Re: Simple books on 4-vectors

From: Danny Ross Lunsford (antimatter33_at_yahoo.com)
Date: 10/08/04 

Date: Fri, 8 Oct 2004 11:20:42 +0000 (UTC)

Oz <oz@farmeroz.port995.com> wrote in message

> Good, that means I am confused....

Everything is impossible until it is obvious.

> >1) Forms in themselves are algebraic objects, and not related to
> >calculus at all. They are generalized from "directed lengths" as the
> >model for vectors.
>
> OK. As usual I have to figure out what you really mean here.
> Am I to guess that they are nD equivalents of vectors?
> A brief description would be gratefully appreciated.

I assume you are familiar with the notion of a vector. Let's defer the
precise definition until later. It's a "directed length" - which
means, it has a direction and a length. Let's call it an "oriented
length" - oriented because you pick the direction from one endpoint to
the other. Let's call it

X

Now we can consider objects which have similar (as yet unspecified)
transformation properties as vectors, which depend on 2 directed
lengths. They mark out a parallelogram in space - which is nothing but
an "oriented surface element" - oriented because we can pick one
length as the "start" and the other as the "finish". Let's call it

X ^ Y

The magnitude of X ^ Y is just the area of the parallelogram.

Now if we reverse "start" and "finish", we have the same parallelogram
but with opposite orientation. By definition it is

Y ^ X

and we express "oppositeness" by the rule

Y ^ X = - X ^ Y

Now suppose X and Y are the same oriented length X = Y. The
parallelogram collapses to a line segment and the area is 0. That is,

X ^ Y = 0 if X = Y

Next is an "oriented space element" - three directed lengths in a
specific order. It is

X ^ Y ^ Z

and

X ^ Z ^ Y = - X ^ Y ^ Z = + Y ^ X ^ Z = - Y ^ Z ^ X = + Z ^ Y ^ X
= - Z ^ X ^ Y

and so on. So we have a hierarchy of objects

X
X ^ Y
X ^ Y ^ Z
X1 ^ X2 ... ^ Xn

They all represent "primitive oriented space elements".

Note that we have to stop inventing new ones when we run out of
dimensions. This is expressed by the rule

X1 ^ X2 ... ^ Xn = 0 if n > d

When you grok this, reply and we'll go into more depth. The thing we
are considering goes by the ten-dollar name of "Grassmann algebra of
multivectors". Today we call them "p-forms".

Grassmann called the formalism he invented "Die Lineale
Ausdehnungslehre" - the "theory of linear extensions". The key point
is that the p-forms are *primitive algebraic objects* and are
irreducible.

-drl