|
|
|
|
|
|
by BaryonBundle
4231 days ago
|
|
|
There is also a similarly titled, much longer (92 pages vs. 29 pages) introduction to tensors by the same author, also hosted by NASA, and released ~3 years later: Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/2005017... The pdf.js in my Iceweasel 31.2 seems to be choking on the pdf document, though it looks fine in other readers. |
|
|
Interestingly, in this longer document the author gives (more or less) an actual definition of "dyad" which isn't the same as "tensor product of two vectors", and with that definition the statement that "every tensor is a dyad" is (more or less) correct.
I need to qualify everything here with "more or less" because the author is trying to do mathematics but fairly clearly isn't actually a mathematician, and so a lot of what he says doesn't really quite make sense, but one can see what he's trying to say. (E.g., he says "a dyad is any quantity that operates on a vector through the inner product to produce a new vector with a different magnitude and direction from the original". Except that actually I'm sure he wants ii+jj+kk to count as a dyad even though it doesn't change either magnitude or direction, and except that he hasn't actually said explicitly what the "inner product" is except for the special case where the dyad is a tensor product of two vectors, and in order to say what the inner product is for a vector and a dyad one already has to know (at least kinda) what a dyad is...)
On the other hand, if you ask a pure mathematician to explain this stuff you'll generally get something nice and elegant but incomprehensible to most physicists and engineers, and much less suitable for symbolic calculations than the physicists' coordinate-based approach. (A tensor field is a section of a product of the tensor product of some tensor power of the tangent bundle of your underlying manifold, and some tensor power of the cotangent bundle. What do you mean you don't know what a section of a tensor power of the cotangent bundle of a smooth Riemannian manifold means? Bozhe moi!)