Re: Re: Wrong behavior of CrossProduct
- To: mathgroup at smc.vnet.net
- Subject: [mg8047] Re: [mg7996] Re: [mg7958] Wrong behavior of CrossProduct
- From: seanross at worldnet.att.net
- Date: Sat, 2 Aug 1997 22:32:55 -0400
- Sender: owner-wri-mathgroup at wolfram.com
Wouter Meeussen wrote:
>
> hi Sean,
>
> your remarks about the cross product set me testing wether I understood
> vectoranalysis well enough.
>
> As I remember,
> Cross[v1,v2,v3] was in my mind equivalent to:
>
> Det[{v1,v2,v3,unit}] where "unit" is the list of unit-vectors.
>
> Since a u[1] + b u[2] + c u[3] + d u[4]
> is to be understood as a vector(-sum), the fact that Det[] produces a
> zero-dimensional result is no objection.
>
> in Mma:
> -------
> v1={a,b,c,d};v2={f,g,h,i};v3={k,l,m,n};
> unit=Array[x,4];
> de=Det[{v1,v2,v3,unit}];
>
> then either:
> Last[CoefficientList[de,#]]&/@unit
>
> or:
> (List@@Collect[de,unit])/unit
>
> let us extract the coefficients of "unit"
>
> BUT:
> when I check it versus Cross[] for different ranks,
> I find an extra factor
> (-1)^r with r the rank of "unit".
>
> Dimensions[unit]
> {4}
>
> Does this correspond in any way to what you called covariant and contravariant?
>
No. After I wrote my message, I went back to my Morse and Feshbach and
looked up covariant and contravariant. They both describe field
vectors, not displacement vectors. A prototypical covariant vector is
like Electric or magnetic field. A prototypical contravariant vector is
the "Del" operator. The designations have to do with the way the
vectors transform under coordinate rotations(covariant transformation
have the primed coordinates in the numerator, contravariants have them
in the denominator). They also have different metric tensors associated
with them, so that the formulas for taking cross products can be
different in non-cartesian coordinate systems. In other words,
DelxE,DelxDel, and ExB represent contravariant x covariant,
contravariant x contravariant and covariant x covariant and all have
different formulae for cross product. In the case of a mixed cross
product, the metric tensors cancel out, but not for the other two.