Clifford Algebra Question on Vectors

[treat2]

I'm a Software Developer by profession, not a Mathematician, or Physicist, so please be patient with my ignorance as I'm about to ask (what I am sure is) a very basic question about Clifford Algebra.

I've been reading some Clifford Algebra books I have, on how C.A. represents and performs math on Vectors. Aside from the aspect of
k-blades to represent dimensions in C.A., I have not yet found ANY difference at all, between "standard" Vector Math that can be found in any basic book about Vectors, and how Clifford Algebra performs mathematical operations on Vectors!

I see inner and outer products being used in any book I pick up and read some pages about math with Vectors. I also see all of the standard mathematical operations on Vectors being the same as I've read so far, in the Clifford Algebra book's Chapter on Vector Math.

The only thing I don't recall seeing in standard math books about Vectors, that I do see in Clifford Algebra is the addition of the inner and outer products to form a geometrical product.

Is the geometrical product, and the use of k-blades, about the only difference I should expect to find, when comparing C.A.'s treatment of Vectors, with how Vectors are operated on in standard Algebra's treatment of Vectors? I'm REALLY confused why I'm not seeing any
huge difference yet. Can you shed any light on this, and should I expect essentially no differnce between Clifford Algebra's treatment of Calculus, and what can be found in any standard Calculus book (aside from some differences in symbols used?

Admittedly, I am not yet done with my reading the section of the Clifford Algebra book having to do with Vectors, but I have not yet seen ANY differences in mathematical operations, including many of the symbols used.

 

[lethe]

Originally posted by treat2
The only thing I don't recall seeing in standard math books about Vectors, that I do see in Clifford Algebra is the addition of the inner and outer products to form a geometrical product.

that s right. this product is unique to geometric algebra, which is an application of clifford algebras.

Is the geometrical product, and the use of k-blades, about the only difference I should expect to find, when comparing C.A.'s treatment of Vectors, with how Vectors are operated on in standard Algebra's treatment of Vectors? I'm REALLY confused why I'm not seeing any
huge difference yet. Can you shed any light on this, and should I expect essentially no differnce between Clifford Algebra's treatment of Calculus, and what can be found in any standard Calculus book (aside from some differences in symbols used?

the thing that you can do with Clifford algebras that you cannot do with other kinds of vectors is describe spinors (these are like vectors but that describe spin-1/2 particles, whereas normal vectors describe spin-1 particles). in fact, i think it would be fair to say that Clifford algebras were originally invented by Dirac and Pauli to do exactly that. they needed an algebra to describe spin, so they made one.

this makes Clifford algebras absolutely indispensable to the physicist (and actually the mathematician too)

because spin is intimately related to rotations, it turns out that Clifford algebras can be used to replace the vector cross product, which is what we used to describe rotations before Clifford algebras came along. there is a fellow who goes by the name of Hestenes who advocates replacing all exterior algebras with Clifford algebras, and all vector calculus in R³ with Clifford algebras as well. his program goes by the name of geometric algebra.

 

[treat2]

To lethe OR ANYONE ELSE that can offer advice on the question below.

Originally posted by lethe
that s right. this product is unique to geometric algebra, which is an application of clifford algebras.

The thing that you can do with Clifford algebras that you cannot do with other kinds of vectors is describe spinors (these are like vectors but that describe spin-1/2 particles, whereas normal vectors describe spin-1 particles). in fact, i think it would be fair to say that Clifford algebras were originally invented by Dirac and Pauli to do exactly that. they needed an algebra to describe spin, so they made one.

This makes Clifford algebras absolutely indispensable to the physicist (and actually the mathematician to)

Because spin is intimately related to rotations, it turns out that Clifford algebras can be used to replace the vector cross product, which is what we used to describe rotations before Clifford algebras came along. there is a fellow who goes by the name of Hestenes who advocates replacing all exterior algebras with Clifford algebras, and all vector calculus in R³ with Clifford algebras as well. his program goes by the name of geometric algebra.

lethe, your a mind reader, and thanks for your response, and speaking English for a lowly Software Developer! I'm reading both of Hestenes most recent books, and have seen him advocate C.A. to replace all Algebras, I'm sure that ROTATION will be of use to me when I write some code to use G.A. for computer simulations of various things.

I am NOT trying to make and exceptional amount of work for myself work for myself. (Well, not quite. I do plan to be independently increasing my higher math and physics skills during the next 5 to 7 years. And I am at the beginning of that long journey.) Which leads me to ask the following question...

