Courses

01:640:251 - Multivariable Calculus

 

General Information:

01:640:251 Multivariable Calculus (4 Credits)  

This course covers multi-variable and vector calculus. Topics include 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:

Thomas' CALCULUS Early Transcendentals, 14/e,  by Joel Hass, Christopher Heil, and Maurice Weir. Pearson Education. ISBN: 9780134639543 

MyMathLab access with etext: ISBN: 9780135901403

MyMathLab access can be purchased directly from Pearson.

Standard Syllabus, and Homework

 

Information for Instructors

Information for instructors

 

Lecture Topics & 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 (matlab labs, workshops, etc.) may be due at times, and additional problems may be suggested.

The text is the 14th edition of Thomas' CALCULUS Early Transcendentals, by Joel Hass, Christopher Heil, and Maurice Weir. Pearson Education. ISBN: 9780134639543 

Lecture Topics and Suggested Textbook Problems for 640:251
LectureTopic(s) and text sectionsSuggested Homework
1 12.1 Three Dimensional Coordinate Systems
12.2 Vectors
12.1/ #3,9,15,16,21,23,39,59
12.2/ #9,19,23,25,31,41,47
2 12.3 The Dot Product 
12.4 The Cross Product 
12.3/ #3,13,18,19,22
12.4/ #7,15,21,23,27,33,35,45
3 12.5 Lines and Planes in Space
12.6 Cylinders and Quadric Surfaces
12.5/ #6,6,9,23,27,31,37,41, 47, 57
12.6/ #1-12, 13, 15, 21, 25
4 13.1 Curves in Space and Their Tangents 13.1/ #1,15,19,23, 31
5 13.2 Integrals of Vector-Valued Functions, Projectile Motion 13.2/ #3,5,8,11,15,19
6 13.3 Arc Length in Space 13.3/ #5,9,13,15
7 14.1 Functions of Several Variables
14.2 Limits and Continuity in Higher Dimensions
14.1/ #6,6,9,15-18,27,31,41,58,59
14.2/ #11,15,20-22,31,47,48,53,59
8 14.3 Partial Derivatives
14.4 The Chain Rule
14.3/ #9,14-18,27,3046,67
14.4/ #7,28,29,33,39,44
9 14.5 Directional Derivatives and Gradient Vectors
14.6 Tangent Planes and Differentials
14.5/ #3-6,9,10,21,28,29,31
14.6/ #5,9,13,19,21,31
10 14.7 Extreme Values and Saddle Points 14.7/ #13,19,29,33,35,43,45,62
11 14.8 Lagrange Multipliers 14.8/ #1,5,9,13,17,21,29
12 15.1 Double and Iterated Integrals Over Rectangles
15.2 Double Integrals over General Regions
15.1/ #7,11,13,18,23,27,29,36
15.2/ #1-8,12,15,18,23,29,35,38,43,53
13 15.3 Area by Double Integration
15.4 Double Integrals in Polar Form
15.3/ #6-8,16,18,21
15.4/ #1-6,9,11,16,23,25,28,35
14 15.5 Triple Integrals in Rectangular Coordinates 15.5/ #3,6,17,21,27,28,37,45
15 Catch up & Review  
16 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 15.7/ #3-7,13-17,25,31,38,47,59,65
17 15.8 Substitution in Multiple Integrals 15.8/ #1,3,6,7,9
18 16.1 Line Integrals of Scalar Functions 16.1/ #1-9,14,15,25,26,29,35,36
19 16.2 Vector Fields & Line Integrals: Work, Circulation, Flux 16.2/ #3,7,11,14,15,18,25,27,29,30,35,39,40,57,59
20 16.3 Path Independence, Conservative Fields, Potentials 16.3/ #3,5,9,11,19,22,25,29,31
21 16.4 Green's Theorem in the Plane 16.4/ #2,5,9,13,16,17,21,29,31,37
22 Catch up & Review  
23 16.5 Surfaces and Area 16.5/ #1,7,11,13,15,27,41,43
24 16.6 Surface Integrals 16.6/#3,5,6,17,23,26,28,43
25 16.7 Stokes' Theorem 16.7/ #5,7,11,19,23,28
26 16.8 The Divergence Theorem & A Unified Theory 16.8/ #9,11,13,15,27,28
27 Catch up & Review  
28 Catch up & Review  

 

Exam Protocols

More details about the exams will be posted on this website as the semester progresses. In principle, the dates are:

Short Exam:  Monday, September 27

Midterm 1:  Monday, October 18

Midterm 2:  Sunday, November 14

Midterm 3:  Monday, December 6

Final Exam: TBA.

