Use of Tensors

Date: 7/23/96 at 0:11:37
From: STU R U?
Subject: Use of Tensors

Hi!

Could you please explain the uses and properties of tensors?

Date: 7/23/96 at 17:52:29
From: Doctor Anthony
Subject: Re: Use of Tensors

We could start by saying that a SCALAR is a TENSOR whose RANK is zero, 
and a VECTOR is a TENSOR whose RANK is one. What we shall need to see 
is what is meant by a tensor of rank 2, or 3 etc.  As you can see, 
TENSOR is an inclusive term, a generalization of the concept of 
vector.

To illustrate a tensor of rank 2, imagine a plane surface area with a 
force acting on it. The total effect depends on two things, the 
magnitude and direction of the force, and the size of the area and its 
orientation. In fact this latter property can be represented uniquely 
by a vector of magnitude proportional to the size of the area, and in 
a direction NORMAL to the area. So the effect of the force upon the 
surface depends on two vectors and is called a TENSOR of RANK TWO.

In fact, if you consider components of force and each of these 
components acting on each component of the area vector, there are nine 
terms in all, which can be displayed as an array, representing the 
total stress, and this quantity is the tensor of rank two. Tensors 
can thus be represented by arrays, and manipulated in a manner 
reminiscent of matrix manipulation. They have particular importance 
in problems involving invariance in changes of a coordinate system, 
and, famously, were used by Einstein in deriving his law of 
gravitation in General Relativity.

The notation which uses double suffixes can not be recommended in 
ASCII, and I think you will need to dig further into this topic with 
the aid of a textbook on advanced topics in vectors.

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