Courses

01:640:251 - Multivariable Calculus

General Information:

01:640:251 Multivariable Calculus (4)

Analytic geometry of three dimensions, partial derivatives, optimization techniques, multiple integrals, vectors in Euclidean space, and vector analysis.

Prerequisite: CALC2 (Math 152, 154, or 192).

Textbook:

Textbook:  For current textbook please refer to our Master Textbook List page

Getting Help

frequently asked questions file is available.

For instructors

Information for instructors

Syllabus & textbook homework for Math 251

This is a very rapid plan of study. A great deal of energy and determination will be needed to keep up with it. Modifications may be necessary. Periodic assignments (Maple labs, workshops, etc.) may be due at times, and additional problems may be suggested.

The text is the 3rd edition of Rogawski's Calculus Early Transcendentals, W.H.Freeman, 2015, ISBN 978-1-319-04911-9.

It has been augmented with some Rutgers "local matter," which is also available for download:  Calculus at Rutgers

Syllabus and Maple labs for 640:251
LectureTopic(s) and text sectionsMaple labs
1 12.1 Vectors in the Plane
12.2 Vectors in Three Dimensions
Lab 0
2 12.3 Dot Product and the Angle Between Two Vectors
12.4 The Cross Product

3 12.5 Planes in Three-Space
4 13.1 Vector-Valued Functions
13.2 Calculus of Vector-Valued Functions
Lab 1
5 13.3 Arc Length and Speed
13.4 Curvature

6 14.1 Functions of Two or More Variables
14.2 Limits and Continuity in Several Variables

7 14.3 Partial Derivatives
14.4 Differentiability, Linear Approximation and Tangent Planes

8 14.5 The Gradient and Directional Derivatives Lab 2
9 14.6 The Chain Rule
10 14.7 Optimization in Several Variables
11 14.8 Lagrange Multipliers: Optimizing with a Constraint
12 Exam 1 (timing approximate!)
13 15.1 Integration in Several Variables Lab 3
14 15.2 Double Integrals over More General Regions
15 15.3 Triple Integrals
16 12.7 Cylindrical and Spherical Coordinates
17 15.4 Integration in Polar, Cylindrical, and Spherical Coordinates Lab 4
18 15.6 Change of Variables
19 16.1 Vector Fields
20 16.2 Line Integrals
21 16.3 Conservative Vector Fields
22 Exam 2 (timing approximate!)
23 16.4 Parameterized Surfaces and Surface Integrals Lab 5
24 16.5 Surface Integrals of Vector Fields
25 17.1 Green's Theorem
26 17.2 Stokes' Theorem
27 17.3 Divergence Theorem
28 Catch up & review; possible discussion of some applications of vector analysis.

Maple labs and workshops

The course has five suggested Maple labs during the standard semester, in addition to a Maple lab 0 which is introductory and should be discussed in the first week or two.
Instructors may also wish to assign some workshop problems so that students can continue to improve their skills in technical writing.

The syllabus omits section 12.6, A Survey of Quadratic Surfaces. The ideas concerning quadratic surfaces are actually addressed in the third Maple lab, and certainly some knowledge of quadratic surfaces is useful when considering the graphs of functions of several variables and studying critical points. Although this section is formally omitted, appropriate examples and terminology should be introduced early in the course.

Q. How can I get help with this course?

A. Free possibilities include asking questions in class, going to either your instructor's or a TA's office hours, and going to the Learning Resource Center or the MSLC Math and Science Learning Center.
A non-free alternative is to go to room 303 Hill Center and ask for their list of tutors for Calc III.

Q. I got a bad grade on the first exam even though I studied. What should I do?

A. If the bad grade is a C or higher, one answer to the your first question could be -- Be glad it's only a fifth or so of your grade. Do better on the rest of the exams, quizzes, etc., and you'll be okay.

A longer answer: it helps if you keep up with the material as we go along. Studying the night before isn't as helpful as keeping up on a weekly or (better) class-by-class basis. When you find something from an earlier semester that you've forgotten, go back to the appropriate section of the text, reread it, and do some of the problems.

Part of this has to do (I theorize) with short term versus long term memory. You want your calculus in long term memory because as an engineer you'll be using it for years in courses like fluids. That means learning the material and then retesting yourself every so often to make sure that it hasn't evaporated. Think of multivariable calculus as a skill, like tennis or piano playing, that has to be learned over time and maintained if you want good results.

Q. How can I use Maple without going to the computer lab?

A1 (recommended). A student version of Maple is available from Maplesoft for a moderate price, and there are frequently special offers giving additional discounts. If you are interested in purchasing a personal copy, look here for more detailed instructions.

A2. From your home computer, you can run Maple (and other software) through the website apps.rutgers.edu. To use this service, please follow the instructions found here.
This service is free, but can be less reliable, and more awkward, than a copy running on your own computer.

If you are not going to install Maple on your own computer, you should complete the lab well ahead of time so that you do not miss the deadline due to forces beyond your control.