 

MyLab - Online Homework

Students are required to purchase access to MyLab Math to complete the online homework, and possibly quizzes and exams. The MyLab assignments are similar to the exercises in the official list of HW exercises. (The official HW exercises are not handed in for grading but instead form a significant, but not exhaustive, portion of your study guide for the course.) Each assignment will have a specific due date set by the professor, and these assignments must be completed online.

How to use MyLab properly:
If you take shortcuts like trying to find answers to MyLab problems from various "homework help" web services without solving all of the problems yourself in their entirety, then your performance on exams will suffer. Instead, use the built-in help tools within MyLab. This online homework exists primarily to give you feedback on your ability to calculate correct answers at early stages of the learning process. The homework is not intended to measure your mastery of the material; only the midterm exams and final exam measure mastery. Without doing well on the exams, it is impossible to pass the course, even with a perfect score on the homework. So be sure to take full advantage of MyLab to get as much feedback as possible on your problem-solving skills.

Getting started with MyLab:

  • You will be able to access your MyLab course directly through your Canvas site for Math 251.
  • In your Canvas site, navigate to MyLab and Mastering and follow the on-screen instructions to create a Pearson account (or link an existing Pearson account) to your Canvas account.
  • You will automatically be enrolled in the MyLab course.
  • If you switch to a different section of Math 251, you can enroll in your new section's MyLab course by following these same instructions.

Student support for MyLab:

  • System Requirements
    MyLab works on a series of pop-up screens. You MUST enable pop-ups when working in MyLab. For help on how to do this, as well as make sure your browser is up to date, use the link above.
  • How to Use MyLab on a Mobile Device
    This video shows you how to set up your mobile device with any necessary browser add-ons and apps to use MyLab properly. 
  • Pearson Support Database
    Use the above link to search Pearson's database for support topics (e.g., resetting password).
  • Contact Support
    Use the above link to contact technical support. Fill out the required form and you will be immediately connected to a support agent based on your issue.
  • Pearson sales representative: melissa.blum@pearson.com
    Melissa Blum is our Pearson Sales representative. If you are having technical issues, please first contact Technical Support. If you are still having issues after contacting Technical Support, please email Melissa Blum with the Incident Number you received from working with Technical Support. You must have an Incident Number for Melissa to be able to help.

Other information about MyLab:

  • MyLab is an interactive, online homework system. The assignments follow the lecture topics.
  • Questions are algorithmically generated to give each student their own random versions of the questions.
  • After entering an incorrect answer, students are given helpful feedback and hints. Most exercises will also include learning aids, such as guided solutions and sample problems.
  • You have three attempts to get an answer correct. If you use all three attempts, you will be told the correct answer and given a new, random version of the same problem. There is no limit to the number of versions of a particular problem you can be given. So you are strongly encouraged to work on a problem until you get the correct answer. There is no penalty for the number of attempts taken.

 

Matlab Computer Labs

We will have some computer labs to show how technology facilitates the learning of certain calculus concepts. These were primarily designed for Matlab, since you can use the Rutgers license to download it for free. However, you may also use Maple or Mathematica to work on these assignmets. More information about them will be given during at the beginning of the Spring Semester.

Grading Weights

ComponentWeight -- 500 points total
Classwork  100 points (20% of grade)
Short Exam  40 points (8% of grade)
Midterm 1 60 points (12% of grade)
Midterm 2 90 points (18% of grade)
Midterm 3 60 points (12% of grade)
Final exam 150 points (30% of grade)

 

 

Solutions to practice exams and additional documents

Extra problems  : if you have finished working on the practice exams and additional review material provided by your instructor, you may want to check this list of problems if you want to do additional practice problems. Some of these are more conceptual in nature, so they may be useful to enhance your understanding of the course material.

Respondus Practice : solutions to the respondus practice assignment

------------

Material related to the September 27 Short Exam

Practice Exam 1 Sep 27

Practice Exam 2 for Sep 27

Practice 3 for Sep27

Short exam guide sep27

 

Information on the Practice Exams and Sep 27 Short Exam

 

Dear students,

Here is the information you need to know about the upcoming short exam.

