Re: Re: Wrong behavior of CrossProduct

• To: mathgroup at smc.vnet.net
• Subject: [mg8044] Re: [mg7996] Re: [mg7958] Wrong behavior of CrossProduct
• From: seanross at worldnet.att.net
• Date: Sat, 2 Aug 1997 22:32:52 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```> Sean Ross wrote:
>
> ...  Looking at the problem you
> gave mathematica in spherical coordinates you specified V=(a1,a2,0},
> which is a displacement vector beginning at the origin and going to the
> point V.  You then wanted to cross it with the vector U={0,0,1}, which
> is a displacement vector beginning at the origin and ending at the
> origin, so you took a cross product between two vectors, one of which
> had a zero magnitude.  The answer given by mathematica was correct for
> DISPLACEMENT VECTORS.   This makes perfect mathematical sense, but is
> ludicrous from a physical standpoint since all cross-products that
> appear in physical equations are for field vectors, not displacements.

Richard W. Finley, M. D. wrote:
>
> Sean,
>
> Regarding the message below, perhaps I missed something....as far as I know a vector with zero magnitude is the zero vector, regardless of whether you consider it a displacement vector or a field vector, and the cross product of any vector with this zero vector should be zero.  This would seem to be the only interpretation that makes mathematical OR physical sense.
>

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.

A look at the standard, tensor notation way of writing cross products
reveals another, often overlooked point:

AxB=g[i,j]epsilon[i,j,k] A[i]B[j]

The metric tensor g[i,j] is the identity matrix in cartesian
coordinates, but has components that are a function of r and theta for
spherical and cylindrical coordinates, so you can't convert cross
products back and forth between cartesian and other coordinate systems
without specifying at what point in space the two vectors exist.  The
mathematica result assumes that the vectors exist at the origin, which
is another reason there is a discrepancy between mathematica cross
products and what might be expected.

```

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