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

Standard Syllabus, Homework, and Maple Labs

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 4th edition of Rogawski's Calculus Early Transcendentals, W.H.Freeman, 2015, ISBN 978-1319323394.

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

Syllabus and Computational Labs for 640:251
LectureTopic(s) and text sectionsLabsSuggested Homework
1 12.1 Vectors in the Plane
12.2 Vectors in Three Dimensions
Lab 0 12.1/ #9,13,15,19,25,43,51,55
12.2/ #11,13,19,29,31,37,40,53
2 12.3 Dot Product and the Angle Between Two Vectors
12.4 The Cross Product
  12.3/ #1,13,21,29,31,68,79
12.4/ #1,5,11,15,19,22,39,41
3 12.5 Planes in Three-Space Lab 1 12.5/ #1,10,13,17,19,27,39,55,57
4 13.1 Vector-Valued Functions
13.2 Calculus of Vector-Valued Functions
  13.1/ #4,11,19,25
13.2/ #4,10,27,28,31,44,45
5 13.3 Arc Length and Speed
13.4 Curvature (optional)
  13.3/ #3,15,19,25
6 14.1 Functions of Two or More Variables
14.2 Limits and Continuity in Several Variables
Lab 2 14.1/ #7,19,20,23,29,30
14.2/ #3,7,8,17,19,30,37
7 14.3 Partial Derivatives
14.4 Differentiability, Linear Approximation and Tangent Planes
  14.3/ #3,19,22,33,49,54,55,57
14.4/ #1,3,7,8,15,19,21,25,32
8 14.5 The Gradient and Directional Derivatives Lab 3 14.5/ #7,13,27,31,33,35,40,41,46,55
9 14.6 The Chain Rule   14.6/ #1,5,7,13,26,29,39,41
10 14.7 Optimization in Several Variables   14.7/ #1,3,10,11,19,21,28,31,32,41
11 14.8 Lagrange Multipliers: Optimizing with a Constraint   14.8/ #2,7,11,14,15,17
12 Exam 1 (timing approximate!)
13 15.1 Integration in Several Variables Lab 4 15.1/ #12,17,27,29,41,43,47
14 15.2 Double Integrals over More General Regions   15.2/ #3,5,11,21,27,32,33,41,43,47,53
15 15.3 Triple Integrals   15.3/ #3,5,9,15,19,21,34,39
16 12.7 Cylindrical and Spherical Coordinates   12.7/ #1,5,25,37,44,55,61
17 15.4 Integration in Polar, Cylindrical, and Spherical Coordinates   15.4/ #1,5,13,17,22,23,27,33,41,43,51,53
18 15.6 Change of Variables Lab 5 15.6/ #1,5,14,1521,29,33,37
19 16.1 Vector Fields   16.1/ #1,3,10,11,13,15,23,27,43,50
20 16.2 Line Integrals   16.2/ #3,15,21,23,35,37,41,43
21 16.3 Conservative Vector Fields   16.3/ #1,5,9,11,15,18,19,23,24
22 Exam 2 (timing approximate!)
23 16.4 Parameterized Surfaces and Surface Integrals   16.4/ #1,5,8,11,13,17,18,23
24 16.5 Surface Integrals of Vector Fields   16.5/ #1,5,6,7,9,12,15,17,23
25 17.1 Green's Theorem   17.1/ #1,3,6,9,10,13,15,16,23,25
26 17.2 Stokes' Theorem   17.2/ #1,5,7,11,13,15,25,27
27 17.3 Divergence Theorem   17.3/ #1,3,11,13,14,15
28 Catch up & review; possible discussion of some applications of vector analysis.

Computational Labs and workshops

The course has five suggested computational labs during the standard semester, in addition to a 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.

Quadratic surfaces

The syllabus omits section 12.6, A Survey of Quadratic Surfaces. The ideas concerning quadratic surfaces are actually addressed in the third computational 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.


Frequently asked questions about Math 251

Q. How can I get help with this course?

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

Q. How can I do the computational labs without going to the computer lab?


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 do the computational labs without going to the computer lab?

A. 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. Student versions of Matlab and Mathematica are available for download via the Rutgers Software Portal.

If you are not going to install one of these systems 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.


Schedule of Sections:

Disclaimer: Posted for informational purposes only

This material is posted by the faculty of the Mathematics Department at Rutgers New Brunswick for informational purposes. While we try to maintain it, information may not be current or may not apply to individual sections. The authority for content, textbook, syllabus, and grading policy lies with the current instructor.

Information posted prior to the beginning of the semester is frequently tentative, or based on previous semesters. Textbooks should not be purchased until confirmed with the instructor. For generally reliable textbook information—with the exception of sections with an alphabetic code like H1 or T1, and topics courses (197,395,495)—see the textbook list.