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 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/9ba03208-6661-4475-a243-206894c1c23dn%40googlegroups.com.
