Re: [Phys-L] physics without cross products

[John Denker <jsd@av8n.com>]

On 08/21/2016 12:18 PM, Derek McKenzie wrote:
> As far as Electromagnetism is concerned, I wasn't sure whether or not
> you could get by with just wedge-products, so that's good news if it's
> right.

Well, you can't get by with "just" wedge products.

As I see it:

  If you have an equation involving a cross product, and that's
  all you have, then the first step should be to replace it with
  a wedge product.  Example:  Angular momentum.

  OTOH if you have two parallel equations, one with a cross product
  and one with a dot product, the first step should be to replace
  both of them with the full geometric product.  This leaves you with
  one equation instead of two ... or in the case of electromagnetism,
  one equation instead of four.

Either way, the first step is usually not the only step.

You can always project out the top-grade piece (wedge) and the
bottom-grade piece (dot) if you want.  That allows you recover
the original equations if/whenever desired.  That upholds the
correspondence principle:  The new model ought to do everything
the old model could do, under appropriate conditions.

=======

I am occasionally called upon to teach angular momentum, up to
and including gyroscopic precession, which requires adding angular
momenta.  Many of the students never took high school physics at
all, or if they did, it was a long time ago.

I have completely given up on teaching them cross products.  I've
tried;  I just can't do it given the constraints on my time and
their patience.

I can however teach them about bivectors ab_initio, and show them
how to add bivectors edge to edge, just as we add vectors tip-to-tail.
Angular momentum is still a challenging topic, but this helps a lot.