"ma122 — Series and Multivariable Calculus"
ma122 — Series and Multivariable Calculus Fall, 2009 Fernando Q. Gouvêa As our ofﬁcial title suggests, this course focuses on two important topics within Calculus. But they will have unequal status in the course. Our dominant concern will be to extend the ideas you learned in your ﬁrst calculus course to the multi- variable setting. In order to do that, however, we will frequently have to rethink the ideas in single-variable calculus, ﬁnding the right way to understand them, the point of view that can be generalized to several variables. Along the way, we will have the chance to add to the single-variable theory as well, by exploring the theory of series. The most important aspect if this is the representation of functions as inﬁnite sums of simpler functions. This is one of the fundamental methods of applied mathematics, and we will focus mostly on those aspects that are most useful. In fact, all the while we will be concentrating on those ideas that have actually been useful in the sciences. We won’t, however, merely focus on recipes for doing things: we want to understand the Calculus, and not just to know how to use it. In particular, I expect you to develop the ability to use your understanding of the theory to apply it in new practical situations. Enough chatter; here’s the practical stuff: Where to ﬁnd me: My ofﬁce is Mudd 412; my phone extension is 5836; my email address is firstname.lastname@example.org. If you need to reach me when I’m not in my ofﬁce, email is the best method. If you prefer, feel free to call and leave a message — but sometimes I take a while to notice the little red light. In any case, see the note on email below. Ofﬁce hours: I’ve separated the following times as ofﬁce hours: • Mondays, 11:00–12:00 • Tuesdays, 1:00-2:00 and 3:00–4:00 • Fridays, 11:00-12:00 and 3:00–4:00 Since I am teaching three courses this semester, you may arrive at my ofﬁce to ﬁnd me busy with another student. Please be patient; I’ll try to make sure that everyone gets a chance to see me. If you can’t come at any of these times, please call and make an appointment. You may ﬁnd me in my ofﬁce at other times, but it is only at the times above that I can guarantee that I’ll be available. I am usually not on campus on Thursdays. Living on Earth may be expensive, but it includes an annual free trip around the Sun. 2 ma122, Gouvêa, Fall 2005 You are encouraged to come see me. It is part of your education, and one of your privileges as a Colby student. How the class will be organized: I will be trying to run the course using a mixture of lectures and discussion and in-class activity. You will often be asked to work in class in a group with others. At any time (and especially in classes where I’m doing most of the talking) you should feel free to interrupt with questions and comments. I will try to prompt this participation by asking you questions too! Text: Our basic texts will be 1. Chapters 9 and 10 of Calculus, by Hughes-Hallett, Gleason, et. al. (third edition) 2. Chapters 12–16 of Multivariable Calculus, by McCallum, Hughes-Hallett, Gleason, et. al. (fourth edition) These books belong to a new generation of calculus texts which are designed to be read, and which put an emphasis on understanding rather than on mechanical proﬁciency. I will be asking you to read sections of the book as we go along, and I will expect you to gain some understanding from this reading. But don’t stop there. Colby has a library, and it contains a great many books about calculus. Some of them will be textbooks, and others are supplementary books of various kinds. Do check them out! Some of you may feel the need to get a book to help you review material from your ﬁrst Calculus course. If you own a copy of your Calculus I text, that’s the obvious thing to use. If not, I suggest that you get a copy of a book called How to Ace Calculus, by Adams, Hass, and Thompson; it’s short, practical, and will probably get the job done. (There’s also a How to Ace the Rest of Calculus, which covers the material in this course. It’s worth a look too.) Some students ﬁnd that Student Solution Guides and Study Guides are helpful. These do exist for our textbook. The Student Solution Guide, for example, includes full solutions (not just answers) for the odd-numbered problems in the book. If you think something like this would help, check with the bookstore. Technology: Calculators and computers are becoming ever more useful to people who need to use mathematics, and it is important to learn how to use these tools. For this class, I will often be using a calculator and will also often ask you to do computations and/or to graph functions on a calculator. If you do not own a graphing calculator, you might want to consider getting one. If you do own one, learn to use it! In addition to using a graphing calculator, I will occasionally make use of a very powerful computer program called Mathematica. This is rather hard to use, but if you are interested in computing and are willing to put in the effort you might want to investigate this program on the computers in the Math&CS Computer Lab and We have enough youth — how about a fountain of SMART? 3 ma122, Gouvêa, Fall 2005 in the Olin Lab. People that use mathematics in their work in the sciences or in economics are more likely to use computer tools than calculators, so it makes sense to begin to learn how to use these tools. email: Email has become a fundamental communication tool, and we will be using it in our course. Your ﬁrst assignment will involve sending me an email message, and I will occasionally use email to communicate with you either individually or as a class. Prerequisites: This is nominally a “second-semester calculus” class, some of you may be worried about what I will be assuming you already know. (Of course, if you’ve taken ma121 at Colby, you should be OK.) I will assume that you know the basics about limits, continuity, derivatives, and integrals of functions of one variable. I will expect you to be familiar with the logarithm and the exponential function in addition to the basic trigonometric functions. As topics arise during the semester, I can offer occasional quick reviews of the necessary background material, but if you need extensive review you’ll need to work on your own. If you have doubts about your background please come talk to me. Quizzes: Approximately every other Friday I will give you a short in-class quiz. These are intended simply to give you (and me) a reference point about how well you are absorbing the material. These will not be announced. Since these are partially a diagnostic tool, I will discard the worst of your quiz scores when I work out your grade. Homework: One can’t learn mathematics without doing mathematics, and doing mathematics, in the context of the calculus, consists in solving non-trivial problems. On the other hand, learning a new subject often requires doing a number of “ﬁve ﬁnger exercises”: relatively simple problems that basically drill you on what you