General Information:
01:640:251 Multivariable Calculus (4 Credits)
This course covers multivariable 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
 Syllabus
 Fall 2022 Technology Requirements
 Lecture Topics
 Exam Protocols
 MyLab  Online Homework
 Grading Weights
 Practice Exams
 There is a Canvas course site for each lecture group where all grades, exam reviews, syllabus, etc., are posted. You can access your Canvas course at https://canvas.rutgers.edu/.
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  

Lecture  Topic(s) and text sections  Suggested 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/ #112, 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 VectorValued 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/ #6,6,9,1518,27,31,41,58,59 14.2/ #11,15,2022,31,47,48,53,59 

8  14.3 Partial Derivatives 14.4 The Chain Rule 
14.3/ #9,1418,27,3046,67 14.4/ #7,28,29,33,39,44 

9  14.5 Directional Derivatives and Gradient Vectors 14.6 Tangent Planes and Differentials 
14.5/ #36,9,10,21,28,29,31 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/ #18,12,15,18,23,29,35,38,43,53 

13  15.3 Area by Double Integration 15.4 Double Integrals in Polar Form 
15.3/ #68,16,18,21 15.4/ #16,9,11,16,23,25,28,35 

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/ #37,1317,25,31,38,47,59,65  
17  15.8 Substitution in Multiple Integrals  15.8/ #1,3,6,7,9  
18  16.1 Line Integrals of Scalar Functions  16.1/ #19,14,15,25,26,29,35,36  
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  
Exam Protocols
More details about the exams will be posted on this website as the semester progresses.
Midterm 1: Monday October 10, 9:20 10:40 pm
Midterm 2: Monday November 14, 9:20 10:40 pm
Final Exam: Friday December 16, 12:00  3:00 pm
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 builtin 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 problemsolving 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 onscreen 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 popup screens. You MUST enable popups 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 addons 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.
Grading Weights
Component  Weight  500 points total 

Classwork  100 points (20% of grade) 
  
Midterm 1  110 points (22% of grade) 
Midterm 2  110 points (22% of grade) 
  
Final exam  180 points (36% of grade) 
Solutions to practice exams and additional documents
Extra problems 251 : 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.

Information on the Oct 10 Exam
Dear students,
Here is the information you need to know about the upcoming exam.
 Date of the exam: Oct 10.
 Duration of the exam: 80 minutes.
 Time of the exam: 9:20 pm 10:40 pm (EDT).The location will be provided by your instructor.
 This exam will be inperson, and will consist of 6 problems.
 Personal Note Sheet: Any formula listed / mentioned in the study guide "Oct10guide" or formulas from previous calculus courses 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 twosided piece of paper, no bigger than 8.5 by 11 inches, and no 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/academicintegritypolicy/
•Academic Integrity Policy  New BrunswickRutgers University  Division of Student Affairs – New Brunswick. Search This Website. Academic Integrity Policy; For Instructors and AIFsnbacademicintegrity.rutgers.edu •), which may lead to a zero on the test. 
•If you have been approved (by ODS) for extra time, please be in contact with them in order to take the exam they will proctor.
 Item #3 in the "Common Hour Exam Policies" website https://scheduling.rutgers.edu/scheduling/examscheduling/commonhourexamschedule/commonhourexampolicies details which situations may allow you to take the makeup exam:
 "Certain scheduled Rutgers activities will take precedence over common hour examinations for students who are formally registered to participate in those activities. Activities that take precedence over common hour examinations will include regularly scheduled Rutgers classes, and scheduled Rutgers athletic events. Students who have conflicts between such activities and common hour examinations must be offered an alternate examination"
 If you have such a situation and need a makeup, please notify your own professor as soon as possible.
At the beginning of this announcement, you will find 3 practice exams: "prac1oct10” , "prac2oct10", "prac3oct10".
Solutions for these practice exams will be posted on Oct 3 at the beginning of this announcement.
On this announcement you will also find a study guide for this exam, called Oct10guide.pdf
The sections of the book for the Oct 10 exam are:
12.1 ThreeDimensional Coordinate Systems
12.2 Vectors
12.3 The Dot Product
12.4 The Cross Product
12.5 Lines and Planes in Space
12.6 Cylinders and Quadric Surfaces
13.1 Curves in Space and their Tangents
13.2 Integrals of VectorValued Functions, Projectile Motion
13.3 Arc Length in Space
14.1 Functions of Several Variables
14.2 Limits and Continuity in Higher Dimensions
 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
Letter Grades for Oct 10 Exam
added mode details to the solutions in problem 5
Dear 251 student,
This is the letter grade scale for the October 10 exam only. This scale does not apply to the next midterm, 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 test.
If X = your exam percentage score (out of 100 points), then the corresponding letter grade is
A: 84<= X <= 100
B+: 78<= X < 84
B: 67<= X < 78
C+: 61<= X < 67
C: 50<= X < 61
D: 42<= X < 50
F: X < 42
Sincerely,
MATH 251 Professors

