Winter 2022: MAT B42 Techniques of the Calculus of Several Variables II
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
 Parker GlynnAdey (he/him)
 Call Me: “Parker” or “Professor Parker”
 EMail : parker.glynn.adey@utoronto.ca (See Communication Policy.)
 Website: http://pgadey.ca/
 Office: IC 344
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 InPerson 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 inperson midterms and an inperson exam. The threat posed by omicron variant of COVID19 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  Inperson with date to be announced 
Exam  Inperson with date to be announced 
Communication Policy
Piazza
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 team@piazza.com.
Find our class signup link at: https://piazza.com/utoronto.ca/winter2022/matb42h3s20221
Please include your name and student number in every email that you send. Mail must be from an official University of Toronto account. To make sure that your email 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 emailing me or any of the course staff. We are not evil trolls. We won’t get angry if you email us. Answering student emails is a part of our job.
However, email is only part of our job. We might not respond to your email 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 wellformatted email:
To: parker.glynn.adey@utoronto.ca
From: leonhard.euler@utoronto.ca
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.
Thanks!
B42Winter2022:1229370
All emails must include the following:
 University of Toronto Email account
 Name and student number
 The magic formula: B42Winter2022:1229370
Grading Scheme
Task  Weight 

Exam  1x50% 
Term Test  1x30% 
Assignments  (61)x4% 
Assignments
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:
 Format: solutions are neatly and correctly assembled with professional look. The graders should not struggle to read your work.
 Completeness: all steps are clearly and accurately explained.
 Content: the written solutions demonstrate mastery and fluency with the content of the course.
Common Questions:

Why does it say (61)x4% for assignments? This is a short way of writing that there are six assignments, but one of them will be dropped. It’s a saying that there are 6  1 = 5 assignments that count for 4% each.

Which assignment will get dropped? Your lowest assignment will be dropped automatically. You do not need to request this, or send an email about it.

