Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
College Linear Algebra
College Physics

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/