Information on the Nov 14 Exam
Prac1 nov14[sol] (more details given, and typos fixed in #6)
Prac2 nov14[sol] (more details given, and typos fixed in #6)
Prac3 nov14[sol] (more details given, and typos fixed in #5)
Dear students,
Here is the information you need to know about the upcoming exam.
 Date of the exam: Nov 14.
 Duration of the exam: 80 minutes.
 Time of the exam: 9:20 pm 10:40 pm (EDT).The location will be provided by your instructor.
 This exam will be inperson, and will consist of 6 problems.
 Personal Note Sheet: Any formula listed / mentioned in the study guide "Nov14guide" or formulas from previous calculus courses or the previous study guide 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 twosided piece of paper, no bigger than 8.5 by 11 inches, and no other exam materials may be accessed while you are taking the exam.
 Other things not allowed: Bathroom breaks, 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 (
If you have been approved (by ODS) for extra time, please be in contact with them in order to take the exam they will proctor.
 Item #3 in the "Common Hour Exam Policies" website https://scheduling.rutgers.edu/scheduling/examscheduling/commonhourexamschedule/commonhourexampolicies details which situations may allow you to take the makeup exam:
 "Certain scheduled Rutgers activities will take precedence over common hour examinations for students who are formally registered to participate in those activities. Activities that take precedence over common hour examinations will include regularly scheduled Rutgers classes, and scheduled Rutgers athletic events. Students who have conflicts between such activities and common hour examinations must be offered an alternate examination"
 If you have such a situation and need a makeup, please notify your own professor as soon as possible.
At the beginning of this announcement, you will find 3 practice exams: "prac1nov14” , "prac2nov14", "prac3nov14".
Solutions for these practice exams will be posted on Nov 7 at the beginning of this announcement.
On this announcement you will also find a study guide for this exam, called Nov14guide.pdf
The sections of the book for the Nov 14 exam are:
14.3 Partial Derivatives
14.4 The Chain Rule
14.5 Directional Derivatives and Gradient Vectors
14.6 Tangent Planes and Differentials
14.7 Extreme Values and Saddle Points
14.8 Lagrange Multipliers
15.1 Double and Interated Integrals over Rectangles
15.2 Double Integrals over General Regions
15.3 Area by Double Integration
15.4 Double Integrals in Polar Form
15.5 Triple Integrals in Rectangular Coordinates
 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

Letter Grades for Nov 14 Exam
Dear 251 student,
This is the letter grade scale for the Nov 14 exam only. This scale does not apply 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 test.
If X = your exam percentage score (out of 100 points), then the corresponding letter grade is
A: 86<= X <= 100
B+: 82<= X < 86
B: 74<= X < 82
C+: 70<= X < 74
C: 62<= X < 70
D: 57<= X < 62
F: X < 57
Sincerely,
MATH 251 Professors

INFORMATION ON THE FINAL EXAM (DEC 16)
 Date of the exam: Dec 16.
 Duration of the exam: 180 minutes.
 Time of the exam: 12:00 pm 3:00 pm (EDT).The location will be provided by your instructor.
 This exam will be inperson, and will consist of 10 problems.
 Personal Note Sheet: Any formula listed / mentioned in the study guide "finalguide" or formulas from previous calculus courses or the previous study guides 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 twosided piece of paper, no bigger than 8.5 by 11 inches, and no other exam materials may be accessed while you are taking the exam.
 Other things not allowed: Bathroom breaks, 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.
If you have been approved (by ODS) for extra time, please be in contact with them in order to take the exam they will proctor.
 Item #3 in the "Common Hour Exam Policies" website https://scheduling.rutgers.edu/scheduling/examscheduling/commonhourexamschedule/commonhourexampolicies details which situations may allow you to take the makeup exam:
 If you have such a situation and need a makeup, please notify your own professor as soon as possible.
At the beginning of this announcement, you will find 3 practice exams: "prac1dec16” , "prac2dec16", "prac3dec16".
Solutions for these practice exams will be posted on Dec 9 at the beginning of this announcement.
On this announcement you will also find a study guide for this exam, called finalguide.pdf. The final exam is cumulative, meaning that the material from the previous midterms can also be evaluated on this exam. However, there will be a bigger emphasis on the material from Chapters 15 and 16 of the book.
The NEW sections of the book for the final exam are:
15.7 Triple Integrals in Cylindrical and Spherical Coordinates
15.8 Substitution in Multiple
16.1 Line Integrals of Scalar Functions
16.2 Vector Fields & Line Integrals: Work Circulation Flux
16.3 Path Independence, Conservative Fields Potentials
16.4 Green's Theorem in the Plane
16.5 Surfaces and Area

16.6 Surface Integrals

16.7 Stokes' Theorem

16.8 The Divergence Theorem and a Unified Theory

 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