19 July 2002

Tensor Than Before

by George E. Hrabovsky, President, MAST

News from MAST

Hello from MAST! Just to let you know, we are expecting to have a shareware periodic table that does lots of neat stuff finished by the end of this month (that means it will be done in the next three months :-) ).

Stress

Last week I rather cavalierly mentioned that something was called the stress tensor, unfortunately I didn't really explain what the word tensor means. I will try to explain about tensors in this column.

So far we have discussed scalars and vectors. Recall that a scalar is simply a magnitude, and a vector is a sequence of magnitudes that conform to certain properties. In fact any sequence that conforms to these rules is a vector.

One can generalize this by stating that a vector is a sum of the products of the components of the vector with the unit vector corresponding to the component. If we are thinking of a vector in Cartesian coordinates the unit vectors are , , and . So the vector becomes,

It is reasonable to ask a question of some relevance for physical phenomena. Is there any phenomena that exhibits an additional level of dependency upon direction. That is, using Cartesian coordinates, is it possible that some property will have an x component that itself has an x, y, and z component.  Take our example of the stress tensor,

At first glance it seems to be very familiar. It looks a lot like the expression for a vector. The difference is that we have a sum of a very special kind of product. Instead of the product being a component scalar and a unit vector, we have a component vector and a unit vector. What kind of product is this? It is not an inner product (or dot product), nor is it an outer (cross) product.

It is a special product called a dyad. In general we can write a dyad as follows

We can also write the dyad as a symbol,

This is called the nonian form of the dyad.

A sum of dyads is called a dyadic. This is what our stress tensor is,

Where the first dyad is the product of the stress vector in the x direction in product with the unit vector in the same direction

If we make this more independent of a specific coordinate system, we can speak of the first direction, second direction, and so on. The first dyad would then have the form,

If we stare at this long enough we can realize that there is a shorter way of writing this,

We can generalize this by writing,

We can put this together for our Stress Tensor,

I think this is enough for this week.

Theory Challenge Answer for Last Week's Column

A tensor represents a higher-order of dependency on components than a vector. For a three-dimensional space a vector will have three terms. A tensor is an object that shows the dependency on the coordinates; a scalar has no such dependency and is thus a tensor of order 0. A vector has a dependency in every direction and is thus a tensor of rank 1. A second order tensor would then have a component in each direction for each direction (a three dimensional order two tensor would have nine components) and so on.

Theory Challenge

Use the dyad and dyadic ideas to construct rules of addition for dyadics.

Every time you answer the Theory Challenge for this column your name will be entered into a database. Every six months a drawing will be made and the winner will receive a prize. The current prize is your choice of either Riley, Hobson, Bence, Mathematical Methods for Physics and Engineering or your choice of any three Schaum's Outline Books. If you respond to four challenges you get four chances to win. If you respond to no challenges, well, you're out of luck! The drawing will take place on 6 September, and the winner will be announced in the next Bulletin after the 6th of September.

Books That I Like

I recommend two books, one is, unfortunately, out of print.

E. A. Fox (1967), Mechanics, Harper and Row. This is a very nice book that introduces the concepts of Continuum Mechanics in a very nice way.

The other is,

Robert C. Wrede, (1963), Introduction to Vector and Tensor Analysis, John Wiley and Sons, Inc. (reprinted by Dover Publications, Inc. in 1972). This is my favorite book about vectors and tensors (from an introductory standpoint).