# Courses

## 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.

## 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 (REMOVED FROM FINAL) 16.6/#3,5,6,17,23,26,28,43
25 16.7 Stokes' Theorem [REMOVED FROM FINAL] 16.7/ #5,7,11,19,23,28
26 16.8 The Divergence Theorem & A Unified Theory [REMOVED FROM FINAL] 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 February 20, 9:20- 10:40 pm

Midterm 2:  Monday April 3, 9:20- 10:40 pm

Final Exam: Thursay May 4, 4:00  - 7:00 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 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.

## Grading Weights

ComponentWeight -- 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 Feb 20 Exam

Midterm 1 guide

Prac1 midterm1

Prac1 midterm1[sol]

Prac2 midterm1

Prac2 midterm1[sol] (typo on limit problem solution fixed)

Prac3 midterm1

Prac3 midterm1[sol]

Dear students,

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

• Date of the exam: Feb 20.
• 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 in-person, and will consist of 6 problems.
• Personal Note Sheet: Any formula listed / mentioned in the study guide "midterm-1-guide" 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 two-sided 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 http://nbacademicintegrity.rutgers.edu/home/academic-integrity-policy/ 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/exam-scheduling/common-hour-exam-schedule/common-hour-exam-policies 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: "prac1-midterm1” , "prac2-midterm2", "prac3-midterm3".

Solutions for these practice exams will be posted on Feb 13 at the beginning of this announcement.

On this announcement you will also find a study guide for this exam, called midterm-1-guide.pdf

The sections of the book for the Feb 20 exam are:

12.1 Three-Dimensional 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 Vector-Valued 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 Feb 20 Exam

Dear 251 student,

Midterm1 A[sol]

Midterm1 B[sol]

This is the letter grade  scale for the February 20 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 score out of 110 points, then the corresponding letter grade is

A:    92<= X <= 110

B+:  86<= X <  92

B:    74<= X <   86

C+:  67<= X <   74

C:     50<= X <  67

D:     46<= X <  50

F:                X < 46

Sincerely,

MATH 251 Professors

## Information on the April 3 Exam

Midterm 2 guide

Prac1 midterm2

Prac2 midterm2

Prac3 midterm2

Prac1 midterm2[sol]

Prac2 midterm2[sol]

Prac3 midterm2[sol]

Dear students,

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

* Date of the exam: April 3.
* 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 in-person, and will consist of 6 problems.
* Personal Note Sheet: Any formula listed / mentioned in the study guide "Midterm-2-guide" 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 two-sided 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/exam-scheduling/common-hour-exam-schedule/common-hour-exam-policies 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: "prac1-midterm2” , "prac2-midterm2", "prac3-midterm2".    Solutions for these practice exams will be posted on March 27 at the beginning of this announcement.   On this announcement you will also find a study guide for this exam, called Midterm2-guide.pdf

The sections of the book for the April 3 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 April 3 Exam

Dear 251 student,

Midterm2 A[sol]

Midterm2 B[sol]

This is the letter grade  scale for the April 3 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 score out of 110 points, then the corresponding letter grade is

A:    92<= X <= 110

B+:  86<= X <  92

B:    74<= X <   86

C+:  67<= X <   74

C:     52<= X <  67

D:     48<= X <  52

F:                X < 48

Sincerely,

MATH 251 Professors

## Information on the May 4 Final Exam

Finalguide

Prac1 final

Prac2 final

Prac3 final  [problem 8 should say: use the outward flux version of Green's theorem to find the area of the region R, instead of "verify Green's theorem]

Prac1 final [sol]

Prac2 final [sol]

Prac3 final cut[sol] [solutions to the Green's problem have been fixed in this version of the solutions, check remark about the statement on this problem]

* Date of the exam: May 4.
* 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 in-person, 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 two-sided 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/exam-scheduling/common-hour-exam-schedule/common-hour-exam-policies 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: "prac1-final” , "prac2-final", "prac3-final".    Solutions for these practice exams will be posted on April 27 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    Integrals

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

* 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 Letter Grades for the Course

Dear Math 251 Student,

The Math 251 instructors have spent the days since Thursday’sfinal 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 on the department website

Classwork         100 pts        [20% of grade]

Midterms [2]    220 pts        [44% of grade]

Final                   180 pts        [36% 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 As, 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 As in their course, the students with the top 5 course scores receive the As, and so on.

We wish you a happy summer and hope that you stay safe and healthy during the upcoming months.

Sincerely,

251 instructors.

## 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.