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Re: [Phys-L] physics without cross products



On Monday, August 22, 2016 7:28 PM, John Denker <jsd@av8n.com> wrote:



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.
This misses a vitally important piece of information. Trivector A^B^C is more than just volume (and shape) of the respective parallelepiped. It also decrees the way we go around the parallelepiped along its edges.
 In fact, this piece is explicitly formulated in the following (correct!) statement about bivectors:
All we need is a sense of circulation around the edge.
But this is in conflict with: 
The faces can be indistinguishable. They can be transparent.
Marking two faces differently is equivalent to distinguishing between two opposite senses of circulation around the edge. The reason is that a bivector is not just a parallelogram made of segments, but a parallelogram made of directed segments (vectors!) and in a way (tail to tip!) creating the sense of rotation. Therefore making the faces transparent would only mask the existing handedness (chirality), but not eliminate it. Transparent sides will not eliminate the difference between A^B and B^A! Generally, I find the table:
scalar point - no geometric extent; grade=0.>vector-length-geometric extent in 1 direction; grade=1>bivector-area-geometric extent in 2 directions; grade=2>trivector-volume-geometric extent in 3 directions; grade=3, etc.   
correct only in its first line. The rest is misleading, because vector is not just the length, it is directed segment, etc. 

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 This is exactly the same. 71 (2) indicates the volume and issue No., while 71, 104 indicates volume and page. If you open the article itself in a printed copy of the journal, you will find 71 (2) at the bottom of p. 104.
Moses Fayngold,NJIT
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