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Topic: Tensor Definition
Replies: 16   Last Post: Jul 16, 2013 8:48 PM

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J.B. Wood

Posts: 45
Registered: 8/29/06
Re: Tensor Definition
Posted: Jul 9, 2013 6:53 AM
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On 07/08/2013 07:48 PM, Mike Trainor wrote:
> The M-W definition is pretty meanigless, IMHO. I think it was put
> there by someone who does not understand it.
>
> While there are many ways that people may look at tensors,
> I am partial to the one that defines vectors (which are really
> rank 1 tensors!) and tensors in terms of their transformation
> properties.
>
> Another fallacy that I have come acorss is "a tensor is nothing
> but a matix".
>
> mt
>

Thanks, Mike. I think most of the stuff in the general-use dictionaries
from well-established publishers (M-W, World and Oxford)has been
well-vetted. Perhaps there's goof or two sometimes. Here's one from
the dict.org line dictionary:

"The ratio of one vector to another in length, no
regard being had to the direction of the two vectors; --
so called because considered as a stretching factor in
changing one vector into another. See Versor.
[1913 Webster]"

That definition seems to imply that we're dealing exclusively with
dyadics (tensors of valence (rank) 2). Historically dyadics were the
first type of tensors other than scalars and vectors formulated to solve
problems of mechanical stress.

I think the most sensible and accurate one comes from Webster's New
World Dictionary:

"An abstract object representing a generalization of the vector concept
and having a specified system of components the undergo certain types of
transformation under changes of the coordinate system."

With regard to a matric representation of tensors, that's OK if you
restrict yourself to 3-D space and scalars, vectors and dyadics. That
suffices in a lot, but not all, applied physics (e.g. electromagnetics)
and engineering applications. So one can think of a matrix (especially
if used as a multiplier) as a tensor subject to these restrictions and
noting that some tensors are not (at least easily) represented as
matrices. Einstein's tensor notation is arguably better since it is
compact and works for any number of spatial dimensions. The other
fallacy is to assume all vectors are valence 1 tensors which clearly
does not apply to axial vectors (e.g. the cross product). It's those
darn transformation rules. Sincerely,

--
J. B. Wood e-mail: arl_123234@hotmail.com



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