Re: G. A. Axiomatic development

[Lanco]

Which page number or formula number?

On Sunday, August 19, 2012 8:03:37 PM UTC+2, Peterlnx wrote:

Can somebody explain how to prove this relation in Doran's book?.

[Peterlnx]

Sorry, page 86, equation 4.10, there is an extra ^ in the right hand side, my bad.

[Lanco]

First page 86, equation 4.10 is a definition that may have been corrected in 2007 in Edition: 1st Pbk. Ed. with Corr.

I have the correct version without  ^ on the right hand side.

lanco

[Peterlnx]

I mean, one can calculate the result below for the tripe product by expanding it, so we can do same for r vectors, and we can arrive at eq 4.10.

[Lanco]

Hi Peterlnx

The 2D and 3D versions of the outer (or wedge) product is just an introduction. In n-Dim the geometric product is first introduced by axioms.
The details in the following are not mentioned in the book, as it is a book in physics.

Here is a possible tedious way to build the outer product.
The outer product is defined by (4.10), where no wedge is on the right side and first relative to a basis e1...en, such that the a-vectors are taken among the basis vectors. Now for each k>1 this product is extended by k-linearity. From this (4.10) follows for arbitrary vectors. Finally further extensions can be made (e.g. sums of k-vectors for different k) and a selection of rules proved.

Dorans book has of course another goal.

lanco

[Lanco]

May be I have misunderstood the problem.
In mathematics a definition can not be proved.

Using axioms and definitions, as in Dorans book, is just a way to escape mathematical work. It sound nice, but for a mathematician these concepts need to be shown consistent. This seems rarely done in books in physics.

lanco

[Peterlnx]

Hello lanco, I agree, a deffinition cannot be proved. Maybe its time to go ahead, learning GA by oneself is difficult.

thanks,

Peterlnx