Course Information and Syllabus
How much calculus is enough? You've all been taking calculus for several semesters by now (several years for a number of you), so why haven't you learned it all by now? Why can't you just stop right now? There's no need to take this class, you're all calculus experts, you know how to take limits, differentiate functions, you know your Fundamental Theorem of Calculus and you can integrate faster than a computer (well, maybe not quite…). But what kinds of functions can you deal with at this point? Sure, you can work with functions like F(x) = 31x12 + 5e2x, or F(x) = 27sin(3x), but how many times in real life do things rely on just a single variable "x"? Why is it that so many times in life the answer to seemingly simple questions is "it depends?" Usually this is because there are many factors to be accounted for, not just a single one. For instance the distance a baseball travels after it's been hit depends on several variables - how hard it was hit and the angle of elevation of its initial trajectory. But then there's also wind speed to factor in and altitude (to take account of possible air density variations) as well as some more factors. So if we wanted to write down a function that modeled the distance a baseball will travel after being hit, then we need to be able to deal with more than one input, i.e. more than one independent variable. This means we are not dealing with a single variable function anymore, and suddenly our old tricks from single variable calculus fail us if we want to do so much as to figure out the impact of a change of wind speed versus a change in trajectory angle, or to account for differences in hitting baseballs in Boston versus in Denver (the altitude does in fact have a noticeable impact).
On the other hand, what can you do with functions that appear to give several outputs at once? For instance how can we work with a function that gives both the height and weight of a person at different times in their life? Here, although there is just one input variable, time, the output is given as two numbers, not just one. Now what? How do we graph such a function? Is there such a thing as differentiation in this case? What about a function that is supposed to track the position of the space shuttle after a launch? Now we'll need three coordinates, to deal with the fact that we live in three-dimensional space.
Aha! This is the reason to take Multivariable Calculus! You don't live in a one-dimensional world - you are all multi-dimensional people, and you need your calculus skills to reflect that reality! What you need now is the knowledge of how to take your single variable calculus expertise and broaden it to take on the multidimensional world we live in. Without Multivariable Calculus, you are stuck in Flatland (or worse, Lineland!).
So… Welcome to Multivariable Calculus! During this semester we will build on all of the calculus fundamentals that you've worked on over the past several semesters, to give your calculus the power to take on any dimension. This will open up your ability to work with the wide variety of situations that you might encounter in applying calculus to economics, biology, physics, chemistry, etc. Your functions will know no dimensional limits - you'll be able to work with functions with 100 variables if you so choose by the end of this semester. To take your calculus skills to higher dimensions you will learn about:
Finally, remember to enjoy yourself this semester! Often students take math because they feel that they have to take it, either as a requirement for a concentration, or because someone else advised them to. Although many of you might think you know exactly what you will be doing with your life, it is rarely that simple. You might think that your current plans do not require you to have to know a terrific amount of math, but who knows, you might end up working in a different area from the one you expected one day and you’ll be thankful to have learned all the math you are about to learn. My wife majored in history, took the minimum number of math courses required as an undergraduate, and then went on to became a molecular biologist at MIT. She is now extremely thankful for the math that she did learn, as it quite useful for much of the biological modeling she does, and yet at the same time she regrets that she didn't take time to study more math while she was in college. Who knows, after 21a you might all want to become mathematicians!
General Course Information:
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 so many 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 in this topic selection. If you are planning on concentrating in BioChemical Sciences, then you are encouraged to enroll in the BioChem section. This section would also be especially useful for those concentrating in economics or social sciences. If you can't schedule the BioChem section into your personal class schedule, however, don’t worry, as you will benefit from taking 21a in a regular section in any case.
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: Go to the Math Department's home page, http://www.math.harvard.edu and click on the "Section for Courses" link in the upper right corner of the page, then follow the instructions listed there. If you are having problems sectioning then please contact Susan Milano via email at milano@math.harvard.edu.
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 better to work on math in smaller groups than in one huge, impersonal lecture setting. This gives you lots of opportunity to ask questions in class, interact with the teacher, and get to know the other students in your class. Make sure you take advantage of this set-up, 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! 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 stay involved during your class.
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 go to classes alongside you, so you can get to know them well during the semester. The problem sessions are an important part of the course and will be devoted mainly to working problems and reviewing material – something which won’t happen during the regular classtimes. Even if you find you're not having difficulty doing the homework problems, you should still make a habit of going to these weekly problem sessions. A schedule of all of the problem sessions will be posted on our 21a course web page after classes start, so that if you have a scheduling conflict with your particular class section's problem session, then you can still attend another problem session.