First, a brief preface: I do intend to understand the math behind the computer simulations that I will write, and to use the most efficient means possible (within reason, as I'm not a mathematician), to perform simulations of many types. So... Does it make sense for me to try to tackle how to do Calculus and Differencial Geometry, etc. using Clifford Algebra, as the basis for computer simulations that I intend to perform a WIDE VARIETY of functions that represent a WIDE VARIETY of virtual system that represent REAL physical systems that will require using a wide variety of math for a waide variety of Dynamic and Static physics?

In other words, I want to do computer simulations. I want to represent real-world Dynamic and Static systems of a wide variety that weill require using equations and math that is used by physicists. If G.A. is cool for rotation, fine. However, if it's going to be essentially a "waste" of time to learn how to use G.A. for Calculus, Differencial Geometry, etc., THEN I certainly do NOT want to waste my time learning standard Calculus AND G.A. Cucluslus, and the same thing again for all the other stuff it's supposed to be able to do. so, the question comes down to this ...

What do you advise? (I follow you regarding rotations and spinors, so you can assume that as a given as a good use G.A., and NOT a "waste" of time), but I don't have the next 20 years to learn 2 ways to do everything. What do you think is best? I've seen Quaternion Algebra as well (which I don't know.) And know of OTHER "Universal Algebras" that have been created very recently.
Whatcha think is the best thing for me to do to make the stuff run in a reasonably efficient manner, but not take 20 ears to learn 2 forms of the math. PLEASE NOT ONE THING. I know all this junk is available as libraries, or in classes, etc. HOWEVER, i DO WANT to KNOW the MATH, AND be capable of writing almost ANY of the types of functions/methods, that I could be expected to write (short of being a Professional Physicist and Mathematician). PLEASE EXCUSE THE STUPIDITY OF MY QUESTIONS! I REALIZE, I'm giving you no idea of what simulations I might run, nor what they might look like, or how fast the H/W is that I'm running the stuff on. HER'S SOME IDEAS I HAVE IN MIND: 3-D representations of: Grids, Planes, Games, Virtual Worlds with REQUIRING many forces Gravity, spin, drag, momentun, inertia, etc., Static worlds, like structures using grids and having various forces applied to materials, "sking on grid" animations/simulations of various kinds of creatures, simulation of water flow. My plan is to use a VERY MINIMAL AMOUNT of public domain or commercial software to accomplish any of the stuff. (I >>>WANT TO<<< write my own stuff physics and math methods/libraries myself, as much as possible, and APPLY the math and physics I will have learned to do it).

Yes, it's a long term project. It's for fun, BUT for REAL as a hobby of mine, being a Software Developer the past 20+ years. Thanks in advance for your suggestions on what to do about Clifford Algebra, or Universal Algebras, or Quaternion Algebra.

 

[lethe]

first of all, let me mention that quaternions are an example of a Clifford algebra.

as i understand it, quaternions are used in computer programming quite a bit to handle 3D rotations. so if this is your interest, you might not have to look any further than the quaternions. and it will certainly not take 20 years to learn.

as far as learning Hestenes' school of geometric algebra, i think it is not really a standard, some people swear by it, and other people think those people are a bit overzealous. but i think there are some neat results there, and it never hurts to learn more math.

i'm not exactly sure what you want to know here. are you asking whether i think you should study Clifford algebras? to that, the answer is: yes, i think you should! Clifford algebras are neat stuff.

 

[treat2]

Lethe, Again, I appreciate your help and willingness to assist me and set me on the right path, although my objectives are not fully clear to even my self. I don't want to waste your time with details that will fall into place (I assure you), as I regardyour time and assistance as valuable. So, I will conectrate on the purely math related question.

Originally posted by lethe
... I'm not exactly sure what you want to know here. are you asking whether i think you should study Clifford algebras? to that, the answer is: yes, i think you should! Clifford algebras are neat stuff.

Well, as you mentioned, the Hestenes folks claim Clifford Algebra can do anything. (As you've mentioned (and I've seen). C.A. seems to be used for rotation and spinors, and not much more.)

HERE;S WHAT i WAS TRYING TO GET ACROSS, AS MY DILEMMA:

On the one hand Hestenes says C.A. can do any kind of math. (I think ther's another aspect of it he calls C.D? or something like that),
along with Hestenes's claim, there do not seem to be a huge number of people using it for "everything", and I thing the number of books on it a qite limited. MOREOVER, i have NO IDEA, if I SHOULD FIRST KNOW how to work with Differencial equations in a way that I would learn in a standard clculus book, as a VERY HIGHLY ADVISABLE THING TO DO, BERFORE, attempting to figure out (from the very limmited C.A. books, Hos Differential Equations would be done in C.A.)
(IN other words, would I have to learn the standard way of doing D.E. FIRST, before I attempt to takcle it usning Clifford Algebra. Meaning that I'd have to learn hopw to do the damn thing 2 differnt ways, that I could not just jump into a C.A. book and tackle Differential Equations from a Hestenes of one of the few other books.

