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Winter 2022: MAT B42 Techniques of the Calculus of Several Variables II

MAT B42 Graphic Syllabus

From the Course Calendar: “Fourier series. Vector fields in $\mathbb{R}^n$, Divergence and curl, curves, parametric representation of curves, path and line integrals, surfaces, parametric representations of surfaces, surface integrals. Green’s, Gauss', and Stokes' theorems will also be covered. An introduction to differential forms, total derivative.”

Course Staff


Professor’s Message

This course is really special. It reminds me a lot of the material that I worked on in graduate school when I studied high dimensional manifolds. The laws of vector calculus power the laws of electromagnetism in physics and they build the path to studying curved spaces. This is the course where we start to do really nice stuff with calculus.

In your introduction to calculus, you learned the fundamental theorem of calculus: $\displaystyle \int_a^b f(x) dx = F(b) - F(a)$. We are going to generalize this to higher dimensions and start building the tools to study manifolds. Along the way, we’ll talk about curves and surfaces which are studied in MAT C63 and Fourier analysis used in MAT D46.

I hope that you’re excited as for this journey as I am!

Policy on In-Person vs Online Delivery

The Ontario government has asked us to remain online until January 31st. We are going to stay online for the whole semester. However, we are planning to have in-person midterms and an in-person exam. The threat posed by omicron variant of COVID-19 is still present, and we want to keep everyone as safe as possible. As we get more information from the Registrar and the Ontario government, we will keep you updated and informed.

Currently, we plan for:

Activity Mode of Delivery
Lecture Online for the full semester
Tutorial Online for the full semester
Term Test In-person with date to be announced
Exam In-person with date to be announced

Communication Policy


This term we will be using Piazza for class discussion. The system is highly catered to getting you help fast and efficiently from classmates, the TAs, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email

Find our class signup link at:


Please include your name and student number in every e-mail that you send. Mail must be from an official University of Toronto account. To make sure that your e-mail does not get lost you must include this magic formula in the body:


Be sure to include the precise question, and the problem or difficulty. If you’re not able to write out the question, take a photo or attach a PDF.

Above all, don’t worry about e-mailing me or any of the course staff. We are not evil trolls. We won’t get angry if you e-mail us. Answering student e-mails is a part of our job.

However, e-mail is only part of our job. We might not respond to your e-mail on the same day that you send it. Generally, give us at least two business days to respond. Parker has limited access to his computer on Tuesdays, Thursdays, and weekends.

Here is an example of a well-formatted e-mail:

Subject: [MAT B42] What is a gradient?

Hi! I am Leonhard Euler (12932188) from MAT B42.
I need help with this question: Find the gradient of f(x,y) = x^2 + xy
My problem is this: I don’t know what the word "gradient" means.



All e-mails must include the following:

Grading Scheme

Task Weight
Exam 1x50%
Term Test 1x30%
Assignments (6-1)x4%


Goal: these assignments give you the opportunity to deepen your understanding of topics covered in this course, and to practice. We use these assignments to determine if you can solve problems slowly, without time constraints.

Procedure: we will be using Crowdmark to grade assignment submissions. You will get a personalized submission link sent to your UToronto email address. Do NOT share this link with other students.

Submission Guidelines: Assignments need to be submitted online through Crowdmark. You will have a week to write the assignment.

Evaluation Criteria: present your solutions in a logical and clear manner. Detailed solutions will be made available shortly after the deadline of submission.

Please pay attention to the following when writing tests:

Common Questions:

Term Tests

Goal: these written tests give you the opportunity to demonstrate your understanding of core concepts and topics in a written format. You will gain experience of communicating mathematical ideas in a logical manner. We write tests in a limited amount of time to assess your fluency with the material.

Procedure: We will hold term tests outside of class time. We will post an announcement on Quercus about where and when to write the term tests. They will be written in-person and invigilated. You will have two hours to write each test.

Evaluation Criteria: In general, you need to present your solutions in a logical and clear manner. Detailed solutions will be made available shortly after the tests.

Common Questions about Term Tests

How Does Grading Work?

Parker sets the assignments and term tests. You, the student, do your best work and hand it in. The TAs then grade your term tests and assignments. The TAs then return your graded work, and you may request a re-grade or further comments if you think the grading is unclear.

Please note, that the professor does not directly look at your homework.

For assignments, the TAs will only grade two of the five questions. This policy of subset grading helps us to save time and energy, and teaches you to evaluate your own work.

For term tests, the TAs will grade all the questions.

Common Questions about Grading


You can use the official MAT B42 Re-Grade Form.

All requests will be read and thoroughly considered by our head TA and Parker Glynn-Adey. We will make sure that you get a response before the final exam. Submit one copy of this form for each regrade request. If you would like three questions regraded, please submit three copies of this form.

Common Questions about Regrading


We will use:

It is available through the library, bookstore, and can be found online. It’s a great read with tonnes of interesting material about the relationship between calculus, geometry, and physics. I recommend getting a paper copy as a reference. I keep mine on my desk all the time and flip through it often.

Week-by-Week Schedule

Recommended Exercises (Up to Week 6)

Schedule of Tasks

Week Task
3 Assignment 1
5 Assignment 2
Reading Week Assignment 3
7 Assignment 4
9 Assignment 5
11 Assignment 6

Exact Dates of Tasks

Task Date and Time of Task Date and Time of Solutions
Assignment 1 Monday January 10th at 13:00 to Thursday January 27th at 13:00 Monday January 31st at 12:45
Assignment 2 Monday January 31st at 13:00 to Thursday February 10th at 13:00 Monday February 14th at 12:45
Assignment 3 Monday February 14th at 13:00 to Thursday February 24th at 13:00 Monday February 28th at 12:45
Assignment 4 Monday February 21st at 13:00 to Thursday March 3rd at 13:00 Monday March 7th at 12:45
Assignment 5 Monday March 7th at 13:00 to Thursday March 17th at 13:00 Monday March 21st at 12:45
Assignment 6 Monday March 21st at 13:00 to Thursday March 31st at 13:00 Monday April 4th at 12:45
Term Test will be announced when the Registrar gives us a date
Exam will be announced when the Registrar gives us a date

To add the dates above to your Google Calendar, import this calendar.

Academic Integrity

The instructional team wants to make sure that everyone has a fair chance to succeed in this course. Therefore, we define an academic integrity violation to be accessing or communicating with any person or resource that gives a unique advantage to some students. For example: participating in private group chats, posting questions and reading solutions on websites, hiring or requesting external help. All of these would give some students advantages that would not be available to other students.

Common Questions:

References on Academic Integrity

Campus Resources

Facilitated Study Groups

Facilitated Study Groups (FSGs) are weekly drop-in collaborative learning sessions for students who want to improve their understanding of challenging content in selected courses at UTSC. FSG sessions give you a chance to discuss the lecture material and important concepts, develop study strategies and fresh approaches, and work through problems as a group to prepare for your assignments and tests.

Research shows that students who regularly attend FSGs gain a deeper understanding of the material and, on average, achieve better grades. It’s also a great way to meet classmates and study in a relaxed, judgment-free space.

Center for Teaching and Learning Academic Learning Support

The Centre for Teaching and Learning provides academic learning support to students through online tutoring, workshops, and peer supports to drive student success. To find out more about all their offerings, see this website:

Math and Stats Support

The Center for Teaching and Learning’s Math & Statistics Support provides free seminars, workshops, virtual tutoring, individual appointments, and small-group consultations to improve students’ proficiency in various subjects of mathematics and statistics. Their main goal is to create a friendly, vibrant environment in which all students can come to learn and succeed.

For their online help offerings see:



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