundamentals of Biostatistics, also available at the Coop, and which we will put on reserve in Cabot. Note that although the Stewart textbook is new this semester, the Rosner textbook has been used for a number of semesters, so you should be able to find someone else who used the book previously and who would be willing to lend or sell you their copy.

Technology: in general, technology is a good thing, but as with everything else, sometimes too much of a good thing can lead to problems. With the advent of graphing calculators and mathematical software programs, such as Matlab and Mathematica it is now possible to do an amazing number of things almost instantaneously that would otherwise take hours or days to do by hand. Visualization of multidimensional surfaces and functions is clearly an important feature of this class, and computers and graphing calculators can often help greatly in this regard. Computers can help you with your multidimensional skills and instincts, however they should not be relied on to the extent that they keep you from developing your own skills. They should be used as an aid, because without a good understanding of the underlying calculus concepts, the computer will quite happily mislead you without your even knowing it.

During the semester we will get a chance to take a look at the power of such computer software by completing several Mathematica Labs. These are computer assignments that you will be able to do from your own computer (with downloaded software available through our 21a website) or on the computers in the Science Center. The assignments will give you an opportunity to witness the power of today's technology. The labs are designed to be self-learning tools, and don't assume any Mathematica knowledge on your part. More will be said about these assignments during the semester.

While we are on the topic of technology, as a general policy, you should feel free to use graphing calculators or computer software during the semester as long as they are used as tools to help you learn and explore math, and not as crutches that keep you from developing your own understanding. To the extent that the main point of the course is for you to develop confidence in your mathematical abilities independent of such tools, we will design the course so that for the most part you won't need to use a graphing calculator to do homework problems. Also, we will not allow the use of calculators on the exams as this puts people with different models of calculators at a possible disadvantage to one another, as well as ends up testing how well you can use a calculator instead of how well you've learned the basic mathematical concepts. We will make sure that the problems on the tests require minimal calculation, to allow you to spend your time demonstrating your mathematical knowledge, not your calculating ability.

Tests: There will be several opportunities during the semester for you to show off your math knowledge. The first two tests will be uniform across all of the math 21a sections. Because different topics are covered in the different sections at the end of the semester, the final will have some appropriate variation depending on whether you took the Regular, Physics, or BioChem sections. Because of the need to have everyone take tests at a common time, something which is practically impossible to do early during the day, the midterms are both scheduled in the evening. It is your responsibility to let your section leader know as soon as possible of any potential conflicts. It is also generally the case that it is your responsibility to resolve any scheduling conflicts - there are only two of these evening tests during the semester, and they should take precedence over any other obligations that you might have.

First Test: Tuesday, March 12^th, from 7:30 to 9:30 pm in Science Center auditorium D

Second Test: Tuesday, April 16^th, from 7:30 to 9:30 pm in Science Center auditorium D

Final Exam: tentatively scheduled for Thursday, May 16^th rooms TBA

Grading: The weights of the various parts of the course are as follows:

Test #1: 20% Test #2: 20% Homework: 20% Computer Assignments: 5% Final Exam: 35%

Typically we don't set absolute point value levels ahead of time (i.e. 92 and above equals A). The reason for this is to take into account the fact that the course and the tests vary somewhat from year to year, and it would be unfair to penalize the class if it turned out that scores on a particular test were lower one semester due to the nature of the test. We will indicate after each test a rough range of grade equivalence, so that you can keep track of how you are doing in the course.