Math 251
Here is the catalog description of the course:

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 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: Wednesday 2 to 3:30 and Thursday from 10 to 11:30 and by appointment (e-mail is best for arranging appointments).

The recitation instructor is V. Nanda.
Office: Hill 608 on Busch Campus; (732) 445-2390 X8608 (there's an answering machine);
His e-mail address is vidit@math.rutgers.edu. That's probably the best way to contact him. His office hours are Monday, 2 to 3 PM..

Grading

• Possible cheating
  Suspected violations of academic integrity (cheating) will definitely be reported. Students should be familiar with Rutgers policies on academic integrity. The penalties for infractions (breaking the rules) can be quite severe.
  Please: if 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.
• 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 on Thursday, May 6, 8-11 AM. 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).
  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.
• Informal quizzes
  Informal quizzes (the Question of the Day) will be given in most lectures. Handing in the quiz will earn full credit. These quizzes will not be major components of the course grade, but may be useful to inform both the instructor and the student regarding progress in the course.
• Formal quizzes
  Most recitations will have formal quizzes. They will reflect closely homework which should be done before that recitation.
• Workshops
  Workshop writeups may be requested, and these writeups will be graded both for mathematical content and for presentation.
• 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:

  	Exam 1	100
  	Exam 2	100
  	Final exam	200
  	Formal quizzes	65
  	Workshops	30
  	Maple	40
  	QotD	25
  	Total	560

  
  The total score will then be converted to a course grade. There are two additional comments on course grades.

  • Students who miss a significant number of classes may have their course grades lowered one step (for example, from a C+ to a C). When I've most recently done this in a calculus course, almost all of the students to whom it applied could not have their grade decreased (think about what this means). Attendance is very useful!
  • Students whose exam grades all are near bare passing or are failing may fail the course in spite of numerical averages: students must show that they can do adequate work connected with this course independently and verifiably.

Maintained by greenfie@math.rutgers.edu and last modified 1/18/2010.