Active Calculus Multivariable

Reviewed by Kyle Salois, Assistant Professor, St. Olaf College on 4/6/26

Comprehensiveness rating: 4 see less

The textbook covers the major topics of a standard Multivariable Calculus course in depth. The authors embed active learning activities in the online version of the book which allow students to measure their progress as they read. There are a reasonable number of additional activities and exercises for students to practice, although I found some sections to be light on exercises, and the WebWork interface can have annoying issues with submitting and checking answers.

Content Accuracy rating: 5

I did not notice any errors in the writing or mathematical content of the book.

Relevance/Longevity rating: 5

Content is up-to-date, and it will be easy to update if there are any major changes that need to be made (although that would be surprising for this field). The book relies heavily on active learning strategies, and it would be more difficult to restructure the textbook if a different approach becomes dominant pedagogy, but that shift would also be unlikely in the near future.

Clarity rating: 5

The book is written well, in accessible language, and with a solid foundation of context from previous courses. Many new topics are introduced through the lens of a similar concept from previous Calculus courses, or by referring back to a previous topic in Multivariable Calculus. The book is written using PreTeXt, which focuses on accessibility.

Consistency rating: 4

I did not find any inconsistencies within the mathematical terminology or notation within the book. As is the case in mathematics, the authors had to make a choice for their notation, which may not be the same notation as other books or fields, but they are consistent with their choice throughout. As a small suggestion, the authors could add a few small remarks about where students might see different notation (in a standard Physics class, for instance).

Modularity rating: 5

Each new concept in a chapter is split into a sub-section. I found it easy enough to swap two sections when I was teaching the course, but there are places where that wouldn't be possible, just from a mathematical standpoint.

Organization/Structure/Flow rating: 5

The topics are presented in a fairly standard order for a Multivariable Calculus course.

Interface rating: 5

The book is easy to navigate, and the in-text activities are straightforward to activate and use. Again, the book was created using PreTeXt, so accessibility is one of the primary concerns by the authors, and it shows.

Grammatical Errors rating: 5

I did not notice any major grammatical errors in the book.

Cultural Relevance rating: 4

The text is neither insensitive nor offensive, partially because the exercises rarely make use of "human" examples with names or pronouns. If anything, the lack of human examples might make students less connected to the content, but as it is, the text does not risk alienating students.

Reviewed by James Collins, Assistant Professor, University of Mary Washington on 4/9/19

Comprehensiveness rating: 4 see less

This book covers most areas of multivariate calculus. However, certain areas, such as Divergence Theorem, and Fundamental Theorem of Line Integrals is not included in the book. I am told it will be added in later additions though.

Content Accuracy rating: 5

I found this book very accurate with few errors

Relevance/Longevity rating: 5

I don't imagine this book will need updating anytime soon, as Calculus will not be changing. Any updates will simply be to improve the activities, not the material itself.

Clarity rating: 5

The book is very clear and easy to understand. The author specifically avoids any confusing mathematical notation in an attempt to make the material accessible.

Consistency rating: 5

The book is very consistent.

Modularity rating: 5

This book does an incredibly good job of subdividing the various sections into subsections. The author of the book has his students read certain sections before class, as do I. It is very easy to tell students to read through this section of the book. Each subsection within a section is usually around a page.

Organization/Structure/Flow rating: 5

I particularly like how one section flows into the next, telling one coherent story.

Interface rating: 5

The online version of this text was very helpful to demonstrate visual elements of the material to the class. With multivariate in particular, the visual elements are complex and hard to draw for those of us with little drawing skill. It was helpful to have various graphs online to demonstrate ideas to the students.

Grammatical Errors rating: 5

I found no grammatical errors in my using of the book.

Cultural Relevance rating: 5

The author uses both male and female pronouns where applicable, though pronouns in general are not used often.

I have used this book for Multivariate Calculus and enjoyed it very much. It is important to stick to doing the activities and having students do the preview activities before class. Some students have trouble using this book as a standalone book, as it is very different from what they are used to. It may be helpful to pair it with a more traditional free textbook, such as APEX.