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\subsection*{About Math 226}



Math 226  and 227 are multivariable calculus classes recommended for Honours students in Mathematics at the University of British Columbia. They are aimed at students who have a deeper interest in mathematics, with more emphasis on rigorous proofs and mathematical maturity than in our general-purpose multivariable calculus courses. Accordingly, less time is spent on computational techniques (such as evaluating integrals) and applications (physics, engineering).

Both courses are taught under specific curricular constraints. Most students come to Math 226 directly from single-variable calculus courses emphasizing computations and applications. They are smart, motivated, and they want to see cool mathematics. Many, however, have had little or no exposure to epsilon-delta proofs (let alone the basic concepts of real analysis) and no formal training in rigorous proof writing. 



Math 226 covers functions of several variables, their continuity, differentiability, optimization problems, and integration. The intention is to study the material in some depth, include selected advanced topics (Implicit Function Theorem, the precise definition of differentiability), and promote correct mathematical reasoning. Our usual textbook is {\it Calculus: Several Variables} (also available as the second half of {\it Calculus: A Complete Course}), by Robert A.~Adams and Christopher Essex, published by Pearson. It is an excellent resource, ambitious yet accessible. A fully rigorous treatment (I'm a fan of Folland's {\it Advanced Calculus}) would be beyond what I can expect to accomplish in this course. 

These notes, based on my lecture scripts (see below), reflect my priorities in class. Calculus is rich in big conceptual ideas, often geometric in nature, but also requires considerable attention to detail in bringing those ideas to life. My goal in teaching is to find both a balance between these two aspects and a path for students to go back and forth between the macro and micro pictures as needed. This goal is more aspirational than attained, given that any balance must be based on compromises, but I hope to give my students the means to find their own way around the subject.


Rigorous proofs on a small scale are introduced early and reinforced throughout the course. At the same time, I often break down longer proofs into components, clarifying the main ideas and presenting a flexible toolkit that students can be expected to learn to use. For the main idea, I might start with a simple model case or a calculation that works under stronger assumptions. Then we move in with additional tools such as reductions and approximations. I follow up with practice and homework problems along similar lines, such as ``prove this theorem in detail in this specific case" or ``assuming we know A, prove B." 

I encourage critical thinking and asking questions such as ``how are these concepts different?" or ``where did we use this assumption?". I have included examples to illustrate what can happen in various counterintuitive situations, and tracked down where the assumptions of the main theorems are used. I also recognize that when a student wants to know why something works, pointing to a formal proof is not always a satisfactory answer. Some of the explanations in these notes evolved over time as I worked to improve my answers to such questions. 

Throughout the course, I emphasize working with inequalities and estimates. 
This is a recurring theme, from evaluating limits to the rigorous concept of differentiability to convergence of improper integrals.
I want my students to be able to construct inequality-based proofs, compare orders of magnitude, use approximations with estimates on the error terms. I hope they will find this useful beyond multivariable calculus.

My classes usually include a few extra topics, such as Jordan area and the Riemann integrability criterion, that Math Honours students find attractive. These are also included here, with suggestions for further exploration. 
On the other hand, some of the conceptually easier parts are abbreviated. Computations are kept reasonably simple to save time. I worked with the remaining material to construct a coherent sequence that I could present in class without rushing.

Most of the problems here are taken from my exams and homework assignments. 
Some are routine (``compute this quantity using the formula provided"); for more practice of this type, I encourage students to use their textbooks as well as online systems such as WebWork. In homework assignments, I often ask questions that are more challenging, proof-oriented, require some creative thinking, and lend well to collaborations for those so inclined. I ask the students to write their solutions clearly, in full sentences, with correct and complete explanations provided. These are the ``proof questions" appearing in most sections of this book. 
The Worked Problems come with solutions; they often include the examples I used in my lectures. The solutions to the Practice Problems are provided in a separate Solutions Manual volume.



\subsection*{About these notes}

I taught Math 226 eight times between 2005 and 2021, and Math 227 seven times between 2006 and 2022. I also taught Math 317,
a more conventional calculus course with scope similar to 227, in 2011 and 2014. Some of this repetition was likely due to the Covid pandemic: as classes went online and workloads skyrocketed, the department made the sensible choice to assign most courses to faculty who had taught them previously. Thus, I taught 226 and 227 online in 2020-21, and in person but with a significant online component in 2021-22. 

By then, I had a full set of handwritten lecture notes for both courses. These were my actual class scripts, with the material selected and presented accordingly. As I taught the online courses, I rewrote the notes one section at a time, cleaned them up, and posted them on the course website. They were still handwritten at that point, with hand-drawn pictures, but with further details and explanations added for student use. I also put together a series of typed practice problem sets with solutions, based on my midterm tests, final exams, and homework questions from previous years.

