Tensor Calculus

Date: 03/06/98 at 18:19:53
From: Ben Saucer
Subject: Tensors and General Relativity

Dear Dr. Math:

I have never been able to understand tensor calculus. I would like a
"layman's" explanation of the Ricci tensor, the Weyl tensor, and the 
geometric interpretation of the components of each. Why are there ten 
components of each? What is the "metric" of space?

I am familiar with scalar and vector calculus, "four-vectors," and 
"Minkowskian" space.

Date: 03/06/98 at 19:11:11
From: Doctor Anthony
Subject: Re: Tensors and General Relativity

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 so on, 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 = pyz.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 rather 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 allowing these arrays to be 
manipulated in an economical way is only too apparent. You must 
consult a textbook to see the notation, as it cannot be represented 
here in ASCII; but, for example, a capital letter with a couple of 
suffixes can be shorthand for a whole system of equations.

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