Bear in mind I have really thought about this stuff since my early 20s :). Most of the areas of physics where this is used would be a metric topological space, which can have a coordinate system (but not necessarily uniquely defined).
I don't get the requirement that points can be labelled. IIUC, this cannot be done in any continuous space anyway - there are uncountably infinite more points in a continuous space than there are possible labels. Cheers On Wed, Nov 12, 2025 at 05:28:55AM -0800, Alan Grayson wrote: > > > On Tuesday, November 11, 2025 at 6:37:20 PM UTC-7 Russell Standish wrote: > > As promised below, my student article on differentiable manifolds can > now be found at > https://www.hpcoders.com.au/docs/differentiableManifolds.pdf > > Apologies for the difficult to read font - this was done on Wordstar > and a dot matrix printer, before laser printers (and LaTeX) became a > thing. > > Cheers > > > Thank you. It's readable by enlarging the font. Tell me if you agree with this > statement; a manifold is a topological space on which a coordinate system > is defined, but to actually define a coordinate system one needs additional > information, namely, that the topological space is also a metric space. But > the > problem is that a metric space requires points to have labels, and I don't see > how points can have labels unless there's a pre-existing coordinate system. > IOW, there's a circularity here that I want to avoid, but I'm not sure how to > do > so. AG > > > On Mon, Nov 10, 2025 at 05:15:53PM +1100, 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. > > > > 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 > /0ac93a9d-2e3b-4713-b0f9-5923cfb811acn%40googlegroups.com. -- ---------------------------------------------------------------------------- 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/aRUO6MMgQPy8ha-X%40zen.
