Subj : Idea - Tensor could be modelled by 2 arrays and a function
To   : alt.math,alt.sci.physics,comp.programming
From : Mr Pixie
Date : Mon Aug 22 2005 08:11 pm

As a programmer, I try to understand objects in terms of data structures, so 
here is my idea for how best to code a tensor object. I would be interested 
to hear comments:

2 arrays, A and B : A is for the real space (covariant) and B is for the 
dual space (contravariant)
The arrays do not have to have the same number of dimensions.
In maths, B would be A*

The to represent a tensor would be something like:

dim A[0..n]    ; first array of n dimensions
dim B[0..m]   ; second array of m dimensions

function tensor_transformation (A, B) real
{
        ' tensor transformation rules go here e.g. the Lorentz 
transformation, or the Euclidean metric (the kronecker delta)

        return transformation_result;
}

So, in maths, if we write a tensor e.g. Ta^b
then, equating the code to the math, we have:  T = 'tensor_transformation' , 
a = 'A'  , b = 'B'
so, in math, when we write down a tensor, we are operating on the level of 
abstraction outside of the function - we are saying nothing about the 
specific detail when we write down a tensor.

The result of the function is/are real number(s). In the case of the Lorentz 
transformation, the function give the relative 'difference' between the 2 
frames - I use the word 'difference' in a very liberal way here - it does 
not necessarily imply subtraction. (Anybody?)

So, for Lorentz the code would be something like:

function lorentz(A, B) real, real, real, real;     - not sure this would 
compile! Anybody know the best way around this?
{
    ' tensor transformation detail for La^b  - L is the lorentz tensor, 
usually written as 'gamma'

    return dx, dy, dz, dt;
}

In the case of the Lorentz transformation, the function returns the real 
numbers dx, dy, dz, dt

Note : in maths there is also another level of abstraction going on : in the 
previous (lorentz) tensor, we had La^b
but a = {x0,x1,x2,x3) and b={x'0, x'1, x'2, x'3)  , so a is actually 4 
numbers and so is b !!!

The Lorentz transformation La^b is the 'relative difference' between the 2 
frames i.e. real number(s)


I think this is all about right... - can anybody spot flaws I have missed?

Can anybody fill be in on the code for the Lorentz transformation detail?? - 
it should be a straight-forward matrix transformation

.