What happens if I miss an assignment deadline? You will not be penalized for missing an assignment deadline, if you submit before the solutions are released. We understand that uploading to Crowdmark is difficult and there are technical mistakes. However, if you submit you assignment after the solutions are released then you will receive zero on the assignment.
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 inperson 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
 When will we know the date and time of the term tests? You will know the date and time of the terms at least one week before the test. We are waiting for information from the Registrar about when and where we can hold term tests.
 Will the tests be hard or easy? This depends on your level of fluency with the material. Some student will find the tests hard, and other students will find the tests easy. We have designed the tests to accurately assess the learning objectives, and believe that they are a fair reflection of the course.
 What should I do to prepare for the test? Practice. Practice. Practice! You can practice via the Suggested Questions from the Reading Guide, ask questioning in lecture, and attending and participating in Tutorials.
 Will all sections of the course write the same test? Yes. All sections of the course will write the same test, so that everyone gets a fair and equal grade. All sections of the course will cover the same material.
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 regrade 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
 Why do you only grade two of the five questions? There are three reasons for this. Firstly, we want to save time and energy. Secondly, we think that if you are doing your best work then any two questions should be representative of your whole assignment. Thirdly, we want you to learn to evaluate your own work. We post solutions to all five problems so that you can see how well you did on the three unmarked problems.
 Can you tell me which two questions will be graded? No. We cannot reveal to you which two questions will be graded.
Regrading
You can use the official MAT B42 ReGrade Form.
All requests will be read and thoroughly considered by our head TA and Parker GlynnAdey. 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
 How will I know that my assignment was regraded? You will receive an email telling you the outcome of your regrade request.
 Can I ask why I got a paricular grade? Yes, you can use the regrade system to ask for more information about why you got a particular grade. If you want clarification about grading please write: “REQUEST FOR CLARIFICATION”.
Textbook
We will use:
 Marsden, Jerrold E., and Anthony Tromba. Vector calculus. Macmillan, 2003.
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.
WeekbyWeek Schedule
 Week 1: Parametric Equations
 Marsden and Tromba: 2.4 Introduction to Paths and Curves
 Marsden and Tromba: 4.1 Acceleration and Newton’s Second Law
 Marsden and Tromba: 4.2 Arc Length
 Week 2: Vector Fields
 Marsden and Tromba: 4.3 Vector Fields
 Marsden and Tromba: 7.1 The Path Integral
 Week 3: Line Integrals
 Marsden and Tromba: 7.2: Line Integrals
 Marsden and Tromba: 8.1: Green’s Theorem (without the vector form)
 Week 4: Vector Valued Functions
 Marsden and Tromba: 4.4 Divergence and Curl
 Marsden and Tromba: 8.1 Green’s Theorem (without the vector form)
 Week 5: Surfaces (Part 1)
 Marsden and Tromba: 7.3 Parametrized Surfaces
 Marsden and Tromba: 7.4 Area of a Surface
 Week 6: Surfaces (Part 2)
 Marsden and Tromba: 7.4 Area of a Surface
 Marsden and Tromba: 7.5: Integrals of Scalar Functions over Surfaces
 Week 7: The Classical Theorems of Vector Calculus (Part 1)
 Marsden and Tromba: 8.2: Stokes’ Theorem
 Marsden and Tromba: 8.3: Conservative Vector Fields
 Week 8: The Classical Theorems of Vector Calculus (Part 2)
 Marsden and Tromba: 8.4: Gauss’ Theorem
 Week 9: Introduction to Differential Forms
 Marsden and Tromba: 8.5: Differential Forms
 Week 10: Fourier Series from HughesHallett
 Fourier Series
 Week 11: Applications Fourier Series to PDEs
 Heat Equation
 Wave Equation
 Week 12: WrapUp
Recommended Exercises (Up to Week 6)
 Week 1:
 2.4 Introduction to Paths and Curves
 Calculations: 2, 3, 4, 5, 6c, 6d, 23
 4.1 Acceleration and Newton’s Second Law
 Calculations: 6, 7, 13
 Demonstration: 9, 10, 20
 4.2 Arc Length
 Calculations: 1, 2, 3, 14
 Curvature: 16, 17, 18, 19
 TNB Frames: 20, 21
 2.4 Introduction to Paths and Curves
 Week 2:
 4.3 Vector Fields
 Calculations: 2, 3, 4, 9, 10, 12
 Demonstration: 20
 7.1 The Path Integral
 Calculations: 1, 2, 7, 8, 9
 Demonstration: 21
 4.3 Vector Fields
 Week 3:
 7.2: Line Integrals
 Calculations: 1, 3, 4, 16
 Demonstrations: 6, 12, 13
 8.1: Green’s Theorem (without the vector form)
 Calculations: 3, 4, 5, 11
 7.2: Line Integrals
 Week 4:
 4.4 Divergence and Curl
 Calculations: 2, 3, 4, 13, 14, 15, 23, 29, 30, 37
 Demonstrations: 27, 33, 34
 4.4 Divergence and Curl
 Week 5:
 7.3 Parametrized Surfaces
 Calculations: 1, 2, 4, 5, 7, 9, 10, 15
 7.4 Area of a Surface
 Calculations: 1, 2, 4, 6, 11
 Demonstrations: 11, 14, 18
 7.3 Parametrized Surfaces
 Week 6:
 7.5 Integrals of Scalar Functions over Surfaces
 Calculations: 1, 2, 3, 6, 8, 9, 14, 22
 Demonstrations: 12, 26, 28
 Dirichlet’s Functional: 23, 24, 25
 7.5 Integrals of Scalar Functions over Surfaces
 Week 7: The Classical Theorems of Vector Calculus (Part 1)
 8.2: Stokes’ Theorem
 Calculations: 3, 8, 11
 Demonstrations: 22, 23
 8.3: Conservative Vector Fields
 Calculations: 1, 3, 7, 19, 21
 Demonstrations: 5, 12
 8.2: Stokes’ Theorem
 Week 8: The Classical Theorems of Vector Calculus (Part 2)
 8.4: Gauss’ Theorem
 Calculations: 1, 3, 7, 9
 Demonstrations: 18, 24, 29
 8.4: Gauss’ Theorem
 Week 9: Introduction to Differential Forms
 8.5: Differential Forms
 Calculations: 1, 3, 4
 Demonstrations: 6, 7, 13
 8.5: Differential Forms
 Week 10: Fourier Series (HughesHallett)
 Fourier Series
 Week 11: Applications of Fourier Series to PDEs (Xiao Jie’s Notes)
 Heat Equation
 Wave Equation
 Week 12: WrapUp
Schedule of Tasks
Week  Task 

1  
2  
3  Assignment 1 
4  
5  Assignment 2 
6  
Reading Week  Assignment 3 
7  Assignment 4 
8  
9  Assignment 5 
10  
11  Assignment 6 
12 
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:
 What resources can I use on assignments? You can use your notes, the textbooks, online references such as videos and Wikipedia, and any online calculators that are freely available. You can also contact Parker and the TAs. If you’re unsure whether you’re allowed to use a particular resource: Ask!
 What calculator do you recommend for this course? When designing the course, and checking the assignments, I used WolframAlpha and Desmos.
 What recources are forbidden durings tests and assignments? Anything that is private or specific to you. You cannot post questions online to websites such as Chegg and other studyhelp sites, participate in group chats, ask for help from online communities. Please note: You can ask the instructional team for help!
 How does the instructional team monitor academic integrity? We grade the assignments and tests. If we spot suspicious answers, we enter them in to a spreadsheet. We also look at popular study help websites to see if our the material from our course is being posted online. At the end of the course, we compile all the academic integrity material and send it to the Office of Academic Integrity.
 What should I do if someone is pressuring me in to cheating? Contact Parker. Send any screenshots / photos / text messages that are relevant. You have the right to succeed in this course without other people trapping you in an academic integrity violation case.
References on Academic Integrity
Campus Resources
Facilitated Study Groups
Facilitated Study Groups (FSGs) are weekly dropin 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, judgmentfree 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:
uoft.me/AcademicLearningSupport
Math and Stats Support
The Center for Teaching and Learning’s Math & Statistics Support provides free seminars, workshops, virtual tutoring, individual appointments, and smallgroup 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: https://uoft.me/MathStats