On Monday, November 10, 2025 at 1:07:13 PM UTC-7 Alan Grayson wrote:
On Monday, November 10, 2025 at 11:58:41 AM UTC-7 Brent Meeker wrote: On 11/9/2025 9:55 PM, Alan Grayson wrote: 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 Why don't you read a book? Russell gave you a definition. I gave you an example. Nobody wants to write a lot of math text online. Brent Why don't you read my comments before replying? I accept Russell's definiiton. Moreover, a tensor can be an inner product, and maps to a real numbers, so a tensor field is not like many arrows but real numbers. So your example is misleading. Like most teachers of tensors, you are averse to giving a precise definition, which Russell did. AG You write "Nobody wants to write a lot of math text online." Do you know that the proof that tensors are invariant to changes in coordinate systems does NOT involve writing any mathematics? It follows directly from the definition of tensors. AG -- 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/442b37b5-dac0-4b7b-ba3f-db5d65aeb324n%40googlegroups.com.
