Tensor

Date: 10/01/97 at 14:13:17
From: MIKE VARELA
Subject: TENSORS/DEFINITION

Dear Dr. Math,

Is there a layman's definition for a tensor?  And is there a simple 
workable example to illustrate this definition?

Thank you!
Mikey V.

Date: 10/01/97 at 17:24:06
From: Doctor Anthony
Subject: Re: TENSORS/DEFINITION

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 = 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 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
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/02/97 at 09:51:39
From: Mike Varela
Subject: Re: TENSORS/DEFINITION

Dr. Anthony,

That was the clearest definition I ever heard for tensors.  Thanks!

However, I don't understand why 3D surfaces need to be defined by 
their normals dSx, dSy, dSz. Does it mean that stress F, with its fx, 
fy and fz components, acting on the xyz components of the surface, is 
being looked at in terms of its perpendicular influence, or pressure?  
Which would mean in that case that F is ALIGNED (parallel) to the dSx, 
dSy aND dSz NORMALS to produce the pressure on the xyz-components of 
the surface in question?  If so, seeing that is how a real force 
influences the surface of an object, then I can see why it's important 
to align such a matrix in terms of the surface's normals. 

Mikey V.

Date: 10/02/97 at 14:21:21
From: Doctor Anthony
Subject: Re: TENSORS/DEFINITION

Hello,

The items pxx, pxy, pxz are forces per unit area (hence the term 
pressure), and these are the pressures arising from fx alone on the 
components of area, dSx, dSy and dSz. In all this work it is much more 
convenient to have forces, velocities, momentums, areas, torsions and 
almost any physical property you can think of represented in component 
form, and because of this the number of possible components and 
possible combinations of components expands so rapidly.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/