Math Question Center: In addition to class, problem sessions, and office hours, the math department operates a Math Question Center in Loker (basement of Memorial Hall) 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 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 exactly the same math problems you're working on now – the point is to learn the material in such a way that you are prepared to use your general math knowledge in the future.
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 (TF) or your 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 2 lowest homework grades. Because of this, please do not try to harass your CA into accepting a late homework - the homework policy is a coursewide 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 different from problems discussed in class. This is not the result of your teacher not covering all the material in class. There will be times when we ask you to read ahead in the textbook so that you are better prepared to get more out of the class. In such cases, you’ll be asked to do a few simple problems based on your reading. Reading math on your own is an important skill. We want you to get used to thinking about the math you're working with 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 (2nd edition). It’s available at the Harvard Coop, but you can purchase this book anywhere you’d like, such as online if you find that to be more convenient. For the BioChem sections, there will be a separate text and/or handouts for the extra sections on probability and statistics which will be available later in the semester.
Technology: in general, technology is a good thing, but as with everything in life, 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 previously have taken 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 short Mathematica projects or "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 15th, 7:00 - 9:00pm, Science Center Hall B
Second Test: Tuesday, April 19th, 7:00 - 9:00 pm, Science Center Hall B
Final Exam: date and place to be announced later in the semester by Registrar’s Office
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 a 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.
Math 21a - Multivariable Calculus - Tentative weekly schedule for Spring 2005
First Topic: Vectors, Multidimensional Spaces and Functions
Feb. 7 (8) 9 (10) 11: (MWF class dates are
in bold, T/TH dates are in parentheses)
Introduction to semester and in Stewart, §9.1 Three-Dimensional
Coordinate Systems, §9.2 Vectors, §9.3 The Dot Product
Feb. 14 (15) 16 (17) 18:
§9.4 The Cross Product, §9.5 Equations of Lines and Planes,
§9.6 Functions and Surfaces
Feb. (22) 23 (24) 25 28: (Note Presidents' Day
holiday on Monday, Feb. 21st - no classes)
§9.7 Cylindrical and Spherical Coordinates, §10.1 Vector
Functions and Space Curves, §10.2 Derivatives and Integrals of Vector
Functions, §10.3 Arc Length
Begin Second Topic: Partial Differentiation
March (1) 2 (3) 4 7:
§10.5 Parametric Surfaces, §11.1 Functions of Several Variables,
§11.2 Limits and Continuity, §11.3 Partial Derivatives
March (8) 9 (10) 11 14:
§11.4 Tangent Planes and Linear Approximations, §11.5 The
Multivariable Chain Rule, §11.6 Directional Derivatives and
the Gradient Vector
First Midterm – Tuesday, March 15th, 7-9pm, Science Center Hall B
Third Topic: Partial Differentiation Equations (PDE’s)
March (15) 16 (17) 18 21:
Introduction to Partial Differential Equations Part I (covered in handouts,
available online),
Partial Differential Equations, Parts II and III
March (22) 23 (24) 25 April 4th:
Begin Fourth Topic: Multiple Integrals
§11.7 Maximum and Minimum Values, §11.8 Lagrange Multipliers (Physics section: tentative switch of sections §11.7 and §11.8 for sections §13.1 Vector Fields and §13.2 Line Integrals), §12.1 Introduction to Double Integrals
Spring Break!! March 26th through April 3rd
April (5) 6 (7) 8 11:
§12.2 through §12.4 Double Integrals over various regions and Iterated Integrals, §12.6 Surface Area
April (12) 13 (14) 15 18:
Begin Fifth Topic: Vector Calculus (note, for the last three
weeks of the semester, the BioChem sectins will not cover the rest of Chapter
13, but will instead include an introduction to probability and statistics)
§12.7 and §12.8 Triple Integrals, §13.1 Vector Fields, §13.2 Line Integrals
Second Midterm – Tuesday, April 19th, 7- 9pm, Science Center Hall B
April (19) 20 (21) 22 25:
§13.3 Fundamental Theorem for Line Integrals (Physics
section: will cover §11.7 and §11.8 on Minimum,
Maximums and Lagrange Multipliers if not covered earlier), §13.4 Green's
Theorem, §13.5 Curl and Divergence
April (26) 27 (28) 29 May 2:
§13.6 Surface Integrals, §13.7 Stokes' Theorem, §13.8
The Divergence Theorem,
May (3) 4 (5) 6:
§13.9 Theorem Summary, extra class time for catch up if necessary
Reading Period, May 7th – 17th, Final Exam date and time to be announced by registrar
Coursewide review sections, office hours TBA