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 most students continuing from those courses should already own a
copy. The book is a first edition of a large math textbook. Although
it was checked carefully for errors, some have been discovered, so
please be a bit cautious.
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 about
$30 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 I will try to avoid tedious use of
elaborate integration techniques in class. There also will be almost
no reference to infinite series in the course. Note, though, that
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 and used 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. One of the early
recitations will be held in a computer-equipped room in ARC (this will
be announced). That meeting time will be devoted to getting acquainted
with Maple. Several homework
assignments will involve use of Maple.
Other software packages (most prominently, Mathematica) have graphic/symbolic/numerical
capabilities similar to Maple. We use
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) 445-2390 X3074 (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: Tuesday 12 noon to 1:00 PM and Thursday from 10:30 to
11:30 AM and by appointment (e-mail is best for arranging
appointments).
Students in these sections may visit either recitation instructor (and the lecturer!) for help
The
recitation instructor for sections 12-14 is G. Bouch.
Office: Hill 624 on Busch Campus; (732) 445-2390 X8608 (there's an
answering machine);
His e-mail address is gbouch@math.rutgers.edu.
That's probably the best way to contact him. His office hours are
Thursday, 9:45 to 11:45 AM.
The
recitation instructor for sections 15-17 is V. Nanda.
Office: Hill 511 on Busch Campus;
His e-mail address is vidit@math.rutgers.edu.
That's probably the best way to contact him. His office hours are 3:00
to 4:30 PM on Thursdays.
Students in these sections may visit either recitation instructor (and the lecturer!) for help
Grading
Sections 12-14 will have a final exam on
Thursday, December 23, from 12 to 3 PM.
Sections 15-17 will have a final exam on
Thursday, December 16, from 12 to 3 PM.
The location of the final exam will be announced later. Some formula
sheets may be used during portions of the exams. If such
formula sheets will be supplied, they 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, please notify the
lecturer as soon as possible using e-mail (preferred: greenfie@math.rutgers.edu),
by phone (at (732) 445-2390 X3074 [equipped with an answering
machine]), or through the Math Department Undergraduate Office (at
(732) 445-2390).
Make-ups will not be given for reasons like "the alarm clock didn't
go off", "not knowing when the exam will be", or "not feeling
prepared".
| Exam 1 | 100 | |
| Exam 2 | 100 | |
| Final exam | 200 | |
| Formal quizzes | 65 | |
| Workshops | 30 | |
| Maple | 40 | |
| QotD | 25 | |
| Total | 560 | |
|---|---|---|
Maintained by greenfie@math.rutgers.edu and last modified 8/31/2010.