  • Dateof the exam:September 27.
  • Durationof the exam: 35 minutes.
  • Time windowto access the exam:9:00 pm -10:10 pm(EDT). 
  • For this exam we will be usingRespondus. Typically it takes between 5 and 10 minutes for the authentication process to be completed before you can start working on the exam.
  • For example, if you finish being authenticated by Respondus at 9:15 pm, then you would be done with the short exam at 9:50 pm, and then you would have 20 minutes (until 10:10 pm) to upload your work to the canvas site. On the other hand, if someone started the exam later and finishes being authenticated by Respondus at 9:35 pm, then you would be done with the short exam at 10:10 pm, and then you would have until 10:30 pm to upload your work to the Canvas site. 
  • We encourage everyone to start the exam process as soon as possible and email your professor immediately if you are running into any technological issues(loosing access to the exam because of poor internet connectivity, etc).
  • Uploading your work and Academic Integrity: after you submit your exam on MyLab, you have 20 minutes to upload your work on your course Canvas site. For example, if you submitted the MyLab exam at 10:00 pm, you must do the upload by 10:20 pm. It will be considered a violation of academic integrity if you change your work after submitting the test on MyLab. In particular, the solutions you show on your uploaded work must agree with the solutions you entered on the MyLab test.
  • Personal Note Sheet: Any formula listed / mentioned in the study guide "short-exam-guide" (mentioned later in this announcement) that you find difficult to memorize may bewritten down by handon your personal "note sheet''.  Your note sheet can be at most atwo-sided pieceof paper, andno other exam materials may be accessed while you are taking the exam
  • Other things not allowed: Phones, calculators, computer algebra systems like Wolfram Alpha, you cannot give or receive help from anyone else, nor “check your answers” on a calculator, online, or with anyone.  If you do not follow these rules, you will risk an Academic Integrity Violation (http://nbacademicintegrity.rutgers.edu/home/academic-integrity-policy/), which may lead to a zero on the test.
  • Late submissions to the exam questions will not be accepted. Emailed submissions will not be accepted.
  • If you have been approved (by ODS) for extra time, you will receive accommodations during the exam. You must provide your own professor the relevant ODS documentation as soon as possible.
  • Students who are in time zones for which the exam time is impractical, or who have an unavoidable conflict/emergency may be able to take a makeup, provided youemail your own professor as soon as possible. 

As of now, you should now be able to access three practice exams for the Sep 27  Short Exam  through the MyLab site of your individual course. The assignments are called "Practice 1 for Sep 27” , "Practice 2 for Sep 27"and "Practice 3 for Sep 27". 

These practice exams will not be graded, so you can practice it as many times as you want, and at any time before the midterm. Solutions for these practice exams can be found on the website 

https://math.rutgers.edu/academics/undergraduate/courses/949-01-640-251

On this website you will also find a study guide for this exam, calledshort-exam-guide.

Likewise, on this website you will find solutions to the respondus assignment "Respondus Practice".Please complete this assignment a few days before the date of the short exam, so that you become familiar with the Respondus environment.

The sections of the book for the Sep 27 short exam are:

  • Section 12.1: Three-Dimensional Coordinate Systems
  • Section 12.2: Vectors
  • Section 12.3: The Dot Product
  • Section 12.4: The Cross Product
  • Section 12.5: Lines and Planes in Space
  • Section 12.6: Cylinders and Quadric Surfaces
  • Section 13.1: Curves in Space and their tangents
  • Section 13.2: Integrals of Vector Functions; Projectile Motion
  • Section 13.3: Arc Length in Space  

Here are some useful things to be aware of:

  • The practice exams and the Sep 27 short exam consists of3 questionseach. 
  • The last two problems will be upload problems,where you will need to upload the solution of these problems to the canvas site of your course in order to receive credit.If you don't upload your work, you will not receive any credit for these problems. On the Canvas site there will be an assignment called “Sep 27 Exam: Upload your work” where you must upload your work. You may upload it as a pdf, jpeg or jpg file. Againyou will have 20 minutes after you complete the exam on Mylab to upload your work to the canvas site.
  • The practice exams do not include every type of possible question you might expect to see on the exam. Therefore, you are still encouraged to work out the MyLab assignments and any other material you find useful which has been provided by your professor. 
  • As you will see on the practice exam, the last two problems have the instructions:

Important:you need to upload your work for this problem to the Canvas site. Remember that you have 20 minutes aftersubmittingthe MyLab exam to do the upload”

  • For the practice exams there will be no need to do such an upload, but we wrote the statement to give you an idea of how the upload questions will look like on the actual exam.

  •  

Sincerely,

Math 251 professors

 

-------------------------------------

Letter Grades for Sep 27 Short Exam

 

Sep 27 EXAM SOLUTIONS

 

Dear 251 student,

 
This is the letter grade scale for the September 27 exam only. This scale does not apply to any other midterm, to the final exam, or to the final course grade – all of which will be determined separately. The scale is intended only to give you an idea of your progress to date based on this one midterm, since the remaining midterms and the final exam will likely have different grade scales.

