Re: vector products

["John S. Denker" <jsd@MONMOUTH.COM>]

Chuck Britton wrote [off list]:
> Tina isn't the only one who is totally unfamiliar with this
> Clifford Algebra thing.

I had never heard of it until about a month ago.
(There's no zealot like a convert :-)

> I understand that the cross product is verboten.

Not verboten.  Just unnecessary.  You're allowed to eat
soup with a fork if you want.  But I don't recommend it,
now that spoons are available.

> Does this imply that the 'Right Hand Rule' doesn't have a place in
> Clifford's algebra?

Correct.  The RH rule is not needed.  The question just
doesn't come up -- which is a good thing IMHO.

> Surely this new math recognizes the difference between left and
> right-handed coordinate systems.

Not really.

Here's the deal:  Physically-relevant physics results depend on
applying the RH rule an !!even!! number of times.  The second
application undoes the mischief done by the first application.
For example, the RH rule is used to define the conventional
magnetic field vector... and the RH rule is used again in the
Lorentz force law.  The Clifford Algebra approach does not have
any RH rule to define the magnetic field bivector, and does not
have any RH rule in the Lorentz force law.  If you want to see
exactly how this all works out, see
  http://www.monmouth.com/~jsd/physics/maxwell-ga.htm



Tina Fanetti wrote:
> What in the world is the wedge product?
> How am I supposed to explain Clifford Algebra when
> 1.  I don't know what it is.
> 2.  Most of my students are in Calc 1.
> 3.  The last time my students saw geometry they were in high school.
>
> I looked in my favorite book, div, grad, curl and all that
> and the wedge product isn't in there.

You could have clicked on the URL at the bottom of the note
that started this thread.  It contains explanations, and links
to some excellent references.

Here they are, one click closer, with commentary:

David Hestenes,
``Oersted Medal Lecture 2002: Reforming the Mathematical Language of
Physics''
  http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf
==> nicely pictorial and pedagogical.

Stephen Gull, Anthony Lasenby, and Chris Doran,
``The Geometric Algebra of Spacetime''
  http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html
==> goes deeper into the physics, but glosses over some of the
    details you need to know if you're actually going to use the stuff.

Richard E. Harke,
``An Introduction to the Mathematics of the Space-Time Algebra''
  http://www.harke.org/ps/intro.ps.gz
==> Wonderfully thorough without being overly-sophisticated.
    Perhaps too detailed to be the optimal starting point, but
    very nice as a reference, as a review, and as a compliment
    to the previous items.

Also, I get 7000 hits from
  http://www.google.com/search?q=clifford-algebra