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

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 Tanu R. Posts: 640 Registered: 12/13/04
Re: Tensor Definition
Posted: Jul 8, 2013 2:02 PM

J.B. Wood wrote:
> On 07/08/2013 11:03 AM, Sam Sung wrote:
>> J.B. Wood write:
>>

>>> Hello, all. Just when I think I've got a good handle on tensors (after
>>> painstakingly reading and working problems in Louis Brand's "Vector and
>>> Tensor Analysis"), I come across the following in Merriam-Webster's
>>> Collegiate Dictionary:
>>>
>>> "A generalized vector with more than three components each of which is a
>>> function of an arbitrary point in space of an appropriate number of
>>> dimensions"
>>>
>>> That definition seems to exclude well-known dyadics (stress tensor,
>>> permeability tensor, etc) having 9-components that are constants (for
>>> the material involved.) That aside, I'm still having a problem grasping
>>> the foregoing definition. Thanks for your time and any comment. Sincerely,

>>
>> You might read this: http://en.wikipedia.org/wiki/Tensor
>>

> Hello, and thanks for responding. I read the Wiki entry prior to my OP.
> I don't see the correlation to the Webster definition.

I read this as an example for the use of tensors.

> Given the
> usual 3-D orthonormal coordinate systems commonly used in physics and
> engineering (rectangular, cylindrical, spherical) I can view unit dyads
> (having "two directions" just as easily as unit vectors having but one
> direction (x, y, or z) and a dyadic having a total of 9 components vice
> 3. Still, that dictionary definition confounds me. Sincerely,

Tensors are/canbe used by more than this kind of stuff; see
(the first sentence of) http://www.wolframalpha.com/input/?i=tensor

Be sure to also know manifolds and dual objects, understand
covectors, Grassmann products, vector fields and their covector
fields, n-forms, volume elements and their integrals, understand
that there is math without reference to any coordinate system,
remember that a tensor Q_a...c^f...h with p lower and q upper
indices for any p,q >= 0 is a multilinear function (or mapping)
of p vectors A,...,C and q covectors F,...,H where
Q(A,...,C; F,...,H) = A^a...C^c Q_a...c^f...h F_f...H_h.