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 ℝ3
  6. Appendices
    1. Geometric Algebra Review
    2. Software
    3. Formulas
    4. Differential Forms
    5. Extending Fields on Manifolds

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

Third Printing

The third printing has no major changes. It corrects all errors known to me in the first printing. There are many improvements in presentation and a small amount of new material. The numbering of equations, theorems, exercises, etc. is mostly unchanged from the first printing. See the preface for an exception.

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.

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.

Price: $35

Computer Exercises

The computer exercises in the book use GAlgebra, a Python module written by Alan Bromborsky. GAlgebra is cross-platform (Linux, PC, Mac), with all components freely available on the web. The module is available at GitHub. Click here.

The software is no longer available at this page.

New! GAlgebra is now available from a jupyter (formerly IPython) notebook. Output is typeset in beautiful LaTeX.

The file GAlgebraPrimer.pdf contains instructions for installing and using the module. It is available here. and also downloads with the module. It will be updated as necessary.

Appendix B of the book also contains instructions for installing and using the module. But the information there will become dated as GAlgebra evolves. So I recommend that you use GAlgebraPrimer.

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

Alan Mcdonald

Alan Macdonald is Professor Emeritus of Mathematics at Luther College in Decorah, Iowa. He received a Ph.D. in mathematics from The University of Michigan in 1970.


  • Errata can be found here. Updated June 1, 2015.

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