Re: non-vectors

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

> 1) The name vector is applied to objects that exhibit certain
> invariances with respect to certain transformations.
>  -- The magnitude of a vector is invariant under rotation.
>  -- Ditto for reflection.
>  -- The direction of one vector relative to another is
>     invariant under rotation.
>  -- Ditto for reflection.

I'm not entirely sure a geometer would totally agree with this.  I recall
sitting in on a differential geometry class where in one of the early
lectures the professor asked what a vector is and how we represent it in
physics.  I basically mentioned magnitude and direction and representation
by an arrow.  He was quite happy and launched into a discussion (details of
which I forget) on how glad he was, despite my being a physics student, that
I hadn't talked about defining it in terms of transformation properties.

IMO little is lost, at least to the undergraduate curriculum to discussing
it in those terms for the advanced course.

> 2) The name pseudovector is applied to objects that exhibit
> certain invariances with respect to certain transformations.
> Pseudovectors behave the same under rotation but change sign
> under reflection.
>
> Vectors (and pseudovectors) in D=3 space can be represented
> by an array of three numbers.  The converse is not true; not
> every array of three numbers is a vector.  It could just be
> a shopping list, with no particular transformation properties.