02 August 2002

Turning One Vector into Another

by George E. Hrabovsky, President, MAST

News from MAST

Well, the news is that there was no column last week! Email problems, the wonders of modern technology, eh?

Also, I am an idiot. The theory challenge should read rules of arithmetic, not rules of addition...  My apologies!

The Jacobian

Last time I introduced the idea of a dyad. This is a special product of vectors described as

We can do away with the summation symbol (this makes these formulas much easier to write) by stating that whenever we see two of the same index in a term that we are summing over that index. This is called the Einstein Summation Convention. So, our expression then becomes

We are almost on to the point where we can talk about tensors. After this column we will be there.

To proceed we must back up a little bit. It is an interesting fact that we can change one vector into any other vector (within the same universe) by a combination of rotating the vector and changing its length (called a boost). When we do this we are transforming one vector into another. The same process can be used to describe changes in an existing vector as some parameter changes (such as time). If we start with vector A and we want to change it into vector B', how do we proceed?

It turns out that we can take the derivative of the coordinates for vector B' with respect to the coordinates of vector A.

We can generalize this to the first, second, third, etc. coordinates,

If we look at the derivative, we see something interesting. Assume any three dimensional space (so that we have components 1, 2, and 3).

This is called the Jacobian Matrix and it is essential for developing coordinate transformations.

Now we must digress a bit. If we have a 2 x 2 matrix,

we can define a relationship between the elements of the matrix

This relationship is called a determinant of rank 2,

For a 3 x 3 matrix

the determinant is a bit more complicated,

where each determinant of rank 2 is called a minor or a cofactor. Each of these determinants is handled as before, so,

If we take the absolute value of the determinant of the Jacobian matrix then we have what is called the Jacobian determinant. This is sometimes just called the Jacobian. The Jacobian of a two dimensional coordinate space gives the differential area element. For a three-dimensional space the Jacobian gives the differential volume element.

Theory Challenge Answer for Last Week's Column

These are simply extensions of the familiar rules, right? Let's see for sure. Let us begin with the commutative property for addition. We have,

Now let's see what happens if we reverse the order,

It looks the same, but let's look a little bit deeper, let's reverse the order of the coefficients in each term (this is allowable since they are real or complex and commutative under multiplication),

This is a column of the first dyad. The reverse order transposes the matrix. So, we can say that,

    Since the order of multiplication does not change, the dyad is associative.
    Let us see what happens if one vector is all 1s, we will define this as the

So the dyad has an identity element.

Theory Challenge

What would the Jacobian for a dyad look like?

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 this out of print book.

E. A. Fox (1967), Mechanics, Harper and Row. This book has a nice section on dyads and their properties.

Converted by Mathematica      August 1, 2002