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,
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.