Re: Tensors explained (was Re: Questions (Space))

In article <fcjfsf$ajg$1$8302bc10@xxxxxxxxxxxxxxxx>, Tim S
<Tim@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

Keith F. Lynch wrote:



Tensors are sometimes called "tensors of the second rank." In that
terminology, scalars (single real numbers, which represent undirected
quantities such as mass, temperature, or voltage) are called "tensors
of zero rank," and vectors (which represent directed quantities such
as distances, velocities, currents, forces, and torques) are called
"tensors of the first rank." There are also tensors of the third,
fourth, etc., ranks. Fortunately, there's not much call for them.

The Riemann curvature is a rank four tensor.

Apparently, the word 'rank' is ambiguous, and can be applied either as
the number of indices (as in Keith Lynch's usage) or as the number of
values each index can take (as in Tim's usage).

According to wikipedia, Tim's usage is currently in vogue:
http://en.wikipedia.org/wiki/Rank_of_a_tensor
It is now preferred, in order to avoid ambiguities, to use the
terminology of "tensor order" to denote the number of indices, and
"tensor rank" to designate the number of simple tensors necessary to
decompose a tensor. Hence the definition of rank is consistent with
Linear Algebra.

So, the Riemann curvature is a tensor that can be written as an 4x4
array of numbers. Keith would call it a rank 2 tensor, and Tim would
call it a rank 4 tensor. Wiki would refer to it as a rank 4 tensor of
order 2.

I don't know if using ordinals ('of the fourth rank' instead of 'rank
4') is commonly used for disambiguation. ('The Riemann curvature is a
second rank tensor of rank 4' is unlikely to provide clarity.)

Tension, apprehension and dissension have begun.

--
David M. Palmer dmpalmer@xxxxxxxxx (formerly @clark.net, @ematic.com)
.

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