From: Deane Yang <yang@fields.waterloo.edu>
Subject: Re: tensors
Organization: The Fields Institute for Research in Mathematical Sciences
X-Useragent: Nuntius v1.1
     <5130@tuegate.tue.nl> <5134@tuegate.tue.nl> <3djygrk@rpi.edu>
Date: Fri, 21 Aug 1992 19:21:33 GMT

In article <3djygrk@rpi.edu> Tom Pierce, pierct@rpi.edu writes:
>I know all about 1st and second order tensor, but what's a third order
>tensor look like?  Also, aren't these the same mathematical critters as
>"matrices", witha the added "square" criterion?
>What would you use a third or highr order tensor for?  Any PHYSICAL
>applications?

Perhaps it's worth distinguishing between tensors and tensor fields.
A tensor is an algebraic object, living inside a tensor product of
vector spaces. A tensor field is analogous to a vector field.
A tensor field is a function with values inside the tensor product,
just like a vector field is a function taking values in a vector space.
However, we're usually sloppy about this, calling a tensor field a tensor.

So a second-order tensor is very much like a matrix,
but a 2nd order tensor field is like a matrix-valued function.

But all of this is a gross oversimplification. It turns out in physics
or differential geometry, when you work with vector fields or
tensor fields, you don't want the vector space or the tensor products
to be a single, fixed object. Instead, each point in the domain
has a vector space of its own. The vector spaces have an abstract
intuitive meaning, e.g. "velocity vectors", a.k.a. tangent vectors,
and when coordinates given on the domain, they induce a natural
basis of each vector space. In any case, working all of this out carefully
leads to the notion of tangent bundles, vector bundles, and tensor
bundles.

The crucial point of tensor fields is
 that the components of the tensor field depend on the 
coordinates you use. Moreover, when the coordinate change, the components 
change linearly, indicating that they represent objects lying in some
abstract vector space. Understanding the behavior of tensor fields under
coordinate changes is important, because one of the guiding principles of
differential geometry and modern physics is that all invariants or
observable phemomena are invariant under coordinate changes. This
is usually called "invariance under diffeomorphisms" and considered
 a special case of "gauge invariance".

It turns out that higher order tensors are important. The easiest way
to see this is that the "derivative" of a k-th order tensor field
is a (k+1)-th order tensor. So the moment you have the desire to 
differentiate a second order tensor, you've got a third order tensor
on your hands.

Also, in differential geometry, general relativity, and in gauge theory,
you work with the curvature tensor, which is a fourth order 
tensor.

Still another natural class of higher order tensors are differential 
forms, which are the correct objects to use for multiple integrals,
since they encode the change of variable formula, as well as Stoke's
theorem in a both elegant and easily computed form.
Deane Yang
Polytechnic University
From: flower@hpcc01.corp.hp.com (Graham Flower)
Date: Thu, 13 Aug 1992 17:02:11 GMT
Subject: Math for Physics References
Organization: the HP Corporate notes server