First year calculus courses introduce for functions of a single variable the fundamental concepts of calculus:  limits, derivatives, and integrals.  Techniques implementing these concepts are studied and applications to one-variable situtations are explored.  In Math 233, we assume that everyone is familiar with this material and proceed to extend each of these fundamental calculus concepts to functions of two or more variables.  To a far greater extent than in one variable calculus, a good working knowledge of geometry is needed.

         Ultimately, multi-variable calculus plays an important role in the study of such extremely complicated phenomena as the behaviour of the U.S. economy, gas dynamics, and brain models.  In these and other examples, the pheomonon under study depends on a huge number of variables, e.g., the buying power of individuals, variables governing corporate activity and govenmental policies;  pressure, volume, and temperature as well as external factors and boundary constraints; molecular configurations, extent of contamination with foreign substances, and status of neural networks.   One cannot simply jump into such difficult applications.  It is the job of scientists to pass from acuumulated data on phenomena to the formation of simplified models involving reasonably good approximation to the phenomena in question by one or  more functions f( x₁, x₂, ..., x_n) of a limited number of variables x₁, x₂, ..., x_n deemed to be fundamental.    Typically, the fundamental variables are obliged to satisfy certain constraints, e.g. scientifically meaningful ranges for their values and equations reflecting conservation laws and other consistency relationships. Once the models have been constructed and the constraints declared, the techniques of Math 233 come into play.

         Crucial to the development and understanding of multivariable calculus is the geometry of  surfaces in n-dimensional space.  Fortuanately, nearly all of the key geometric ideas arise in the study of curves and surfaces in  three dimensional space.  Math 233 therefore begins with a detailed study of lines, planes, curves, and surfaces in  R³, i.e., in  three dimensional Euclidean space. Vectors are introduced and used both for calculations and for concise notational descriptions of geometric objects.  We are then ready to move on to the study of the partial derivatives of a function f of n variables:  this involves fixing all but one of the variables and seeing how f changes as the remaining variable changes, in essence taking the ordinary derivative of f with respect to the variable allowed to change.  Instead of the graph of f being a curve in R² as in one variable calculus with derivatives defining tangent lines, the graph of f is now an n-dimensional surface in Rⁿ⁺¹ which can be closely approximated by n-dimensional tangent hyperplanes constructed using the family of  partial derivatives of  at each point.  As we will see, once one understands how this works for n =2, there is essentially nothing new in passing to values of n in the millions.  For applications, the ability to pass from contorted surfaces to tangent plane approximations is of huge importance;  essentially all computer models rely on tangent plane approximations for calculations.  We will also see how to find the local maximum or local minimum values of f by suitable generalizations of one variable methods.  Our study of geometry will enable us to understand the settings when these techniques can and can't be applied.

         Just as in one variable calculus, the process of integration is the reverse of differentiation.  We will look at a variety of multi-variate integrals: double (area) integrals, triple (volume) integrals, line integrals, and surface integrals.   We'll see how one variable techniques are used to calculate every type of integral and the manner in which there are generalizations of the Fundamental Theorem of Calculus linking multi-variable integrals and partial derivatives.

         It is common for calculus students to ask their mathematics instructors to present more "real-life" applications and dispense with the over-simplified, tidy, and non-realistic applications typically treated in textbooks and lectures.  All of us would like to bypass years of hard work with nifty short-cut techniques for beating the stock market, examining the inner workings of black holes, and forecasting the earth's ecology.  Sadly, no such short-cuts are possible.  We mentioned above that calculus techniques come into play only after scientists have constructed mathematical models.  Modeling is both difficult and controversial with practical models often of such complexity that detailed descriptions of the model are cloaked in many pages of computer code.  In order to construct useful models, one needs a wealth of experience in a scientific field plus great sophistication in statistics, data analysis, and high level programming.  We urge you to ask your instructors in advanced natural and social science courses to give you a taste of modeling in their disciplines.  The role of Math 233 is to prepare you for later meaningful dialog about modeling with scientific experts.   We do this by emphasing the beautiful and intricate way in which calculus and geometric ideas are interwoven and illustrating this with relatively simple examples in low-dimensional settings.  This in itself is a large agenda.

            EXPECTATIONS AND ADVICE

         You'll probably find that the material in most university courses, including this one, is covered much more quickly than it would be in high school.  You'll probably also be asked to have a greater command of the material than before, especially in understanding the ideas (not just the techniques) and applying them in new situations. This may take some adjustments in your style of learning.

         The instructors and are here to help you understand and learn the material, but actually learning it is your responsibility.  The lectures are designed to try to highlight the important ideas and give you some perspective on the material, so that you can digest and really learn the material as you think, read the text, discuss the material with other students, and practice on problems. You should expect that more of the learning will occur outside the classroom than in it.  On the practical side, the average student should expect to spend at least a couple of hours on calculus for every hour in the classroom.  If you do this for all your courses, then being a student is the equivalent of a full-time job.

