Name: Vector and Geometric Calculus
ISBN: 9781480132450

Vector and Geometric Calculus

This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus. This is the printing of April 2025. The book is a sequel to the text Linear and Geometric Algebra by the same author. That text is a prerequisite for this one. Its web page is at faculty.luther.edu/ macdonal/laga. 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. Download the book's table of contents, preface, and index at the book's web site: faculty.luther.edu/ macdonal/vagc. 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.

What Hacker News says

_ktndMar 2024

My unworthy opinion:

Herbert Gross' “Calculus Revisited” three-part series.

https://ocw.mit.edu/courses/res-18-006-calculus-revisited-si...

https://ocw.mit.edu/courses/res-18-007-calculus-revisited-mu...

https://mitocw.ups.edu.ec/resources/res-18-008-calculus-revi...

The above is more or less “engineering math”.

The below is “normal college” math:

Edward Frenkel's Multivariable Calculus lectures:

https://www.youtube.com/playlist?list=PLaLOVNqqD-2GcoO8CLvCb...

Dr. Valerie Hower's Linear Algebra lectures:

https://youtube.com/playlist?list=PLpcwHaLYiaEXW5fLNOlItPH4A...

Interactive Linear Algebra (online “college level” textbook):

https://textbooks.math.gatech.edu/ila/

Tutorial-like Linear Algebra (interactive online textbook):

https://immersivemath.com/ila/index.html

Sheldon Axler's Linear Algebra Done Right book:

https://linear.axler.net/

Sergei Treil's Linear Algebra Done Wrong book:

https://www.math.brown.edu/streil/papers/LADW/LADW-2014-09.p...

Linear algebra and vector calculus revamped:

— Linear and Geometric Algebra by Alan Macdonald: http://www.faculty.luther.edu/~macdonal/laga/index.html

— Vector and Geometric Calculus: http://www.faculty.luther.edu/~macdonal/vagc/index.html

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_ktndJan 2024

Below is a list of resources that I personally like, because most of them emphasize derivations/motivations (e.g., deriving the cross product out of 3 assumptions/axioms) over (mechanical) memorization (e.g., just showing the mechanics of matrix multiplication/determinants without derivations/motivations):

   - Bari A., Algorithms, YouTube (playlist),
  https://www.youtube.com/playlist?list=PLDN4rrl48XKpZkf03iYFl-O29szjTrs_O

  - Frenkel E., UC Berkeley: MATH 53 - Multivariable Calculus with Edward Frenkel, YouTube (playlist) https://www.youtube.com/playlist?list=PLaLOVNqqD-2GcoO8CLvCbprz2J0_1uaoZ

  - Hower V., Linear Algebra Full Course, YouTube (playlist), https://www.youtube.com/playlist?list=PLpcwHaLYiaEXW5fLNOlItPH4ATorKjBuc

  - Lockhart P., Measurement

  - Axler S., Linear Algebra, Springer, 2024, https://link.springer.com/content/pdf/10.1007/978-3-031-41026-0.pdf

  - Ström et al., Immersive Linear Algebra, https://immersivemath.com/ila/index.html (online book)

  - Margalit et al., Interactive Linear Algebra, https://textbooks.math.gatech.edu/ila/ (online textbook)

  - Macdonald A., Linear and Geometric Algebra

  - Macdonald A., Vector and Geometric Calculus

  - Ramirez et al., From Vectors to Geometric Algebra, 2018, https://doi.org/10.48550/arXiv.1802.08153

  - Hertzmann et al., Computer Graphics Lecture Notes, Computer Science Department - University Toronto, 2005, https://www.dgp.toronto.edu/~hertzman/418notes.pdf

Further, see Sergey Trail's "Linear Algebra Done Wrong" and watch Grant Sanderson, Easy Theory (theoretical CS), Jacob Sorber (operating systems), Graphics in 5 Minutes, Sam Buss (computer graphics), Dr. Daniel Page (theory of computation, data structures and algorithms), First Principles of Computer Vision on YouTube.

Mr. Hertzmann has some more resources (that I haven't checked out yet):

https://www.dgp.toronto.edu/~hertzman/research.html

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thechaoNov 2020

I cannot recommend Alan MacDonald's two books enough (Linear and Algebra; and, Vector and Geometric Calculus). They literally replace reams of texts on complex numbers, quaternions, octonions, spinors, etc. under a single unifying concept. (They don't replace those concepts; and, for sure, if you want to learn quantum you're going to need them; but they make getting your head around them so much easier.)

One issue is that most of the "online" references are crap. MacDonald's presentation is a wonderful slow ramp: it starts with really basic linear algebra and then just slowly segue's into geometric algebra.

I firmly believe that in 100 years we just won't be teaching 'bare' linear algebra, complex, quaternions, ... we'll just teach GA and then introduce the physical concepts separately.

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SAI_PeregrinusFeb 2020

In addition to gibsonf1's recommendations (not books), the following books are good. The first 2 are more pure introductions to the math, the third is applying it to physics, the fourth to CS.

Linear and Geometric Algebra, by Alan Macdonald: http://www.faculty.luther.edu/~macdonal/laga/

Vector and Geometric Calculus, by Alan Macdonald: http://www.faculty.luther.edu/~macdonal/vagc/index.html

Application to physics: New Foundations for Classical Mechanics by David Hestenes: https://books.google.com/books/about/New_Foundations_for_Cla...

Geometric Algebra For Computer Science by Dorst, Fontijne, and Mann: http://www.geometricalgebra.net/index.html

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nimishApr 2019

Linear and geometric algebra

Vector and geometric calculus

Both by Alan macdonald, both very good.

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happy4crazySep 2016

For another resource, Alan Macdonald has some nice videos that accompany his books, "Linear and Geometric Algebra" and "Vector and Geometric Calculus": https://www.youtube.com/channel/UCymE67THrWoeTABxzJm1wdg/pla...

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