tensors (was: field transformations)

["John S. Denker" <jsd@MONMOUTH.COM>]

On 04/03/2003 05:41 PM, Joe Heafner wrote:
> But I don't understand tensors completely yet because no one has ever
> been able to explain them to me. Most of the text books aren't helping
> either, just spouting definitions in terms of transformations and
> all...very confusing to me.  I don't really feel too badly about it
> thought since many practicing physicists (including one author of a
> widely read relativity text) tell me they don't understand tensors
> either. :-)

In general, a tensor is just mathematical machinery.
It is a way of expressing a linear relationship.
It is completely general, which is its strength.
But this comes at a price:  because it is so general,
it can represent lots of different physics.  I doubt
will ever be an all-purpose physical interpretation
of tensors in general.

A tensor of rank (1,0) i.e. one contravariant index
is a plain vector.  You can model this as a stick with
a direction marked on it.

A tensor of rank (0,1) i.e. one covariant index is a
one-form.  You can visualize this as the level curves
on a contour map.

  There is a correspondence between one-forms and plain
  vectors if you have a dot product (e.g. geometry) and
  not otherwise (e.g. thermodynamics).

A tensor of rank (1,1) might be a linear operator such
as a rotation.  Typically it is neither symmetric nor
antisymmetric.  It takes a vector as input and produces
a vector as output.

An antisymmetric tensor of rank (2,0) might be the
representation of a bivector.  You can model this as
a piece of surface with a direction of circulation
marked on it.

A symmetric tensor of rank (0,2) might be a bilinear
operator such as the tensor of inertia, or the stress
tensor, or whatever. It takes two vectors and produces
a scalar.  You might visualize this as an ellipsoid.

Et cetera.

As far as I can tell, trying to understand the physical
significance of "tensors in general" is futile.  It's
like trying to understand the physical significance of
"arithmetic in general".