This mention of oriented areas makes me think of "geometric
algebra" as developed in a couple of new texts I've been looking
at: "New Foundatiions for Classical Mechanics" by David Hestenes
and "Geometric Algebra for Physicists" by Doran & Lasenby. They
make use of, for example, a "geometric product" of two vectors
a and b such that ab = a.b + a^b Here a.b is the scalar dot
product and a^b is the outer product, which they interpret as
an oriented area and call a "bivector". ba = -ab only if a and
b are prependicular. This generalizes to higher order products
(oriented volumes, etc.) and higher dimensions in a very clean
fashion. Pauli spin matrices arise naturally, and the problems
with cross products (polar vs axial vectors) disappear when
the cross product is regarded as disguised bivector.
I have hopes of developing some J tools to work within this
geometric algebra framework. I think the consistency of this
new treatment might appeal to those who love J for its set of
consistent rules.
Patrick
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