Tensor Calculus

Date: 10/10/2002 at 07:55:07
From: Lachieze-Rey
Subject: Tensor calculus

Could you please remind me of the general definition of a determinant 
for a tensor with more than 2 indices, like T_m1_m2..._mk ?

Thank you.

Date: 11/30/2002 at 01:48:17
From: Doctor Nitrogen
Subject: Re: Tensor calculus

Hi, Lachieze-Rey:

Depending on the rank of the tensor and the number of dimensions 
involved, calculating the determinant for a general tensor can get 
pretty cumbersome indeed, especially for a rank greater than two and 
for dimensions exceeding four.

To illustrate, let

   T^a1a2a3.....ar_b1b2.....bs,


be a general (non Cartesian) tensor of rank r + s, contravariant in

a1, a2,......ar, covariant in b1, b2, ......,bs. Suppose the rank is 
r + s = 7, so that the tensor is actually, for one instance:

   T^a1a2a3a4_b1b2b3.

Suppose you choose a chart on a four-dimensional manifold (dimension 
4). This tensor will have a matrix form after you expand all the terms 
out. After you expand all the terms out to get the matrix form:

 T^a1a2a3a4_b1b2b3

 = (&x^a1/&u^p)(&x^a2/&u^q)(&x^a3/&u^r)(&x^a4/&u^s)

   X (times) (&x_t/&u^b1)(&x_w/&u^b2)(&x_v/&u^b3)T^pqrs_twv.

Each one of those partial derivative terms above (in the parenthesis 
with the "&" symbols) would have to be expanded out as a matrix. If 
you are on a four-dimensional manifold, they would be 4x4 matrices, 
each having 16 elements. Then you would have to express EACH of these 
matrices as a 4x4 determinant, compute its value, and multiply all the 
values together, not before computing all the partial derivative 
terms. Do you begin to see the tedium involved? 

In General Relativity Theory, the rank four tensors expanded out, 
such as the famous Riemann-Christoffel Covariant Curvature Tensor, 
have a lot of terms, so if you intend to expand out a determinant for 
a tensor with more than two indices and with a coordinate system with 
dimension greater than three, you had better be prepared to do a lot 
of careful calculating.
                       
I hope this helped answer the questions you had concerning your 
mathematics problem.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/