Math 13, Winter 2022

Multivariable Calculus


This course is a sequel to MATH 8 and provides an introduction to calculus of vector-valued functions. Topics include differentiation and integration of parametrically defined functions with interpretations of velocity, acceleration, arc length and curvature. Other topics include iterated, double, triple and surface integrals including change of coordinates. The remainder of the course is devoted to vector fields, line integrals, Green's theorem, curl and divergence, and Stokes' theorem.


Math 8 or equivalent.

Lecture Information

Instructor:Peter MuchaJustin MillerMike Wong
Time:(10) 10:10-11:15 AM(11) 11:30 AM-12:35 PM(12) 12:50-1:55 PM
Location:007 Kemeny007 Kemeny007 Kemeny
Email:Peter MuchaJustin MillerMike Wong
Office:240 Kemeny Hall315 Kemeny Hall320 Kemeny Hall
Office Hours:T 11:30-1:00T 12:15-1:05, W 2:00-3:00T: 1:20-2:50

Please contact your instructor if you'd like to make an appointment to meet.


The course shall use two textbooks in addition to online materials found on this site. You will only need one of the texts, not both.

"Openstax Calculus Volume 3", a free learning resource provided by Rice University. It is strongly recommended that students consider this before spending money on a textbook.

"Calculus Early Transcendentals Multivariable", by Rogawski & Adams, 3rd Edition, ISBN: 978-1464171758


There will be midterm exam and a final exam. Unless otherwise stated, all exams are to be completed without any outside assistance, including but not limited to communicating with other people, electronics, the internet, calculators, or notes.

Midterm2/08Moore Hall B13 Filene Auditorium5-9pm (2 hours)
Final3/12Carpenter 133pm


Homework will consist of short WebWork problem sets assigned each lecture day, and longer written assignments assigned weekly. Written assignments will be due Thursdays by 11:59pm. Collaborating with and assisting your fellow students is strongly encouraged, however all students must submit their own work.

For written assignments, you are required to show your work for full credit. We strongly discourage the use of calculators on written homeworks, but they are not prohibited. Calculators will be prohibited on exams, so not using them on homeworks will better prepare you to succeed on the exams.

Office Hours

Dr. Mucha's office hours are from 11:30am-1:00pm on Tuesdays.

Dr. Wong's office hours are from 1:20-1:50 on Tuesdays.

Dr. Miller's office hours are held virtually from 12:15-1:05 on Tuesdays and from 2:00-3:00 on Wednesdays. They can be accessed via the Zoom link here: Link


Please see the Canvas page for information on tutorials and TA office hours.

Honor Policy

Academic integrity is at the core of our mission as mathematicians and educators, and we take it very seriously. We also believe in working and learning together.

Cooperation on homework is permitted (and encouraged), but you are not allowed to simply copy the answers from someone else. You can share your thoughts with other students, but afterwards make sure to write only your own understanding of the problem.

On WeBWorK assignments, each person in the class will receive similar problems, but the numbers will differ slightly. You can work together to come up with ideas to apply to the problems. Then, once you understand how to do the problems, log in to your own account and do the problems assigned to you by yourself.

On written homework, you are encouraged to work together and you may get help from others, but you must write up the answers yourself. If you are part of a group of students that produces an answer to a problem, you cannot then copy that group answer. You must write up the answer individually, in your own words. This applies to working with tutors as well: students are welcome to take notes when working with tutors on general principles and techniques and on other example problems, but must work on the assigned homework problems on their own.

We ask that you include with each written homework problem a list of "collaborators" with whom you worked on the problem or from whom you got help. If you worked alone, write "no collaborators". The main reason for this requirement is for you to get in the practice of acknowledging the people you work with. If you found any additional on-line resource helpful, also list that. You must not simply copy verbatim from on-line sources.

On exams, you may not give or receive help from anyone. Exams in this course are closed book, and no notes, calculators, or other electronic devices are permitted. Also, since students may be taking their exams at different times, it is important not to discuss the problems with other students until everybody has taken the exam.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards. If you have any questions as to whether some action would be acceptable under the Academic Honor Principle, please speak to your instructor beforehand.

