Vector and Geometric Calculus

Alan Macdonald

Second printing, corrected and slightly revised.

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.

Differential geometry is used today in many disciplines. A final chapter is devoted to it.

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

The table of contents, preface, and index are available here.

The second 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 unchanged from the first printing, with the exception of four problems.

      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

Notice. Due to publisher error the shading in several of figures is washed out in some copies of the book.
The errata file shows the figures as they should be.

The computer exercises in the book use Python, a cross-platform language freely available on the web, its library sympy, and a module ga written by Alan Bromborsky. The module is available here. (Updated Sepember 2, 2013.) It works with Python 2.7. (Python 2.5 and 2.6 might also work.) The module is cross-platform: Linux, PC, and Mac. Instructions for ga are in VAGC Appendix B.pdf. (Updated September 2, 2013.) I use Geany, a free cross-platform IDE, to write my programs.

Here are other files promised in the book: VAGC Instructor, URLs.txt, Symbols.pdf,

Errata can be found here. Updated March 23, 2015.

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

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. Other than geometric algebra, his research interests include the foundations of physics and generalized Riemann integration. His web page is here.


Alan Macdonald
Professor Emeritus of Mathematics
Luther College
Decorah, IA 52101
macdonal at luther dot edu
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