# Vector and Geometric Calculus

## Contents

- Preliminaries
- Curve and Surface Representations
- Limits and Continuity

- Derivatives
- The Differential
- Tangent Spaces
- The Gradient
- Extrema

- Integrals
- Integrals over Curves
- Multiple Integrals
- Integrals over Surfaces

- The Fundamental Theorem of Calculus
- The Fundamental Theorem of Geometric Calculus

- Differential Geometry
- Differential Geometry in ℝ
^{3}

- Differential Geometry in ℝ
- Appendices
- Geometric Algebra Review
- Formulas
- Differential Forms
- Extending Fields on Manifolds

Read a detailed table of contents, the preface, and the index.

This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus. It is a sequel to my *Linear and Geometric Algebra*. That text is a prerequisite for this one.

Linear algebra and vector calculus have provided the basic vocabulary of mathematics in dimensions greater than one for the past one hundred years. Just as geometric algebra generalizes linear algebra in powerful ways, geometric calculus generalizes vector calculus in powerful ways.

Traditional vector calculus topics are covered, as they must be, since readers will encounter them in other texts and out in the world.

The book can be used for self study by those comfortable with the theorem/proof style of a mathematics text.

#### Videos

I have created a five video YouTube playlist
Geometric Calculus, about 53 minutes in all, taken from the book.
It is a sequel to my
Geometric Algebra playlist.
Some knowledge of vector calculus is a prerequisite for the videos,
but no knowledge of geometric calculus is assumed.
The *book* assumes no knowledge of vector calculus.

## Fifth Printing

The fifth printing has no major changes. It corrects all errors known to me in the fourth printing. There are many improvements in presentation and a small amount of new material. Some material in Section 11.2 has been moved to Section 5.5, causing some changes in the numbering of equations, theorems, etc. in those sections. Also, some page numbers have changed due to moving material.

## What People Are Saying

From a review of *Linear and Geometric Algebra*:

"Alan Macdonald's text is an excellent resource if you are just beginning the study of geometric algebra and would like to learn or review traditional linear algebra in the process. The clarity and evenness of the writing, as well as the originality of presentation that is evident throughout this text, suggest that the author has been successful as a mathematics teacher in the undergraduate classroom. This carefully crafted text is ideal for anyone learning geometric algebra in relative isolation, which I suspect will be the case for many readers."

— Jeffrey Dunham, William R. Kenan Jr. Professor of Natural Sciences, Middlebury College

## Available at Amazon

*Vector and Geometric Calculus* is available at Amazon.
See the strong
reviews there.

Price: $35

## Computer Exercises

**GAlgebra.** The computer exercises in the book use GAlgebra,
a Python module written by Alan Bromborsky.
It is available here.
GAlgebra is cross-platform (Linux, PC, Mac), with all components freely available on the web.
The software is no longer available at this page.

GAlgebra is written in Python 2. One of its components (Sympy), will stop supporting Python 2, probably by the end of 2019. Alan Bromborsky is converting GAlgebra to Python 3.

**Notebooks.** GAlgebra works in Jupyter (formerly IPython) and Jupyter Lab notebooks.
Output is typeset in beautiful LaTeX.

GAlgebraPrimer.pdf contains instructions for installing and using the module. It also downloads with the module. Updated 1/24/18.

The book promises these files: VAGC Instructor, URLs.txt, Symbols.pdf, Template.py.

Stephen Abbott has written a free symbolic geometric algebra program GAwxM, based on Maxima. It runs on both Windows and Ubuntu Linux. The program and instructions for installing and using it are available at GitHub. I have not used it, so I am announcing, not recommending the software. I am interested in feedback about GAwxM.

Kurt Nalty has written free symbolic geometric algebra software based on GiNaC, a C++ library. It has been tested on Linux. Its chief virtue is that it is "much faster" than GAlgebra. Its output is not typeset by LaTeX. It does not run within Jupyter. The program and instructions for installing and using the program are available at GitHub. I have not used the software, so I am announcing, not recommending, it. I am interested in feedback about the program.

## Errata

The latest errata file is dated Updated January 23, 2019.

Please email me corrections, typos, or any other comments about the book. I will post them here as appropriate.