         Read the text. Your instructor can't possibly say everything that's there. An hour's lecture wouldn't be enough even to read the day's assignment aloud, let alone try to highlight or clarify. Your instructor's job is to complement the text so that you can learn from it.  Simply relying on your lecture notes will (and should!!!) probably only give you the highlights of the material.  To ignore the text is to start out looking for a minimal understanding.

         Reading a math text requires active involvement and is often a slow process. You might spend a few hours reading an assignment of 100 pages in another subject.  A reading assignment of 10 pages of mathematics might take just as much time.  Read with scratch paper at hand and fill in missing steps. When the author simply asserts  "...this expression simplifies to...", simplify it!  When you read "you should check that...", check it!  Try to work some of the examples in the text before reading them.  Play with the material: ask yourself  "what if I changed the problem to...?".  Write down questions that come to mind.  If you can't answer them, ask your friends. Try to understand why the assertions in the text are true.

               One of the marvelous features of mathematics is that you need not (perhaps should not!) trust the author.  If a
               physics book refers to an experimental result, it might be difficult or prohibitively expensive for you to do the
               experiment yourself.  If a history book describes some events, it might be highly impractical to find the original
               sources (which may be in a language you do not understand).  But with mathematics, all is before you to verify.
               Have a reasonable attitude of doubt as you read; demand of yourself to verify the material presented.
               Mathematics is not so much about the truths it espouses but about how those truths are established.  Be an
               active participant in the process.
                                                       E.R., Scheinerman, Mathematics, A Discrete Introduction
                                                       Brooks/Cole, Pacific Grove, CA, 2000, p. xviii-xix.
 

         Ideally, you should read ahead.  The lectures will be much more valuable if you have tried to digest some of the material already.  Also, you'll be able to judge that some things on the board needn't be written down.  For example, you'll know that those formulas the instructor is writing on the board are just the ones listed in the text.  That frees you to think about what's being said and not be always frantically transcribing material from the board.

         If you choose to ignore the advice about reading ahead, then at least read the material in the text before trying to do the assigned problems.  Some students start out with the problems and when they get stuck, they page back through the text looking for a formula or example to help.  This puts the focus in the wrong place.  Simply being able to "do the problems"  represents at best a minimal strategy for the course. You do have to do problems, and you do want to have certain technical skills that the problems sets can help develop, but you also want to understand the material.  Carefully reading the text is essential for that.

         You can't learn to speak French by merely sitting and listening, and you can't learn mathematics by watching someone else do problems. You need to do as many problems as you can realistically find time to do: the more, the better.  It's useful to work with other people on homework as long as you're involved as an active partner. Compare solutions; ask each other questions; make up quizzes for each other; agree on some extra problems to try together--such as the review problems at the end of each chapter.  Prod each other to do the assignments!  Do whatever helps!  But all of this is to improve your understanding: your study group isn't there at quizzes, tests, or when you need your math in another course.  See the syllabus section Homework Policy and Suggestions.

         There are facts and formulas you need to have at your fingertips: that is, there are some things you simply need to memorize.  But that's only the beginning.  Formulas are just tools.  Some routine problems, admittedly, are designed merely to be sure  you really can handle the tools, but when attacking a non-routine problem, don't just try to hunt
around for a formula. To quote an old but accurate text,

          "(the) time wasters are the formula worshipers ... who spend more time hunting a magic formula than they would need to analyze the problem piece by piece using simple familiar methods and calculations ... it cannot be said too often that the ability to understand and solve problems does not come by memorizing formulas ... formulas are not substitutes for thought."

         Stay on top.  Sometimes it's hard to keep up with everything, but it's important not to fall behind in the course.  A lot of the material we cover is interdependent, and if you're not comfortable with the material from two days ago, today's lecture might be totally incomprehensible.  It's better to be a bit ahead--the lectures will mean more and you've got a buffer when an emergency strikes.

         Also, there's a limit to how fast you can digest material and learn actually to use it in your thinking.  Avoid cramming just before tests by staying on top of things.  It makes life much more pleasant.

         Make use of your instructor..  Office hours are set aside specifically to see students; those times are for you!  Most instructors are also willing to set up special meetings with students as time permits. Your responsibility is not to waste these times.  Try to solve your own problems first, together with your friends. What you learn that way will be more valuable than the same answer "given" by your instructor. Come to office hours or your discussion section prepared with specific questions.  Bring questions to your instructor quickly, as soon as you realize there's some difficulty you can't resolve.  (It's hard for any instructor to cope with  "I haven't understood anything for three weeks"; we aren't magicians.)

         Finally, one last word.  Some of you have had some multivariable calculus in a previous course: this can be a blessing or a curse.  It's a curse if it  makes you think you know more than you do and traps you into slacking off until you suddenly realize you're in trouble. Even if you do already understand what's being covered in class and the homework, this shouldn't make you think there's nothing to learn.  Try instead some of the harder problems in the text so that the time's not wasted.  If you're bored and want to talk about some harder problems or find some extra reading, talk to your instructor.  If you're serious, your instructor will probably be delighted!  And, if you're on top of the current material, then you have a great opportunity to be helpful to some of your friends who are still working on it!