From Vectors to Multivectors (Part 1 of 3)
Channel: Nick Okamoto
Date: 2016-09-02

Geometric Algebra (GA) is proposed as the culminating step in the development of a universal mathematical language for all of physics. When first encountering such a bold claim (cf. http://geocalc.clas.asu.edu/html/Evolution.html and http://geometry.mrao.cam.ac.uk/2000/01/a-unified-mathematical-language-for-physics-and-engineering-in-the-21st-century/) I was skeptical, and found that the only way to dispel that skepticism was to actually learn GA. It is a deeply satisfying experience to see a babble of mathematical formalisms, designed for diverse physical applications, all be subsumed and seamlessly integrated into one unified system. Witnessing this variety of specialized mathematical tools and "tricks that get the job done" suddenly acquire intuitive and edifying geometric meanings, is a delight I naturally want to share with any who are interested. And that provides the motivation leading me to create these videos. 

Lesson 1, From Vectors to Multivectors (FV2M), is split into three parts. Links to the other two parts, as well as other videos in this series (as they are posted), can be found near the bottom of this video description. This foundational video of our GA tutorial series demonstrates how a vector space (given a non-degenerate scalar product) may be extended into a multivector system called the geometric algebra of that vector space. 

Prerequisites: An introduction video for this GA tutorial series is being developed with a wider audience in mind, but this FV2M video is intended for viewers who are already familiar with basic vector algebra, including geometric understanding of operations like vector addition, multiplication by scalars, as well as the dot and cross products. For example, it will be assumed that viewers are familiar with material similar to what is covered in this Khan Academy video "Dot and cross product comparison/intuition" [https://www.youtube.com/watch?v=tdwFdzVqito]. Knowledge about vector space dimension and basis would be helpful as well. 


--Some GA introductory papers (pdf links)--

A unified mathematical language for physics and engineering in the 21st century (Joan Lasenby, Anthony Lasenby, & Chris Doran, )
http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/00RSocMillen.pdf

Imaginary Numbers Are Not Real - the Geometric Algebra of Spacetime (Stephen Gull, Anthony Lasenby, & Chris Doran)
http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf

Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics (David Hestenes)
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Introduction to Clifford's Geometric Algebra (Eckhard Hitzer)
http://erkenntnis.icu.ac.jp/gcj/publications/SICEJintro/SICEJintro.pdf

A Survey of Geometric Algebra & Geometric Calculus (Alan MacDonald)
http://faculty.luther.edu/%7Emacdonal/GA&GC.pdf

Geometric Algebra (Eric Chisolm)
http://arxiv.org/pdf/1205.5935v1.pdf


--Other videos in this GA tutorial series--

00 GA tutorial Intro & Overview 
[UnderConstruction]

01 From Vectors to Multivectors 
Part 1: https://youtu.be/P-IKemH3jsg 
Part 2: https://youtu.be/FUivWb4k_bE 
Part 3: https://youtu.be/b0K451IxLBQ 

02 Geometric Product of Vectors (redux)
[UnderConstruction]

03 Grades and Blades
[UnderConstruction]

04 Q&A #01
[UnderConstruction]

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*Note: I'm very busy right now wrapping up my PhD dissertation (which does involve multivector analysis), and I do not have much time to work on these videos, so I cannot predict when the future videos planned for this series will be completed. Subscribing to my youtube channel will help keep you aware of any new videos I post. In the mean time, here are some other GA related youtube videos I recommend.

Mathoma's "Geometric Algebra" playlist is a really good (undergraduate friendly) first introduction to GA: 
https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K

"The Vector Algebra War" (J. Chappell, A. Iqbal, J. Hartnett, D. Abbott)
https://www.youtube.com/watch?v=_AaOFCl2ihc

"Tutorial on Clifford's Geometric Algebra" (E. Hitzer)
https://www.youtube.com/watch?v=sHjXccDAIzw&list=PL8mAte7RusZp7_-TnLdizhTpWfKdWuYzr

Alan Macdonald's "Geometric Calculus" tutorial: 
https://www.youtube.com/playlist?list=PLLvlxwbzkr7i6DlChcYEL7nJ8R9ZuV8JA


***Last updated: 10 October 2016***