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
Standard Syllabus, Homework, and Maple Labs
 Syllabus
 Homework will be assigned via Sapling Plus. To sign in to your Sapling account (or to create one), go to www.saplinglearning.com/login. You will need the class key, which will be provided by the lecturer.
 All sections of Math 251 have either a Canvas or Sakai site, where all grades, exam reviews, syllabus, etc., are posted. You can access your Canvas course at https://canvas.rutgers.edu/, and your Sakai course at https://sakai.rutgers.edu/portal.
 Computational Labs in Math 251
Getting Help
A frequently asked questions file is available.
Reviews
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 9781319323394.
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  

Lecture  Topic(s) and text sections  Labs  Suggested 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 ThreeSpace  Lab 1  12.5/ #1,10,13,17,19,27,39,55,57  
4  13.1 VectorValued Functions 13.2 Calculus of VectorValued 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 nonfree 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) classbyclass 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.