General information for Math 251:05-10, spring 2006

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. 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).

The text is Calculus: Early Transcendentals, 5th edition, Brooks/Cole, 1999, ISBN 0-534-39321-7, by James Stewart. This is a standard U.S. calculus textbook, better than some. This is the text used in Math 151 and 152 so I hope that most students already own a copy.
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, 4th edition (paperback) by H. M. Schey. I especially recommend it for students interested in physics and in mechanical or chemical engineering. The cost is $29.25 on

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 the knowledge that 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.

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 devoted to getting acquainted with Maple and 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.

The lecturer is S. Greenfield. The recitation instructor for sections 5, 6, and 7 is Mr. Liviu Ilinca and the recitation instructor for sections 8, 9, and 10 is Mr. Alexander Zarechnak (e-mail:


Office hours
My office is in Hill Center: Hill 542, telephone number: (732) 445-3074. My formal office hours will be announced soon. You certainly can also make an appointment at a mutually convenient time. I usually check e-mail several times a day so it is probably the best way to communicate with me: You can ask also questions via e-mail and I'll try to answer them.

Maintained by and last modified 1/16/2006.