On Monday, November 10, 2025 at 1:15:42 AM UTC-7 Russell Standish wrote:
On Sun, Nov 09, 2025 at 10:34:38PM -0800, Alan Grayson wrote: > > > On Sunday, November 9, 2025 at 11:16:05 PM UTC-7 Russell Standish wrote: > > On Sun, Nov 09, 2025 at 09:55:15PM -0800, Alan Grayson wrote: > > > > > > Someday I might find a teacher who can really define tensors, but that > day has > > yet to arrive. Standish seems to come close, but does every linear > multivariate > > function define a tensor? I'm waiting to see his reply. AG > > Well I did say multilinear function, but the answer is yes, every > multilinear function on a vector space is a tensor, and vice-versa. > > > How does one prove that every multilinear function on a vector space is > invariant under a > change in coordinates? What exactly happens to its matrix representation? And A vector is a geometric quantity having direction and length. As such it is independent of any coordinate system that might be applied to the space (although the list of numbers representing the components of the vector in a given coordinate system must vary covariantly with the coordinate system varying). A function operating on vectors, and returning vectors or scalars must therfore also be independent of the coordinate system. *I think you deserve a special "THANK YOU!" Now, after many years, I finally* *understand why tensors are invariant under changes in coordinate systems.* *The problem has always been the reluctance or inability for most physicists* *to explicity define the MATHEMATICAL definition of a tensor. They usually omit this* *necessary definition by just asserting they're mathematical entities which* *are invariant under changes in coordinate systems. The explanation is totally* *simple; namely, tensors are defined on vectors in vector spaces, and these* *vectors are invariant under changes in coordinate systems, so the function* *defining tensors must likewise have this property since these vectors are* *the domains on which these functions operate. NOT COMPLICATED! I think* *you nailed it because your core orientation is more in mathematics than physics. * *Finally, please post your summary of GR when you have time. I am anxious to* *read it. **AG* > I did write an 8 page article appearing in our student rag "The > Occasional Quark" when I was a physics student, which was my attempt > at explaining General Relativity when I was disgusted by the hash job > done by our professor. I haven't really thought about it much since > that time, though. I can also recommend the heavy tome by Misner, > Thorne and Wheeler. > > I could scan the article and post it to this list, but not today - I > have a few other things on my plate before finishing up. >--------------------------------------------------------------------------- > > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders [email protected] > http://www.hpcoders.com.au > ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/935f6240-93df-4eaa-8a8a-4782dd38b80fn%40googlegroups.com.