Helpful Learning Resources

COVID-19 Information


You are expected to attend class in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19.

For the health and safety of our class community, please: do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home.


In accordance with current College policy, all members of the Dartmouth community are required to wear a suitable face covering when indoors, regardless of vaccination status. This includes our classroom and other course-related locations, such as labs, studios, and office hours. If you need to take a quick drink during class, please dip your mask briefly for each sip. Eating is never permitted in the classroom. (The only exception to the mask requirement is for students with an approved disability-related accommodation; see below.) If you do not have an accommodation and refuse to comply with masking or other safety protocols, I am obligated to assure that the Covid health and safety standards are followed, and you will be asked to leave the classroom. You remain subject to course attendance policies, and dismissal from class will result in an unexcused absence. If you refuse to comply with masking or other safety protocols, and to ensure the health and safety of our community, I am obligated to report you to the Dean’s office for disciplinary action under Dartmouth’s Standards of Conduct. Additional COVID-19 protocols may emerge. Pay attention to emails from the senior administrators at the College.


This course is meant to be welcoming and accessible to students from all backgrounds. Please schedule a meeting as soon as possible to discuss any potential conflicts arising from religious observances, planned medical procedures, or other life events. There is no guarantee that accommodations can be made if you wait until the last minute, so please be proactive.

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible, or before the end of the second week of the term—at the latest, to discuss appropriate adjustments.

All students are encouraged to take care of their well-being. Dartmouth's Counseling Center and Wellness Center are available for student use. Please do not hesitate reach out to me about any difficulties you have in the course, or other stressors external to the course if you need someone to talk to.

Sexual respect, safety, and well-being are critical to Dartmouth's environment. Any and all forms of sexual assault, gender-based harassment, domestic violence, dating violence, and stalking are unacceptable and hostile to other members of the community. For more information, please visit Dartmouth's Sexual Respect website. Please note that, as a faculty member, I am a mandatory reporter under Title IX. That means that I must share any disclosures regarding conduct governed by Title IX to Dartmouth's Title IX coordinator. The Title IX office provides many resources to assist students who need them, which you can view here.

Mental Health

Even without the global pandemic, the academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. In the midst of a global pandemic, with all the uncertainty surrounding every aspect of our lives, these challenges take on an extra toll. There are a number of resources available to you on campus to support your wellness, including your undergraduate dean, Counseling and Human Development, and the Student Wellness Center.

Disability and COVID-19 Accommodations

Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Getting Started with SAS webpage;; 1-603-646-9900) and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

Course Schedule

WebWork Assignments are due by 11:59 PM on the listed date, except for the assignment due 2/08, which is due by 5pm. (I.e., it is due before the midterm.)

DateSubjectRogawski-Adams SectionOpenStax SectionHomework Due
1/07Iterated integrals15.15.1
1/10More general regions15.25.2WW - Introduction
1/12Triple integrals15.35.4WW - Iterated Integrals
1/13Homework #1
1/14Triple integrals 215.35.4WW - More General Regions
1/17MLK Jr. Day (No class)
1/18 (11, 12)Polar and Cylindrical Coordinates11.3, 15.41.3, 2.7, 5.3, 5.5WW - Triple Integrals
1/19 (10)Polar and Cylindrical Coordinates11.3, 15.41.3, 2.7, 5.3, 5.5
1/19 (11, 12)Spherical Coordinates15.45.5WW - Triple Integrals 2
1/20 (10)Spherical Coordinates15.45.5Homework #2
1/21Spherical Coordinates 215.45.5
1/24Change of variables15.65.7
1/25WW - Polar and Cylindrical Coordinates, WW - Spherical Coordinates
1/26Change of variables cont.15.65.7WW - Spherical Coordinates 2
1/27Homework #3
1/28Vector fields16.16.1WW - Change of Variables
1/31Scalar line integrals16.26.2WW - Change of Variables 2
2/02Vector line integrals16.26.2WW - Vector Fields
2/03Homework #4
2/04Fundamental Theorem of Line Integrals16.36.3WW - Scalar Line Integrals
2/07Midterm Review15.1-16.35.1-6.3WW - Vector Line Integrals
2/08Midterm15.1-16.35.1-6.3WW - Fundamental Theorem of Line Integrals
2/09Conservative Vector Fields16.36.3
2/11Green's theorem17.16.4
2/14Curl and Divergence16.16.5WW - Conservative Vector Fields
2/16Parametric surfaces16.46.6WW - Green's Theorem
2/17Homework #5
2/18Surface integrals16.46.6WW - Curl and Divergence
2/21Surface integrals on vector fields16.56.6WW - Parametric Surfaces
2/23Stokes' theorem17.26.7WW - Surface Integrals
2/24Homework #6
2/25Stokes' theorem cont.17.26.7WW - Surface Integrals on Vector Fields
2/28Divergence theorem17.36.8WW - Stokes' Theorem
3/02Divergence theorem cont.17.36.8WW - Stokes' Theorem 2
3/03Homework #7
3/04Review and Advanced Topics16.1-17.36.1-6.8WW - Divergence Theorem
3/07Final Review15.1-17.35.1-6.8WW - Divergence Theorem 2
3/12Final Exam15.1-17.35.1-6.8