The reception was excellent, with both students and myself relying on these notes to a greater extent than I had expected. I suggested to the department that, with appropriate support, I could make my course materials available to all students and instructors as a free supplementary resource. I asked for a teaching assistant to type up the handwritten part and help compile the files. I also requested a one-course teaching release to compensate for the additional work I was undertaking, and I asked to continue to teach Math 226 and 227 for the duration of the project (initially, I expected 2 years; it took 3, from 2023-24 to 2025-26).  The department agreed that there was a need for such a resource, and granted the support I requested.


The TA assigned to this project was Caleb Marshall, a doctoral student at UBC at the time. He worked on it with me from September 2023 to August 2025.  Caleb's main responsibility was to type up, in \LaTeX, my handwritten notes and use computer graphics to make professional-looking versions of my hand-drawn pictures. Transcribing my handwritten notes involved a significant amount of work beyond simple typesetting: some text needed to be added, parts of the material needed to be reorganized, and it took some time (especially at the beginning) to establish a workflow and sort out various technical issues. 

The text Caleb produced was lively and engaging, and I have kept many of the expository comments he added. I talked with him frequently about my pedagogical approach, the organization of each section, the level of mathematical rigour, the choice of examples and exercises, and much more. There were many times when a problem needed to be modified or a proof needed clarification. I often asked Caleb for input in such cases so that we could find a solution together, and then let him work out the details. I checked and edited the files before making them available to students. 

In the first year (2023-24), we focused on getting the typed content on the page, so that the quality of the illustrations was less of a concern. In the second year, Caleb had more time to work on the graphics, and he trained himself to be an excellent mathematical illustrator. He used Geogebra (a freely available online program) as well as occasional iPad drawings, and became highly competent at both. 
He also wrote the first drafts of Sections 1.1.5 and 1.1.7 (on proofs and quantifiers), and contributed a few completely new problems with solutions. These include Practice Problem 2 in Section 2.5 (harmonic functions), Practice Problems 2 and 3 in Section 3.1 (linearization), Practice Problem 6 in Section 4.1 (critical points), Practice Problem 4 in Section 4.3 (Lagrange Multipliers), and 
Practice Problem 7 in Section 5.2 (centroid of a triangle).


We both edited various parts of the text during the 2024-25 academic year, improving the presentation and taking student feedback into account. (The reader might notice two different writing styles.) In Summer 2025, Caleb compiled the Math 226 notes and problem solutions into two volumes, this one and the separate Solutions Manual. Math 227 notes were compiled later in 2025 by another TA, Mihai Marian. In 2025-2026, the last year of the project, I had more time to reassess the pedagogical approach and review the notes for overall consistency. I reorganized parts of the material, made the final adjustments, and added a new batch of problems and solutions from the 2023-26 exams and homework assignments.


We had minimal resources, with no copy editors or production team. The formatting is often {\it ad hoc}, based on whatever worked for us for that part of the material. There are no strict rules as to what gets framed in a box or how the examples are numbered. The presentation can be a little bit informal. Nonetheless, my students found these notes useful, and I hope that so will their future readers.



\subsection*{Acknowledgements}

This book was written on the Point Grey campus of the University of British Columbia, located on the traditional, ancestral, and unceded territory of the %xwməθkwəy̓əm 
Musqueam people.


I am grateful to the Department of Mathematics at the University of British Columbia, and especially to Daniel Coombs (Head) and Fok-Shuen Leung (Undergraduate Chair), for approving and supporting the project. I owe many thanks to Caleb Marshall for the two years of working on it together. Krysten Casumpang, a staff member in the department, did a great job designing the cover pages.

I also would like to acknowledge all Teaching Assistants who helped me run these courses over the years. In Math 226, this includes Matthew Brassil, Nathan Lawrence, Manuel Ruivo de Oliveira, Ethan White, Emmanuel Michta, Summer Sheng, Yuveshen Moorogen, Yuze Cui, Sebastian Gant, Nikou Lei. Unfortunately I have not kept TA records from 2016 and earlier, but nonetheless I would like to thank any of my former TAs who might be reading this.  


Last but not least, I'm grateful to all undergraduate students who took Math 226 with me. They have provided valuable and actionable feedback, asked challenging questions, and otherwise pushed me to improve this course. They have also pointed out many typos and errors in earlier versions of these notes. It was a pleasure to work with them.


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\noindent
\hfill Izabella {\L}aba

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\hfill Vancouver, July 2026




