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\(\newcommand{\R}{\mathbb{R}} \newcommand{\va}{\mathbf{a}} \newcommand{\vb}{\mathbf{b}} \newcommand{\vc}{\mathbf{c}} \newcommand{\vC}{\mathbf{C}} \newcommand{\vd}{\mathbf{d}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vi}{\mathbf{i}} \newcommand{\vj}{\mathbf{j}} \newcommand{\vk}{\mathbf{k}} \newcommand{\vn}{\mathbf{n}} \newcommand{\vm}{\mathbf{m}} \newcommand{\vr}{\mathbf{r}} \newcommand{\vs}{\mathbf{s}} \newcommand{\vu}{\mathbf{u}} \newcommand{\vv}{\mathbf{v}} \newcommand{\vw}{\mathbf{w}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vy}{\mathbf{y}} \newcommand{\vz}{\mathbf{z}} \newcommand{\vzero}{\mathbf{0}} \newcommand{\vF}{\mathbf{F}} \newcommand{\vG}{\mathbf{G}} \newcommand{\vH}{\mathbf{H}} \newcommand{\vR}{\mathbf{R}} \newcommand{\vT}{\mathbf{T}} \newcommand{\vN}{\mathbf{N}} \newcommand{\vL}{\mathbf{L}} \newcommand{\vB}{\mathbf{B}} \newcommand{\vS}{\mathbf{S}} \newcommand{\proj}{\text{proj}} \newcommand{\comp}{\text{comp}} \newcommand{\nin}{} \newcommand{\vecmag}[1]{|#1|} \newcommand{\grad}{\nabla} \DeclareMathOperator{\curl}{curl} \DeclareMathOperator{\divg}{div} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

PrefacePreface

Coordinated Multivariable Calculus is adapted from Active Calculus: Multivariable + Vector by Steve Schlicker, Mitchel T. Keller, and Nicholas Long, originally released under a Creative Commons license. Much of the spirit of the original text is preserved, though we have added many worked examples and simplified some of the presentation. We have also trimmed the material to match the material of Math 208 at UNL; mostly we have moved the extra material to Extra Topic sections at the end of chapters, but in some cases we have eliminated the material entirely to avoid confusion to students.

This is an ongoing project, and in the future we hope to add many more interactive graphics to assist students as they work through the text.

For completeness we include below the preface to the Vector addition to Active Calculus: Multivariable.

Zach Norwood and Audrey Goodnight
University of NebraskaLincoln
August 2023
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 (Section4.14.4) can be combined with Section4.6 on Green's Theorem. We would encourage you to explore the section on curl4.5 before proceeding to Green's Theorem, but only the first portion on circulation density would be terribly beneficial. Section4.5 does make some passing backward references to the development of divergence in Section4.9, 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 Chapter4 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. The ongoing COVID-19 pandemic has delayed a final release of these materials. Perhaps by August 2022, they will be ready in a complete, public form, but we hope you continue to find them useful until then.

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