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\subsection*{About Math 227}



Math 226  and 227 are multivariable calculus classes recommended for Honours students in Mathematics at the University of British Columbia. 
Math 227 is the second course in the sequence, with Math 226 as a required prerequisite. It covers curves in two and three dimensions, surfaces in three dimensions, line and surface integrals, vector fields, and the main theorems of vector calculus: Green's Theorem, Divergence (Gauss's) Theorem, Stokes's Theorem. As in Math 226, there is more emphasis on rigorous proofs and critical mathematical thinking, and less on computations and applications. 

A major difference between Math 227 and its more conventional counterparts is that the last few weeks are devoted to an introduction to differential forms. This is a general theory that unites many of the geometric concepts introduced earlier in class, extends them to arbitrary dimensions, and provides a glimpse into the beauty of more abstract mathematics. The class ends with the Generalized Stokes Theorem, a common framework for the main theorems of vector calculus.

We spend less time on epsilon-delta proofs and local regularity issues, having covered that type of questions quite extensively in Math 226. Instead, we usually assume enough regularity to proceed with the main constructions, and then focus on the geometry involved. The key new skills include visualizing 3-dimensional geometric structures (such as surfaces or moving frames of vectors), writing up geometric arguments involving descriptions and manipulations of such structures, and, eventually, extrapolating arguments of this type to higher dimensions. 

These notes do not aim to be a complete treatment of the subject. I recommend using them together with a textbook for additional examples, applications, and further details of some of the proofs that are only sketched here.
In Math 226 and 227, we have been using {\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 one of the few calculus books that cover differential forms, including in higher dimensions.  For a fully rigorous treatment, I would refer the interested student to Folland's {\it Advanced Calculus}. 

Most of the problems here are taken from my exams and homework assignments. The Worked Problems come with solutions; the solutions to the Practice Problems are provided in a separate Solutions Manual volume. Some are routine computations: even as we develop a more general theory, it still needs to be grounded in specifics. I encourage students to use their textbooks and online systems such as WebWork for more practice of this type. The proof questions may require more time and some creative thinking. I ask my students to write up the solutions clearly, in full sentences, with correct and complete explanations provided. 





\subsection*{About these notes}

The making of this book follows the same story that was described in the preface to the Math 226 volume, so I will be brief here. This volume is, again, based on the class notes that I developed while teaching the course repeatedly over the last two decades. 
I first made the notes available to students for the online and partially-online editions of Math 227 between 2020 and 2022. They were mostly handwritten then, except that I typed up parts of the differential forms material. 
 All remaining notes (including most of this book) were initially typeset by Caleb Marshall. Caleb has also prepared the professional-quality illustrations based on my hand-drawn drafts, and worked with me on some of the problems and examples (for instance, he contributed Example 2.2.3 in Section 2.2). Between 2023 and 2025, we both edited various parts of the initial text.
Mihai Marian compiled the Math 227 files into two volumes, this one and the separate Solutions Manual. 
I made the final revisions, and added the more recent exam and homework problems, in the Spring and Summer of 2026.







\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 other Teaching Assistants who helped me run these courses over the years. In Math 227, this includes Sebastian Gant, Elliot Cheung,  Manuel Ruivo de Oliveira, Angel Cruz, Chenjian Wang, Yuveshen Mooroogen, Hao Ouyang, and Noureldein Hanafi. 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 227 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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\hfill Izabella {\L}aba

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\hfill Vancouver, July 2026



