Definition of Cartesian Tensor

Date: 12/18/98 at 15:01:19
From: Martin Beard
Subject: Definition of Cartesian Tensor

What is the precise defintion of a CT?

Date: 12/18/98 at 15:58:55
From: Doctor Anthony
Subject: Re: Definition of Cartesian 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 so on, 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 = pyz.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 which 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/