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Active Calculus - Multivariable

Nicholas Long, Mitchel T. Keller, Steve Schlicker

Preface Features of the Text

Similar to the presentation of the single-variable Active Calculus, instructors and students alike will find several consistent features in the presentation, including:
Motivating Questions
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At the start of each section, we list motivating questions that provide motivation for why the following material is of interest to us. One goal of each section is to answer each of the motivating questions.
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Preview Activities
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Each section of the text begins with a short introduction, followed by a preview activity. This brief reading and the preview activity are designed to foreshadow the upcoming ideas in the remainder of the section; both the reading and preview activity are intended to be accessible to students in advance of class, and indeed to be completed by students before a day on which a particular section is to be considered.
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Activities
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Every section in the text contains several activities. These are designed to engage students in an inquiry-based style that encourages them to construct solutions to key examples on their own, working either individually or in small groups.
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Exercises
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There are dozens of calculus texts with (collectively) tens of thousands of exercises. Rather than repeat a large list of standard and routine exercises in this text, we recommend the use of WeBWorK with its access to the National Problem Library and its many multivariable calculus problems. In this text, each section begins with several anonymous WeBWorK exercises, and follows with several challenging problems. The WeBWorK exercises are best completed in the .html version of the text. Almost every non-WeBWorK problem has multiple parts, requires the student to connect several key ideas, and expects that the student will do at least a modest amount of writing to answer the questions and explain their findings. For instructors interested in a more conventional source of exercises, consider the freely available APEX Calculus text by Greg Hartmann et al.
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Our Fall 2025 preview release includes at least some WeBWorK exercises in each of the sections of our Second Edition preview, but we welcome suggestions of additional WeBWorK exercises that faculty would consider good inclusions and feedback on any aspect of this edition. The best way to provide 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 repository. We will occasionally give some updates through the blog on https://activecalculus.org/.
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Graphics
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As much as possible, we strive to demonstrate key fundamental ideas visually, and to encourage students to do the same. Throughout the text, we use full-color graphics to exemplify and magnify key ideas, and to use this graphical perspective alongside both numerical and algebraic representations of calculus. Because so much of vector calculus relies upon visualizing things in three dimensions, we elected to produce almost all of the 3D graphics using SageMath. This allows for a degree of consistency between the two-dimensional graphics, which are typically static, and the three-dimensional graphics, which are almost always interactive. Thus, you can grab a three-dimensional plot, rotate it, zoom in on it, etc.
  • To rotate a three-dimensional graphic, click and drag with your mouse.
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  • To zoom on an interactive three-dimensional graphic, use your mouse’s scroll wheel or make your operating system’s scroll gesture on your touchpad.
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  • To move a three-dimensional graphic instead of rotating, hold down the space bar while clicking and dragging.
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Summary of Key Ideas
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Each section concludes with a summary of the key ideas encountered in the preceding section; this summary normally reflects responses to the motivating questions that began the section.
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Links to technological tools
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Many of the ideas of multivariable calculus are best understood dynamically, and we encourage readers to make frequent use of technology to analyze graphs and data. Since technology changes so often, we refrain from indicating specific programs to use in the text. However, aside from computer algebra systems like Maple, Mathematica, or Sage, there are many free graphing tools available for drawing three-dimensional surfaces or curves. These programs can be used by instructors and students to assist in the investigations and demonstrations. The use of these freely available applets is in accord with our philosophy that no one should be required to purchase materials to learn calculus. We are indebted to everyone who allows their expertise to be openly shared. Below is a list of a few of the technological tools that are available (links active at the writing of this edition). Of course, you can find your own by searching the web.
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