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Re: Tensor Definition
Posted:
Jul 9, 2013 6:53 AM


On 07/08/2013 07:48 PM, Mike Trainor wrote: > The MW 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 generaluse dictionaries from wellestablished publishers (MW, World and Oxford)has been wellvetted. 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 3D 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 email: arl_123234@hotmail.com