have just learned. The odd-numbered problems in the book, which have answers in the back, are a good way to provide yourself with this kind of practice. You will receive a weekly homework assignment every Tuesday, and it will be due the following Tuesday. These assignments will often contain problems that require a little creativity to solve. It is by solving these types of problems that you will really begin to understand what the mathematics is about, and it is also this kind of problem that you will meet in the “real world” when it comes time to use your knowledge. So, while these problems will be difﬁcult, solving them will be worth the effort. Since these assignments are difﬁcult, I will separate some class time to discuss homework problems. We will do this on Fridays. This means you should make sure to take a ﬁrst stab at the problems before the Friday class so that you know what questions you need to ask. Please don’t leave your assignments for the last day, because you will probably not be able to do them in one day. The problems I’ll assign will require time for Thesaurus: ancient reptile with an excellent vocabulary. 4 ma122, Gouvêa, Fall 2005 their solution, and you should plan to put in that time. (Read this paragraph again. You have been warned!) Since homework is so important, it will have a relatively heavy weight in your grade. On the other hand, since homework is also the place to make mistakes and learn from them, I will discard the worst homework score from each half of the course. Outside of class: You should expect to have to study two to three hours outside of class for every hour you spend in class. The best use of this time is to spend a part of it working with a group of friends. This will add a social dimension to your study, and will also help you resolve difﬁculties by using the differing strengths of people in your group. The ideal sequence seems to be to work the problems on your own ﬁrst, then to work in a group to resolve any difﬁculties and reconcile any differences (when two people in the group get different answers, it can be a great learning opportunity). An important resource to help you with studying and homework is Calculus After Hours, a Calculus lab where you can ask questions, work on homework, and generally learn with others. Watch for the handout with more information about this program. While you are free to work on homework with a group, you should write your ﬁnal draft by yourself, so that it reﬂects your state of understanding of the mate- rial. In other words, getting help from others to understand, then writing a solu- tion that reﬂects that understanding is completely acceptable, and even desirable. Copying someone else’s solution without understanding is not acceptable, and will be treated as a form of academic dishonesty. When you write up your solutions, be careful: use good grammar and full sentences, explain your reasoning, justify your approach, make it readable. Writing well requires real understanding; by requiring yourself to write well you will make sure that you really do understand what is go- ing on. If you have questions about what kind of writing I’ll be looking for, please feel free to ask. Writing Assignments: There will be one or two writing assignments during the semester. These will probably involve working with a group of others to ﬁnd some- thing out, then writing up the results. I will provide more information on these soon. Tests: We will have three tests: two midterm tests and a ﬁnal exam. The midterms will be on October 7 and November 17. In order to allow you more time and to let two of my courses take them together, the midterms will be in the evening: from 7:00 to 8:30 pm, in Olin 1. Please mark your calendars. All of the tests will consist of problems similar in style (but not in difﬁculty!) to the problems assigned for homework. Cheating: While you are encouraged to cooperate with others in class and in your If bankers can count, how come they have eight windows and only four tellers? 5 ma122, Gouvêa, Fall 2005 homework assignments, your other work is expected to be your own. In particular, you are forbidden to get or give help during quizzes and tests. Please refer to the Colby policy on academic honesty as stated in the College Catalogue. Attendance: Class attendance is required. Should you need to miss a class, please talk to me in advance to see if your absence can be excused. Missing too many classes will result in academic warnings, grade penalties, and eventually dismissal from the class. Please note: If you miss a Friday class during which a quiz was given, you will not be offered an opportunity to make up that grade. Grading: Your grade will be computed as follows: quizzes 10% writing assignments 15% homework 20% midterm tests 25% ﬁnal exam 30% An outline: Rather than giving you a week-by-week schedule (which I’d end up not following), here’s an outline of what I’m planning to do, in roughly the order we’ll want to do it. As we go along, I’ll point you to the relevant bits in the textbook. 1. Functions of one and several variables. 2. Linear functions, linear approximations. 3. Derivatives as linear approximations, differential. 4. Geometry of R2 and R3 ; vectors. 5. Differentials and directional derivatives, the gradient. 6. Improving the approximation: Taylor polynomials in one and several vari- ables. 7. Optimization, constrained optimization. 8. Sums and integrals, inﬁnite series. 9. Integration of functions of several variables. 10. If time allows, other topics. The careful application of terror is also a form of communication. 6 ma122, Gouvêa, Fall 2005 About Me Students often wonder about their professors. You can ﬁnd out more about me by looking at my home page, at http://www.colby.edu/˜fqgouvea But, just for fun, here are some factoids: • I was born in Brazil. • I have been at Colby since 1991. • I have two sons. One has a PhD in Political Science and works at a non-proﬁt consulting ﬁrm. The other is a PhD student in Classics at the University of Chicago. • My main research interests are in number theory and in the history of math- ematics. • I’m also interested in lots of other things. (The web site has a partial list.) • I’m a Christian; currently I am a member of the Lutheran Church of the Resurrection in Waterville. • I sing, but only in church. • According to philosopher Richard Rorty, I am “frightening, dangerous, vi- cious. . . ” • I have had a beard since 1980, but it has only been gray for the last few years. • I vote. • My house is the best place to eat in Waterville, but it’s only open by invitation. If you want to have a taste, come to the Mathematics Colloquia. • I own a rather excitable dog called Jelly Bean. She’s about one year old and is lots of fun. • On my bookshelves, you’ll ﬁnd many books by (among others): Gene Wolfe, D. J. Enright, Dorothy L. Sayers, Jaroslav Pelikan, Reviel Netz, Barry Mazur, Gordon Fee, Robert Jenson, G. K. Chesterton, Walt Kelly, Mark Helprin, David B. Hart, Niccolò Tartaglia. 93.5% of all statistics are made up.