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.
Exam 1 | 100 |
Exam 2 | 100 |
Final exam | 200 |
Formal quizzes | 65 |
Maple | 40 |
Textbook homework | 35 |
Total | 540 |
---|---|
Academic integrity
Last year I taught Math 151-2. Several incidents of alleged violations
of academic integrity (suspected cheating) occurred. I had not
noticed such things for more than a decade. I reported these
violations, some students were found to have committed them, and
severe penalties were imposed. Here
is a link to the Rutgers academic integrity policy.
Please: you are uncertain about what the policy requires in any
situation specific to this course, then discuss this uncertainty with
the lecturer. The course should support efficient and comfortable
learning of the material. Grading, and any sort of assessment of
student progress, will be done accurately, carefully, and honestly.
Maintained by greenfie@math.rutgers.edu and last modified 9/1/2008.