Preface How to Use this Text
Because the text is free, any professor or student may use the electronic version of the text for no charge. For reading on laptops or mobile devices, the best electronic version to use is the .html version of the text, but you can find links to a pdf and hard copy of the text at https://activecalculus.org/. Furthermore, because the text is open-source, any instructor may acquire the full set of source files, which are available on GitHub.
As of August 2026, you can find the official release of the second edition of ACM at activecaclulus.org. We will strive to not make changes to this edition during the academic years. You can find a preview version of an updated version (typos and other fixes) of the Second Edition of ACM at
https://nelong.github.io/active-calculus-vector/multi/frontmatter.html and the instructor version at https://nelong.github.io/active-calculus-vector/multi-instructor/frontmatter.html. For a complete description of the differences between the first and second editions see https://activecalculus.org/acm/acm-1e-vs-2e/
This text may be used as a stand-alone textbook for a standard multivariable calculus course or as a supplement to a more traditional text. Chapter 9 covers the precalculus ideas necessary for multivariable calculus: coordinates, vectors and common graphs (planes, lines, quadric surfaces, and cylinder surfaces) in three dimensions. Chapter 10 covers vector-valued functions of one variable and introduces how to use vector tools and calculus on these functions. Chapter 11 introduces functions of more than one variable and studies differentiation of functions of several independent variables in detail, addressing the typical topics including limits, partial derivatives, and optimization. Chapter 12 motivates and defines double and triple integrals.
Chapter 13 covers vector fields and vector calculus topics through Stokes Theorem and the Divergence Theorem. The 2nd edition of ACM has evolved because in reality, “multivariable calculus” can mean one of three different courses: one that follows chapters 9-112 of the 2nd edition and omits vector calculus entirely; one that introduces students to line integrals, the Fundamental Theorem of Calculus for Line Integrals, and Green’s Theorem; and one that proceeds through all of that material plus Stokes’s Theorem and Gauss’s Divergence Theorem. As such, the extent to which certain topics in chapters 9-12 are necessary varies depending on the course’s final destination.
- Electronic Edition
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Because students and instructors alike have access to the book in electronic format, there are several advantages to the text over a traditional print text. One is that the text may be projected on a screen in the classroom (or even better, on a whiteboard) and the instructor may reference ideas in the text directly, add comments or notation or features to graphs, and indeed write right on the projected text itself. Students can do the same when working at the board. In addition, students can choose to print only whatever portions of the text are needed for them. Also, the electronic versions of the text includes links to on-line programs, so student and instructor alike may follow those links to additional resources that lie outside the text itself. Finally, students can have access to a copy of the text anywhere they have a computer.Beginning with the Second Edition, we have written activities and the presentation of material with the html as the primary focus. For example, almost every 3D plot will be interactable in the html version of the text (can rotate, zoom, or use interactive tools like sliders and buttons) which offers students the option to literally look at the objects from a different perspective.Note: In the pdf version, there is sometimes not an obvious visual indicator of the html links, so some available information is suppressed and there are often links or QR codes to view an interactable plot. If you are using the text electronically in a setting with internet access, please know that it is assumed you are using the html version.
- Activities Workbook
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Each section of the text has a preview activity and at least three in-class activities embedded in the discussion. As it is the expectation that students will complete most of these activities, it is ideal for them to have room to work on them adjacent to the problem statements themselves. A separate workbook of activities that includes only the individual activity prompts, along with space provided for students to write their responses, is in development and should be widely available in the 26-27 academic year.
- Community of Users
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Because this text is free and open-source, we hope that as people use the text, they will contribute corrections, suggestions, and new ideas for material. At this time, the best way to communicate such feedback is through the Active Calculus - Multivariable Google Group. We have also set up a Google Form to collect feedback. If you are comfortable with GitHub and/or PreTeXt, you can submit your suggested changes as a Pull Request or Issue to the ACM repository. We will occasionally give some updates through the blog on activecalculus.org.
