Mathematical Tools for Physics

12—Tensors 360

The conductivity tensor relates current to the electric field:

~=^11 σ

(

E~

)

.

In general this is not just a scalar factor, and for the a.c. caseσis a function of frequency.

D

DF

cut

A

The stress tensor in matter is defined as follows: If a body has forces on it (compression
or twisting or the like) or even internal defects arising from its formation, one part of the
body will exert a force on another part. This can be made precise by the following device:
Imagine making a cut in the material, then because of the internal forces, the two parts will
tend to move with respect to each other. Apply enough force to prevent this motion. Call it
∆F~. Typically for small cuts∆F~will be proportional to the area of the cut. The area vector
is perpendicular to the cut and of magnitude equal to the area. For small areas you have
differential relationdF~=^11 S

(

dA~

)

. This functionSis called the stress tensor or pressure tensor.
There is another second rank tensor called the strain tensor. I described it qualitatively in section9.2and
I’ll simply add here that it is a second rank tensor. When you applystressto a solid body it will developstrain.
This defines a function with a second rank tensor as input and a second rank tensor as output. It is the elasticity
tensor and it has rank four (^22 E).
So far, the physically defined tensors have been vector-valued functions of vector variables, and I haven’t
used then-linear functional idea directly. However there is a very simple example of such a tensor:

work =F~.d.~

This is a scalar valued function of the two vectorsF~ andd~. This is of course true for the scalar product of any
two vectors~aand~b
0
2 g

(

~a,~b

)

=~a.~b. (9)

0
2 gis a bilinear functional called the metric tensor. There are many other physically defined tensors that you will
encounter later. In addition I would like to emphasize that although the examples given here will be in three
dimensions, the formalism developed will be applicable to any number of dimensions.

12.2 Components
Up to this point, all that I’ve done is to make some rather general statements about tensors and I’ve given no
techniques for computing with them. That’s the next step. I’ll eventually develop the complete apparatus for