## General information for Math 251:22-24, fall 2008

Math 251
Here is the catalog description of the course:

 01:640:251 Multivariable Calculus (4) Prerequisite: CALC2. Analytic geometry of three dimensions, partial derivatives, optimization techniques, multiple integrals, vectors in Euclidean space, and vector analysis.

The course extends calculus to the analysis of functions which depend on more than one variable or which have more than one output (that is, domain or range with dimension>1). Although the course will concentrate on functions of two or three variables, the techniques discussed are applicable to functions depending on any number of variables. The ideas are basic for almost all of modern applied science and engineering. For example, most upper-level engineering courses use partial derivatives and multiple integrals in their modeling of physical situations. The notation and language of 251 are required for advanced study in chemistry (640:251 is required for physical chemistry) and physics, and are also very useful in computer science (it's hard to analyze algorithms depending on more than one variable without the ideas of 251).

Text
The text is the first edition of Rogawski's Calculus Early Transcendentals, W.H.Freeman, 2008, ISBN-10: 0-7167-7267-1. It has been augmented with some Rutgers "local matter," which is also available here. This is the text used in the most recent semesters of Math 151 and 152 so I hope that most students already own a copy. The book is a first edition of a large math textbook. Although it has been checked carefully for errors, some may have still not been detected and corrected. Please tell me about any errors you notice in the problems or the text.
An excellent supplementary text for the vector calculus portion of the course (the last segment) is Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, fourth edition (paperback) by H. M. Schey. I especially recommend it for students interested in physics and in mechanical or chemical engineering. The cost is \$20.25 on Amazon.com. Most of the student reviews of the book posted there are quite positive.

Background
Certainly the course needs both of the beginning semesters of the calculus sequence although ... here's some honesty: I'll try to avoid tedious use of elaborate integration techniques in class. There also will be almost no reference to infinite series in the course (although improper integrals turn out to be very natural in certain physical applications).
So what will we need from the two semesters of calculus? The second semester of the calculus sequence we give is very computational. We will certainly need familiarity with properties of functions which occur in calculus, and this familiarity is part of what any successful survivor of second semester calculus has. Math 251 will compute "things" but the course also deals with many new big ideas. These ideas echo some of the foundational concepts of calculus. The derivative in the first semester is a number which is tied to a local linear approximation of a function. With several variables, the ideas connected with local linear approximation turn out to be important. The one dimensional integral does compute "things" (area, arc length, mass, etc.) but one version of the Fundamental Theorem of Calculus connects the definite integral as an "accumulation function" of the derivative with the net {gain|loss} at each end of an interval. It is this version of the FTC which gets generalized in vector calculus, and this version which is applied very powerfully to ideas of heat flow, diffusion, etc., which are analyzed in physics and engineering. But the ideas get quite elaborate, so:

Derivative
There will be at least seven different notions of derivative (vector-valued derivatives, partial derivative, differentiability and linear approximation, gradient, directional derivative, divergence, and curl).
Integral
There will be at least eleven different notions of integral (vector-valued integral, double and triple integral, iterated integrals, integrals in polar, cylindrical, and spherical coordinates, line integrals of various types, and surface integrals of various types).

Technology
Pictures help me a great deal with many of the ideas and computations in this course. There are few hand-held devices which can give really useful pictures in two and three dimensions. The software package Maple is very useful, and I urge you to learn to use Maple. The first recitation meeting will be held in Records Hall, in the PC Instructional Microcomputer Lab and this time will be devoted to getting acquainted with Maple. There will be several homework assignments which will involve use of Maple.
Other software packages (most prominently, Mathematica) have graphic/symbolic/numerical capabilities similar to Maple. But I'll refer to Maple in this course, since it is installed on almost every large computer system at Rutgers. Notice that many Maple capabilities can be accessed through a Matlab toolbox.

Instructors
The lecturer is S. Greenfield.
Office: Hill 542 on Busch Campus; (732) 445-3074 (there's an answering machine);
My e-mail address is greenfie@math.rutgers.edu. I usually check e-mail several times a day, so that's probably the best way to leave a message.
Office hours: My formal office hours will be announced soon. You certainly can also make an appointment at a mutually convenient time. You can ask also questions via e-mail and I'll try to answer them.

The recitation instructor is J. Dibble.

Grading I believe that the last day to withdraw with a W this semester is Monday, October 27.

• Formal exams
Several formal exams will be given during classes. These exams will be announced in advance. There will be a three-hour final exam which is already scheduled. The exact location of the final exam will be announced later. Some formula sheets may be used during portions of the exams. These sheets will be available in advance. Almost surely there will be special review sessions and office hours before each exam. There are many old Math 251 exams on the web, and links to these exams will be provided.
Make-up exams will be given only in the case of illness, a major emergency, or a major outside commitment. Verification of each of these should be done through the appropriate Dean's office, and a written note from the Dean requesting a make up exam should be presented to the lecturer. You will need some form of proof (like a doctor's note, a police report, a towing bill etc.). The lecturer must approve each request for a make-up exam.
If the reason for the make up is known in advance you must ask for permission before the exam. In all other cases, you must notify the lecturer using e-mail (preferred: greenfie@math.rutgers.edu), by phone (at (732) 445-3074 [equipped with an answering machine]), or through the Math Department Undergraduate Office (at (732) 445-2390) as soon as possible.
No make ups will be granted for reasons like "the alarm clock didn't go off", "not knowing when the exam will be", or "not feeling prepared".
• Homework
Students should do homework. Students at or above C-level should be able to successfully complete most of the suggested problems. Some homework problems may be collected.
• Formal quizzes
Formal quizzes may be given in recitations. They will reflect closely homework which should be done before that recitation.
• Maple
There will likely be 4 Maple labs. The "lab reports" will be graded.
• A precise formula?
I don't have an exact formula for grades yet. Something resembling the following will be used:

Total 540 Exam 1 100 Exam 2 100 Final exam 200 Formal quizzes 65 Maple 40 Textbook homework 35

The total score will then be converted to a course grade.