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
 Spring 2022 Technology Requirements
 Lecture Topics
 MyLab  Online Homework
 Matlab Assignments
 Grading Weights
 Exam Protocols
 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/.
 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  

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  
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.
Short Exam: Feb 7
Midterm 1: Feb 28
Midterm 2: April 11
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 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.
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.
Matlab PrimerCharnley by professor Matt Charnley
Grading Weights
Component  Weight  500 points total 

Classwork  100 points (20% of grade) 
Short Exam  30 points (6% of grade) 
Midterm 1  100 points (20% of grade) 
Midterm 2  100 points (20% of grade) 
Final exam  170 points (34% 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.

Information on the February 7 Short Exam
Dear students,
Here is the information you need to know about the upcoming short exam.
 Date of the exam: February 7.
 Duration of the exam:45 minutes.
 Time window to access the exam: 9:20 pm 10:20 pm (EDT).
 This exam will be online, and will consist of two nonupload and two upload problems.
 For this exam we will be using Respondus. 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 9:30 pm, then you would be done with the short exam at 10:15 pm, and then you would have 20 minutes (until 10:35 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 (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 10:10 pm, you must do the upload by 10:30 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 "shortexamguide" (mentioned later in this announcement) 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, and no other exam materials may be accessed while you are taking the exam.You must show both sides of your note sheet to the camera during the Respondus login process.
 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/
nbacademicintegrity.rutgers.edu
 Late submissions to the exam questions will not be accepted, and will result in you not obtaining any points on that problem, even if the solution on mylab is correct. 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 two practice exams for the February 7 Short Exam through the MyLab site of your individual course. The assignments are called "Practice 1 for Feb 7 Short Exam” , "Practice 2 for Feb 7 Short Exam".
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 will be posted on February 1st at the beginning of this announcement.
On this website you will also find a study guide for this exam, called shortexamguide.
Likewise, on this website you will find solutions to the respondus assignment "Respondus Practice Sp 22". 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 February 7 short exam are:
 Section 12.1: ThreeDimensional 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
Here are some useful things to be aware of:
 The practice exams and the February 7 short exam consists of4 questions each.
 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 “February 7 Exam: Upload your work” where you must upload your work. You may upload it as a pdf, jpeg or jpg file. Again, you will have 20 minutes after you complete the exam on Mylab to upload your work to the canvas site.
 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).
 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 after submitting the 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 Feb 7 Short Exam
Dear 251 student,
This is the letter grade scale for the Feb 7 short exam only. This scale does not apply to any 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 test, 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: 94<= X <= 100
B: 80<= X < 94
C: 66<= X < 80
D: 60<= X < 66
F: X < 60
Sincerely,
MATH 251 Professors
Information on the February 28 Exam
Prac1 feb28 ans [fixed typos highlighted]
Prac2 feb28 ans [fixed typos highlighted]
Dear students,
Here is the information you need to know about the upcoming exam.
 Date of the exam: February 28.
 Duration of the exam: 80 minutes.
 Timeof the exam: 9:20 pm 10:40 pm (EDT).Thelocation will be provided by your instructor.
 This exam will be inperson, and will consistof6 problems.
You should think of each problem as an "upload" problem, in the sense that you should provide an explanation of your work in order to receive credit. Thus, you are supposed to give a detailed solution to each problem of the exam.
 Personal Note Sheet: Any formula listed / mentioned in the study guide "Feb28guide" or the "shortexamguide" (mentioned later in this announcement) 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, 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/
nbacademicintegrity.rutgers.edu
 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: "Practice 1 for Feb 28 Exam” , "Practice 2 for Feb 28 Exam", "Practice 3 for Feb 28 Exam"
Solutions for these practice exams will be posted on February 21st at the beginning of this announcement.
On this website you will also find a study guide for this exam, called Feb28guide.
The sections of the book for the February 28 short exam are: ***
 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
 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
*** the topics regarding the epsilondelta definition for limits, computing error bounds using the linear approximation and the concept of "differentials" will not be tested on the exam
 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 February 28 Exam
Dear 251 student,
This is theletter grade scale for the Feb 28 exam only. This scale does not apply to the next 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 test, since the remaining midterm and the final exam will likely have different grade scales.
If X = your exam numerical score (out of 100 points), then the correspondinglettergrade is
A: 87<= X <= 100
B: 71<= X < 87
C: 55<= X < 71
D: 50<= X < 55
F: X < 50
Sincerely,
MATH 251 Professors
solutions
Prac exam1 april11(typos) (highlighted, prob 1)
Prac exam4 april11(typos) (prob 1, fixed sol)
Dear students,
Here is the information you need to know about the upcoming exam.
 Date of the exam: April 11.
 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 "April11guide" or 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, 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/
nbacademicintegrity.rutgers.edu
 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.
Solutions for these practice exams will be posted on April 4 at the beginning of this announcement.
On this announcement you will also find a study guide for this exam, called April11guide.
The sections of the book for the April 11 exam are:
14.7 Extreme Values and Saddle Points
14.8 Lagrange Multipliers
15.1 Double and Iterated 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
15.7 Triple Integrals in Cylindrical and Spherical Coordinates
15.8 Substitution in Multiple Integrals
 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 April 11 Exam
Dear 251 student,
This is the letter grade scale for the April11 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, since the final exam will likely have a different grade scale.
If X = your exam numerical score (out of 100 points), then the corresponding letter grade is
A: 87<= X <= 100
B: 71<= X < 87
C: 55<= X < 71
D: 50<= X < 55
F: X < 50
Sincerely,
MATH 251 Professors
Information on the May 5 Final Exam
Dear students,
Here is the information you need to know about the upcoming (final) exam.
 Date of the exam: May 5.
 Duration of the exam: 180 minutes.
 Time of the exam: 4:00 pm 7:00 pm (EDT).The location will be provided by your instructor.
 This exam will be inperson, and it will have between 8 and 10 problems.
 Personal Note Sheet: Any formula listed / mentioned in the study guide "finalguide" or 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, 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/
nbacademicintegrity.rutgers.edu
 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.
 The website https://sasundergrad.rutgers.edu/degreerequirements/policies/finalexam 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.
Solutions for these practice exams will be posted on April 28 at the beginning of this announcement.
On this announcement you will also find a study guide for this exam, called Final Examguide.
The final exam is cumulative, thus every section of the book from the previous midterms is included on the final.
In particular, you should also check the previous practice exams to have more practice to work on. It is also important to notice that there is a bigger emphasis on Chapters 15 and 16 of the book. The NEW sections of the book for the final exam are: *
 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
Information on the Course Grades
Dear Math 251 Student,
The Math 251 instructors have spent the days since Thursday's final exam grading the final exam and compiling the course grades. Many of your instructors have posted the final exam grades and have submitted the course grades for their sections. If you have not received your final course grade from WebReg and/or "unofficial transcript site", you will be able to see them very soon.
Your course grade is determined via the table posted on the department website
Classwork 100 pts [20% of grade]
Short Exam and Midterms [4] 230 pts [46% of grade]
Final 170 pts [34 % of grade]
The number of A's, B+'s, B's, etc., that an instructor can give as the letter grade for the course are determined by a committee of instructors after grading is done.
Once each instructor determined the A’s, B+’s, B's, etc., they could assign in each class, the course score was computed using the weights mentioned previously.
For example, if the instructor was allowed to give 5 A’s in their course, the students with the top 5 course scores receive the A’s, and so on.
We wish you a happy summer and hope that you stay safe and healthy during the upcoming months.
Sincerely,
251 instructors.