ON THE OTHER HAND: If what I happened to want to learn is Differencial equations, i WOULD GUESS THAT it WOULD be a much easier math to learn from any of the dozen Calc books I have, and ONLY USE THE "standard" method for dealing with D.E.

iT REALLY BBOILD DOWN TO 2 QUESTIONS: 1) If I were to try to do "everythin in C.A. would that pretty much require that I REALLY SHOULD FIRST learn the standard way to accomplish ALL OF THOSE SAME FORMS of math and operations using the standard way that THEY WOULD BE TAUGHT IN AN AVERAGE TEXT BOOK?

The last question of the two are:
Do you any idea if there are enough C.A. books available to permit me to learn all forms of calculus operations, diff, intgrl, multi-variable, etc., and after that, learn Differencial Geometry, as well, via a C.A. book?
It's a matter of no knowing if to abandon them standard waybecause C,A, is goingto be easier to learn, program, and there are availiable books for the "every knind of math" that C.A. is supposed to be able to do? I don't know if to bet the farm on C.A. or learn standard this, standard that. (I gotcha as far as rotations and spinors, though).

 

[lethe]

Originally posted by treat2


On the one hand Hestenes says C.A. can do any kind of math. (I think ther's another aspect of it he calls C.D? or something like that),
along with Hestenes's claim, there do not seem to be a huge number of people using it for "everything", and I thing the number of books on it a qite limited.

OK, that is right. only a small number of people subscribe to Hestenes ideas. it is not the standard approach.

MOREOVER, i have NO IDEA, if I SHOULD FIRST KNOW how to work with Differencial equations in a way that I would learn in a standard clculus book, as a VERY HIGHLY ADVISABLE THING TO DO, BERFORE, attempting to figure out (from the very limmited C.A. books, Hos Differential Equations would be done in C.A.)
(IN other words, would I have to learn the standard way of doing D.E. FIRST, before I attempt to takcle it usning Clifford Algebra. Meaning that I'd have to learn hopw to do the damn thing 2 differnt ways, that I could not just jump into a C.A. book and tackle Differential Equations from a Hestenes of one of the few other books.

to be honest, i don't know all that much about what Hestenes does. does he use Clifford algebras for differential equations? that strikes me as quite odd.

i do not think you should take that approach. i think it is better to learn differential equations from a standard calculus text.

however, i don't think that differential equations should be a prerequisite for learning Clifford algebras, so if you do not already know how to solve differential equations (either the standard way, or Hestenes way), i don't think that should keep you from your interest in Clifford algebras.

ON THE OTHER HAND: If what I happened to want to learn is Differencial equations, i WOULD GUESS THAT it WOULD be a much easier math to learn from any of the dozen Calc books I have, and ONLY USE THE "standard" method for dealing with D.E.

iT REALLY BBOILD DOWN TO 2 QUESTIONS: 1) If I were to try to do "everythin in C.A. would that pretty much require that I REALLY SHOULD FIRST learn the standard way to accomplish ALL OF THOSE SAME FORMS of math and operations using the standard way that THEY WOULD BE TAUGHT IN AN AVERAGE TEXT BOOK?

the standard vectors are taught in many calculus textbooks. you should probably master normal vector spaces, dot products and cross products and such before learning Clifford algebras.

this is only my opinion, and it is not infallible. for example, Hestenes advocates learning Clifford algebras first. this approach requires nothing of you as a prerequisite except for arithmetic and a bit of high school algebra.

more advanced texts on Clifford algebras (e.g. Michaelson + Lawson) require you to be quite at home with many concepts from abstract algebra and manifold theory. so when it comes to Clifford algebras, you can go as deeply or as shallowly as you want (or have time for)

The last question of the two are:
Do you any idea if there are enough C.A. books available to permit me to learn all forms of calculus operations, diff, intgrl, multi-variable, etc., and after that, learn Differencial Geometry, as well, via a C.A. book?
It's a matter of no knowing if to abandon them standard waybecause C,A, is goingto be easier to learn, program, and there are availiable books for the "every knind of math" that C.A. is supposed to be able to do? I don't know if to bet the farm on C.A. or learn standard this, standard that. (I gotcha as far as rotations and spinors, though).

you might take a look at the book by Pertti Lounesti. it starts off in the first couple of chapters with basic vectors, including the standard approach, and shows how Clifford algebras fit in. the first couple of chapters are pretty elementary, and so you should find them relatively easy to read

but he does go on to treat some more advanced topics, so the book may have some usefulness for you for a few years.