Your final grade will be calculated using the following breakdown of your WebWork, written homework, and exam scores.

AssignmentPercent of Final Grade
Written Homework20%
Final Exam30%

Late assignments will not be accepted and extensions will not be granted without a university-approved excuse.


WebWork will be assigned at 8am on the day of the lecture covering the relevant material. Assignments are due at 11:59pm two lectures later. For example, assignments assigned Monday at 8am are due Friday at 11:59pm, and assignments assigned Wednesday at 8am are due by Monday at 11:59pm. WebWork assignments are composed of three questions and students have unlimited attempts to complete each one. If you are struggling to answer a question, you are strongly encouraged to seek help from an instructor, TA, or tutor before the deadline in order to maximize your points and ensure you understand the material. The lowest two scores will be dropped.

Written Homework

There will be written homework assignments each week (except for the week of the midterm), due Thursdays at 11:59pm. Worksheets questions will cover the previous week of lecture material. For example, the worksheet due on 1/20 will cover the lecture material from 1/10, 1/12, and 1/14. The lowest worksheet score will be dropped. For written assignments, you are required to show your work for full credit. We strongly discourage the use of calculators on written homeworks, but they are not prohibited. Calculators will be prohibited on exams, so not using them on homeworks will better prepare you to succeed on the exams.

Each homework problem will be graded out of three points. You will receive three points for a correct or nearly correct solution to a problem. (For example, minor mistakes like missing a minus sign or copying down a number incorrectly should still result in the full three points.) You will receive two points for a solution which demonstrates a healthy attempt to solve the problem. You will receive one point if your solution is significantly incomplete or shows no understanding of what the problem is asking, but does show a good-faith attempt to find an answer. Zero points will be awarded for blank or woefully incomplete answers.


All assignment grades will be available through the course's Canvas page.

Lecture Notes

3/02 = Divergence Theorem 2

2/28 - Divergence Theorem

2/25 - Stokes' Theorem 2

2/23 - Stokes' Theorem

2/21 - Surface Integrals on Vector Fields

2/18 - Surface Integrals

2/16 - Parametrized Surfaces

2/14 - Curl and Divergence

2/11 - Green's Theorem

2/09 - Conservative Vector Fields

2/04 - Fundamental Theorem of Line Integrals

2/02 - Vector Line Integrals

1/31 - Scalar Line Integrals

1/28 - Vector Fields

1/26 - Change of Variables II

1/24 - Change of Variables

1/21 - Spherical Coordinates II

1/19 - Spherical Coordinates

1/18 - Polar and Cylindrical Coordinates

1/14 - Triple Integrals 2

1/12 - Triple Integrals

1/10 - Double Integrals Over More General Regions

1/7 - Iterated Integrals

1/5 - Introduction and Review


Solutions to WebWork assignments, written homeworks, and exams will be posted to Canvas: