Vector and Geometric Calculus

Vector and Geometric Calculus


  1. Preliminaries
    1. Curve and Surface Representations
    2. Limits and Continuity
  2. Derivatives
    1. The Differential
    2. Tangent Spaces
    3. The Gradient
    4. Extrema
  3. Integrals
    1. Integrals over Curves
    2. Multiple Integrals
    3. Integrals over Surfaces
  4. The Fundamental Theorem of Calculus
    1. The Fundamental Theorem of Geometric Calculus
  5. Differential Geometry
    1. Differential Geometry in ℝ3
  6. Appendices
    1. Geometric Algebra Review
    2. Formulas
    3. Differential Forms
    4. 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.


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.

Current Printing

The latest printing is from May 2020. It has several minor improvements and corrects all errors known to me.

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

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,

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.


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.