On Sunday, November 9, 2025 at 8:01:50 PM UTC-7 Brent Meeker wrote:
On 11/9/2025 5:24 PM, Alan Grayson wrote: On Sunday, November 9, 2025 at 6:16:15 PM UTC-7 Alan Grayson wrote: On Sunday, November 9, 2025 at 5:12:54 PM UTC-7 Russell Standish wrote: On Sun, Nov 09, 2025 at 03:56:16PM -0800, Alan Grayson wrote: > > If it's a map, how can an ordinary vector in Euclidean space be a tensor? > Such vectors are NOT maps! See my problem? AG I did explain that in my post if you read it. In an inner product space, every vector is isomorphic to a linear map from the space to its field. Eg R^n->R in the case of the space R^n. That linear map is the rank 1 tensor. In mathematics, something walks and quacks like a duck is a duck. Even the inner product operation is an example of a bilinear map, hence a rank 2 tensor. In Minkowski spacetime, the inner product is known as the Levi-Civita tensor. So a tensor is nothing more than a multi linear map to the reals? But if we represent a tensor by a matrix, will it be automatically invariant under coordinate transformations? Do we need an inner product space to define a tensor? TY, AG If the tensor, represented by a matrix, is "unchanged" under a coordinate transformation, does this mean its determinant is unchanged? AG No, in general it transforms like a density. So it's only unchanged if the determinant of the transformation matrix is 1 Brent 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 -- ---------------------------------------------------------------------------- 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/ca01bcc3-ef25-43cc-9e37-dde88e8d97ecn%40googlegroups.com <https://groups.google.com/d/msgid/everything-list/ca01bcc3-ef25-43cc-9e37-dde88e8d97ecn%40googlegroups.com?utm_medium=email&utm_source=footer> . -- 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/8e16e934-4e9a-47eb-a042-67c1516c6c09n%40googlegroups.com.