If X = your exam numerical score (out of 100 points), then the corresponding letter grade is

A: 85<= X <= 100
B: 70<= X < 85
C: 50<= X < 70
D: 40<= X < 50
F: X < 40
Furthermore, for this exam only, you will have a grade replacement option: if your score on the final exam
is higher than your score on this exam, then your final exam grade will replace your grade on this exam.
Sincerely,

MATH 251 Professors

 

Information on the Practice Exams and October 18 Exam

 

Practice 1 Oct 18

Practice 2 Oct 18

Practice Exam 3 Oct 18

OCT 18 STUDY GUIDE

 

Dear students,

Here is the information you need to know about the midterm. 

  • Date of the examOctober 18.
  • Duration of the exam: 55 minutes.
  • Time window to access the exam: 9:00 pm -10:30 pm(EDT). 
  • This midterm will again use Respondus.
  • We encourage everyone to start the exam process as soon as possible and email your professor immediately if you are running into any technological issues (losing access to the exam because of poor internet connectivity, etc).
  • Uploading your work and Academic Integrity: after you submit your exam on MyLab, you have 20 minutes to upload your work on your course Canvas site. For example, if you submitted the MyLab exam at 10:00 pm, you must do the upload by 10:20 pm. It will be considered a violation of academic integrity if you change your work after submitting the test on MyLab. In particular, the solutions you show on your uploaded work must agree with the solutions you entered on the MyLab test.
  • Nature of the Uploaded Work:the uploaded work should not be interpreted as "scratch work". Rather, it is supposed to be a detailed solution to the problem of the exam (similar to the one you would write if the exam were in person).  
  • Personal Note Sheet: Any formula listed / mentioned in the study guide "october-18 midterm"  that you find difficult to memorize may be written down by hand on your personal "note sheet''.  Your note sheet can be at most a two-sided piece of paper, and no other exam materials may be accessed while you are taking the exam
  • Other things not allowed: Phones,ipads or tablets, calculators, computer algebra systems like Wolfram Alpha, you cannot give or receive help from anyone else, nor “check your answers” on a calculator, online, or with anyone.  If you do not follow these rules, you will risk an Academic Integrity Violation (http://nbacademicintegrity.rutgers.edu/home/academic-integrity-policy/), which may lead to a zero on the test.
  • Late submissions to the exam questions will not be accepted. Emailed submissions will not be accepted.
  • If you have been approved (by ODS) for extra time, you will receive accommodations during the exam. You must provide your own professor the relevant ODS documentation as soon as possible.
  • Students who are in time zones for which the exam time is impractical, or who have an unavoidable conflict/emergency may be able to take a makeup, provided you email your own professor as soon as possible. 

As of now, you should now be able to access three practice exams for the Oct 18 midterm  through the MyLab site of your individual course. The assignments are called "Practice 1 for Oct 18” , "Practice 2 for Oct 18" and "Practice 3 for Oct 18". 

 

These practice exams will not be graded, so you can practice it as many times as you want, and at any time before the midterm. Solutions for these practice exams can be found on the website 

https://math.rutgers.edu/academics/undergraduate/courses/949-01-640-251

On this website you will also find a study guide for this exam, calledOct-18 guide.

 

 

The sections of the book for the Oct 18 midterm are:

  • Section 14.1: Functions of Several Variables
  • Section 14.2: Limits and Continuity in Higher Dimensions 
  • Section 14.3: Partial Derivatives
  • Section 14.4: The Chain Rule
  • Section 14.5: Directional Derivatives and Gradient Vectors
  • Section 14.6: Tangent Planes and Differentials
  • Section 14.7: Extreme Values and Saddle Points

Here are some useful things to be aware of:

  • The practice exams and the Oct 18 midterm consists of 4 questions each. 
  • The last two problems will be upload problemswhere you will need to upload the solution of these problems to the canvas site of your course in order to receive credit. If you don't upload your work, you will not receive any credit for these problems. On the Canvas site there will be an assignment called “Oct 18 Exam: Upload your work” where you must upload your work. You may upload it as a pdf, jpeg or jpg file. Againyou will have 20 minutes after you complete the exam on Mylab to upload your work to the canvas site.
  • The practice exams do not include every type of possible question you might expect to see on the exam. Therefore, you are still encouraged to work out the MyLab assignments and any other material you find useful which has been provided by your professor. 
 

  •  

Sincerely,

Math 251 professors

 

 

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.