Re: [Phys-L] physics without cross products

[John Denker <jsd@av8n.com>]

On 08/22/2016 02:40 PM, Moses Fayngold wrote:

> If I understand correctly, the wedge product a^b of 2 vectors a, b is
> synonym of their outer product as defined in geometric algebra (see,
> e.g., Fig. 2 in [1]).

There are two problems with that.

First of all, Hestenes's terminology is highly nonstandard and ambiguous.
There is a centuries-old definition of outer product, i.e. tensor product,
i.e. V ⊗ W.

The wedge product V ∧ W is more properly called the /exterior/ product,
not the outer product.  This is discussed in more detail at
  https://www.av8n.com/physics/clifford-intro.htm#sec-outer

This conflict drives students crazy.


Secondly, the wedge product is not, by itself, a drop-in replacement
for the cross product.

As a super-important example, the volume of a parallelepiped is given
by the trivector A∧B∧C which is equal to A×B·C ... definitely not A×B×C.

The recipe for replacing cross products is spelled out at
  https://www.av8n.com/physics/clifford-intro.htm#sec-cross-out-cross

> The fact that by-vector a^b lives in (a, b)-plane does not eliminate handedness.

I disagree.

> Just as the line perpendicular to (a, b) may have two opposite directions,
> the parallelogram representing by-vector a^b is a 2-sided plane segment.
> Calling the opposite sides "black" and "white" respectively, we have to decide which one of them will face, 

In the immortal words of Smith & Dale:  Don't do that.  We do not need
to mark the faces.  All we need is a sense of circulation around the edge.
The faces can be indistinguishable.  They can be transparent.

There is some serious physics here.  I have presented it in a way
that makes clear what the answer is supposed to be ... but there
are other ways of formulating the question that make it much more 
mysterious.  Here is a riddle posed by a young student:
  https://www.av8n.com/physics/pierre-puzzle.htm






> Reference [1]: D. Hestenes, Am. J. Phys., 71 (2), 2003  

There's something wrong with that reference.  I assume it refers to
  David Hestenes
  "Oersted Medal Lecture 2002: Reforming the mathematical language of physics"
  Am. J. Phys. 71, 104 (2003)
  http://dx.doi.org/10.1119/1.1522700
  http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf