Skip to main content

Preface Vector Calculus Preface

We are glad to have the opportunity to contribute the chapter on Vector Calculus to expand Active Calculus - Multivariable. This chapter is currently in draft form, but ready for testing. We have tried to adhere to the style of the Active Calculus series of texts as much as possible. Sections begin with “Motivating Questions” and end with a summary that answers those questions. Each section has a preview activity and several activities intended for classroom use.

Recognizing that not all institutions will cover all the material in this chapter, we have intended that the overview of vector fields, line integrals of vector fields, and Fundamental Theorem of Calculus for Line Integrals (Section 12.1–12.4) can be combined with Section 12.7 on Green's Theorem. We would encourage you to explore the section on curl before proceeding to Green's Theorem, but only the first portion on circulation density would be terribly beneficial. Section 12.6 does make some passing backward references to the development of divergence in Section 12.5, but it is not necessary to cover divergence in order to do the basics of circulation density and then Green's Theorem.

Observant faculty may notice that the graphics in our chapter deviate from those in the previous 11. Because so much of vector calculus relies upon visualizing things in three dimensions, we elected to produce almost all of the 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.
  • 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.
  • To move a three-dimensional graphic instead of rotating, hold down the space bar while clicking and dragging.

Notice that Chapter 12 is a work in progress in a few ways:

  • Answers and solutions to many activities are not yet written.
  • WeBWorK exercises have not yet been added.
  • Some sections lack exercises altogether.
  • Sections on line integrals and surface integrals of scalar functions are yet to be added.

We welcome suggestions of WeBWorK exercises that faculty think would be good inclusions as well as suggestions for exercise topics. We also welcome feedback on any aspect of this chapter. We have set up a Google Form to collect feedback. We plan to make updates before January 2021 for further classroom testing, with the first official release in August 2021.

Mitchel T. Keller and Nicholas Long
Sioux City, Iowa, and Nacogdoches, Texas