So far, the books I've seen do not provide explicit numerical examples.
One of the advantages, we are told, of learning to use GA is that it is a coordinate free system for handling many of the mathematical problems in physics.
However, without showing a single example of how to calculate a numerical answer to any problem, it has not been shown how to use this algebraic system in calculating an answer.
My own experience is one of frustration. For example, I needed to rotate a vector about the (x,y,z) = (1,1,1) axis the other day. Certainly not a coordinate free calculation! So I thought I'd use the GA way of doing it using "rotors".
Given a vector a you can rotate it using a rotor R by the equation c = RaR†
O.K., so what is R?
The rotor R is given by R = nm where n and m are vectors in the plane in which the vector a is to be rotated and such that the angle between n and m is twice the angle of the desired rotation.
So then I was left with trying to calculate 2 vectors (n and m) in the plane of rotation such that....
Well, I thought, maybe I can find an example like this in one of the books on GA that I have. Nope. No explicit example at all! That's when I realized that the GA books are all missing a big component: Explicit examples of numerical calculation!
I gave up trying to rotate my vector using GA.
I actually used, as I usually do, the quaternion method to do the rotation.
Yes, quaternions are embedded in GA, so you might argue that I ended up calculating the rotor after all. But this isn't exactly correct because the quaternion approach uses a vector perpendicular to the plane of rotation while the GA approach uses two vectors in the plane of rotation. But there is a relation between the two. (I don't know how to translate quaternion vector rotation into GA rotor rotations, yet.)
So, be warned! If you want to learn how to do explicit numerical calculations, there isn't a book out there that will show you.
And there might not be one for some time as this seems to be counter to the (one of the) big selling points of GA: That it is a coordinate free algebraic system for "doing" physics calculations. So, selecting a particular coordinate system and carrying out an explicit numerical calculation seems to go against the grain.
But this is exactly what is going to be needed if GA is to be used in intro physics and engineering and computer science (vision, robotics, graphics) classes!! I suspect this is why GA has not been used in such intro classes.
Another annoying point (actually, the same point) is the way that the geometric product of vectors is introduced.
Consider two vectors a and b. The geometric product can be written as
(See p. 30 of Hestenes' book "New Foundations for Classical Mechanics", 2nd ed.)
(equ. 6.1) ab = a.b + a^b
(equ. 6.2) ba = b.a + b^a = a.b – a^b
I had never seen the wedge product "^" of vectors before getting into GA, so I didn't understand what this "really" meant (how to numerically calculate with it). I had hoped that there would be an explicit numerical example in the book shortly after equ. 6.1 that would give me a clue. Nope. No numerical example.
Instead, a couple of pages later, we get
"Now observe that by taking the sum and the differences of equation (6.1) and (6.2), one gets
a.b = (1/2)(ab + ba),
and
a^b = (1/2)(ab – ba)."
That's not very helpful! It is a circular definition. The geometric product of 2 vectors is defined by the inner (scalar) product and the wedge (outer) product of the 2 vectors. But the inner and wedge products are defined as combinations of the geometric product.
This occurs in all the GA books and papers that I've seen and not just in Hestenes' book.
The "problem" here is not with these books and papers but with my expectations. I expected a new (better? easier?) way of calculating geometric and physical numerical answers to my geometry, physics, computer graphics problems. What I've seen (so far) is an algebraic system for setting up and manipulating the equations for the problems and with a very nice geometric interpretation of the algebraic elements. But, within that context, there is not yet sufficient (any) explicit numerical assistance by way of numerical calculations.
Someone needs to write an Introduction to College Level Physics Using Calculus and Geometric Algebra! Only then might it start to become familar to students.
Copyright 2007 by R.W. Gray
Home Page: www.rwgrayprojects.com