Definition of a Tensor

Date: 05/20/2001 at 10:00:41
From: M Innes
Subject: Clear concise definition of a tensor

If a vector can be described as "any quantity having magnitude and 
direction" why can't a tensor be defined as "any..." in just a 
couple of lines?

Many thanks.

Date: 05/20/2001 at 17:32:42
From: Doctor Anthony
Subject: Re: Clear concise definition of a tensor

Tensors are a further extension of ideas we already use when defining 
quantities like scalars and vectors.

A scalar is a tensor of rank zero, and a vector is a tensor of rank 
one. You can get tensors of rank 2, 3, ... and their use is mainly in 
manipulations and transformations of sets of equations within and 
between different coordinate systems.

If you consider a force F with components fx, fy, fz and you have an 
element of area whose NORMAL has components dSx, dSy, dSz, then fx 
itself has components acting on these three elements, and the PRESSURE 
of fx ALONE is denoted by its three components

     pxx, pxy, pxz

Similarly fy will produce pressures pyx, pyy, pyz and fz will produce 
pressures pzx, pzy, pzz.

The product pxx.dSx gives the FORCE acting upon dSx by fx ALONE.

It follows that:

     fx = pxx.dSx + pxy.dSy + pxz.dSz

     fy = pyx.dSx + pyy.dSy + pyz.dSz

     fz = pzx.dSx + pzy.dSy + pzz.dSz

and the total STRESS F on the surface dS is

     F = fx + fy + fz

which is given by the sum of the three equations (nine components) 
shown above.

So we see that stress is not just a vector with three components (in 
three-dimensional space) but has NINE components in 3D space. Such a 
quantity is a TENSOR of rank 2.

In general if you are dealing with n-dimensional space, a tensor of 
rank 2 has n^2 components.

Unlike a vector, whose components can be written in a single row or 
column, the components of a tensor of rank 2 will be written as a 
square array.

In n-dimensional space a tensor of rank 3 would have n^3 components. 
The need for a convenient notation that allows these arrays to be 
manipulated in an economical way is only too apparent. You must 
consult a textbook to see the notation, it cannot be represented here 
in ASCII, but a capital letter with a couple of suffixes can be 
shorthand for a whole system of equations.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/21/2001 at 20:59:18
From: Doctor Schwa
Subject: Re: Clear concise definition of a tensor

Hi Mark,

My approach would go like this:

You know what a vector is. Let's think of that as being a column 
matrix: an n by 1 matrix.

Then a covector will be a row matrix: a 1 by n matrix. Another way to 
define a covector is that it's a linear function that takes each 
vector to a number: in other words, you can multiply a covector by any 
vector and get a number. You could write that as c(v) = a number.

A matrix takes both a covector and a vector to produce a number: cMv 
(row times matrix times column) will produce a number. Matrix 
multiplication also has certain associative/distributive laws, and 
some properties of how it acts when one of those vectors (c or v) are 
multiplied by a scalar. Those are called "linearity rules" or "linear 
properties" or things like that.

Furthermore, if you "feed" the matrix only a vector, multiplying Mv, 
you'd be left with a vector. If you "feed" it only a covector, 
multiplying cM, you're left with a covector, still "hungry" for a 
vector to produce a final numerical answer.

A tensor is like a matrix, but it can have more than two dimensions: 
it takes some collection of covectors and/or vectors, and produces a 
number, following the same kind of linearity rules as you'd expect 
from matrix multiplication. And if you "feed" it less than its full 
complement of desired vectors and covectors, you'll be left with a 
tensor that's still "hungry" for a few things. It only makes sense to 
equate tensors that have "hunger" for the same things, just like you 
can only equate vectors with other vectors and matrices with other 
matrices.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/