Re: Re: Re: Wrong behavior of CrossProduct

• To: mathgroup at smc.vnet.net
• Subject: [mg8088] Re: [mg8044] Re: [mg7996] Re: [mg7958] Wrong behavior of CrossProduct
• From: seanross at worldnet.att.net
• Date: Tue, 5 Aug 1997 03:22:58 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```Mark Evans wrote:
>
> >
> > You raise a good point and put your finger on a subtlety that escapes
> > most people.  The vector v={0,b,c} in spherical coordinates represents a
> > vector of zero length for displacement vectors.  Consider,however, the
> > case of a gradient field, such as an electric field.  Certainly we could
> > conceive of an electric field that, at some point in space, had no
> > radial component, but only a theta or phi component.  The magnitude of
> > the field would not be zero because the radial component was zero.  Most
> > mechanics and electodynamics textbooks pass over this subtlety because
> > physical cross products that occur in nature don't involve displacement
> > vectors, they involve field vectors and vector differential operators
> > which occur at a local point in space.
> >
>
> Sean,
>
> Mathematica's vector math is implemented in a package.  Do you think
> there should be two packages, one for displacement vectors, another for
> field vectors?  What do you think Wolfram Research should do to resolve
> these subtleties?  How would you go about resolving them?
>
> Mark

I wouldn't resolve them.  Anyone who actually needs to calculate numbers
does it in Cartesian coordinates after doing symbolic math in whatever
other system.  If I ever got lazy and wanted Mma to do my symbolic cross
products, I would write it out myself in tensor notation:

g[i,j] epsilon[i,j,k] A[i] B[j].
The last three factors are the same as the cartesian CrossProduct[A,B]
command, so I would only have to put in the correct metric tensor
myself.  It would take a decent package to correctly include all the
subleties of covariant, contravariant and displacement vectors and
include all the metrics for all the possible combinations.  If I were to
"fix" Mma, I would eliminate the non-cartesian cross-products altogether
and let people construct their own with matrix multiplication.

```

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