Tensors

[Anonymous No.12074350]

What are tensors? I still don't understand them. Can someone here explain it to me?

[Anonymous No.12074374]

array of numbers

[Anonymous No.12074387]

>>12074350
It an overly convoluted and complex tool used to solve ideas, i.e. to fit abstract ideas into mathematical framework to prove that the model works but you cannot confirm it by observations because muh infinity. Do not bother yourself with this. It is not useful to solve real life problems.

[Anonymous No.12074408]

https://www.youtube.com/watch?v=bpG3gqDM80w

[Anonymous No.12074411]

>>12074374
>>12074387

t. Brainlets

[Anonymous No.12074415]

>>12074387
Theory of Relativity BTFO'd

[Anonymous No.12074428]

>>12074374
It's an array of arrays ya dingdong.

[Anonymous No.12074432]

>>12074408
Thanks for the video, anon.

[Anonymous No.12074513]

>>12074350
If you're a brainlet tier physicist:
>a tensor is something that transforms like a tensor, hyukhyukhyuk
If you're a regular non retarded person:
>it's just a multilinear map of vector/dual vectors into the real numbers

[Anonymous No.12074524]

>>12074350
It depends on the field. But, in psychology, a "tensor" is defined as "any such socio-psychological interaction as can be said to have made one feel, directly, or indirectly, the experience of emotional discomfort (i.e. tension)."
And, in material sciences, [1] "the physical mechanism, or specific axial load, by which tensile stress is exerted on a given material or component" [2] "the process upon which an observed change in a material's microstructure is contingent"

[Anonymous No.12074530]

I thought tensors and matrices were interchangeable

[Anonymous No.12074532]

>>12074350
A system of systems

[Anonymous No.12074533]

>>12074524
Sorry anon, I should have clarified that I meant physics/math.

[Anonymous No.12074554]

>>12074533
Uh, yeah?
>in material sciences, [1] "the physical mechanism, or specific axial load, by which tensile stress is exerted on a given material or component" [2] "the process upon which an observed change in a material's microstructure is contingent"

[Anonymous No.12074560]

[Anonymous No.12074709]

>>12074530
Matrices are a system,

A tensor is a system of systems

[Anonymous No.12074719]

>>12074554
Please don't copy and paste wikipedia here if you don't actually know what you're talking about

[Anonymous No.12074783]

>>12074350
A tensor is an abstract geometric mathematical object. In a given coordinate system, you can write it down in a matrix representation, but this matrix representation is different based on what coordinates you're using. The tensor is the abstract geometric object, which is independent of coordinates. Once it's in a matrix rep, it's just a regular ol linear transformation matrix that acts on vectors to produce new vectors. The stress tensor, for example, basically takes in a surface normal, and spits out the force acts on a surface with that normal.

[Anonymous No.12075518]

>>12074350
There are some 3 definitions I know of. A tensor is a geometrical object, with it's coordinates having a predictable transformation law, based entirely on a certain contraction like this-[math]A_p^n T^{pq}=T^{nq}[/math]. There are geometric objects that don't transform like tensors and it is worth knowing some of these. This mode of thought tends to lay into the hands of playing with indices and contractions and what not. In turn, with p indices at bottom and q on top we have a (p,q) rank tensor. It isn't simply a matrix as the coordinate transformations for the 'row parts' and 'column' parts are different, making it a (1,1) rank tensor. It should be noted the variety of manners coordinate transformations may change objects-the Levi-Civita pseudotensor doesn't transform like a tensor but adds a sign change. The Christoffel symbols are geometric objects but are not like tensors. having additional components then the 'tensor transformation'-this is critical in getting non-zero christoffel symbols from simple coordinate changes, like cartesian to polar.

[Anonymous No.12075563]

>>12075518
An alternative definition is a multilinear map on products of q vector spaces V with field F and p duals of that space and this makes it a (p,q) tensor. Suppose we have a given (0,2) tensor, g. Then, g(u,v) is a number in the field. We can actually turn this into a (1,1) tensor since each fixed u=c defines a function g(c,v)->F, a dual vector in the dual space. In this manner, the vectors of u act as a function on the v and in this type of prescription, it is not a (0,2) tensor anymore but a (1,1) tensor. This duality between the vector and dual vectors is part of the intuitive meaning of this definition.

The third definition is a lot more abstract-it uses the tensor product. In essence, this constructs a kind of 'universal' object in the following sense: if we have a multilinear map V x W into a given space Z, that there is a map from the tensor product [math]V \otimes W[/math] to Z, in a manner that given the map from the tensor product to V x W, have factored it. In this sense, the tensor product represents the most universal object in these types of objects under multilinear maps. A tensor is then the tensor product of p vector spaces V and q dual spaces of that.

[Anonymous No.12075717]

>>12074350
It's a type of bandage

[Anonymous No.12076224]

>>12074350
Do you want me to define it in C#?
It's just an array of numbers, anon. Don't worry about it too much.

[Anonymous No.12076409]

numbers

[Anonymous No.12076467]

>>12074428
an array of arrays is just an array of numbers

[Anonymous No.12076588]

>>12074350

[Anonymous No.12076797]

tensors are GL(n)-equivariant (in a suitable sense) mappings from the set of frames into (suitable cartesian power of) the set of coordinates

[Anonymous No.12076905]

>>12076797
Oh, that makes sense.

[Anonymous No.12076911]

>>12076905

[Anonymous No.12076913]

>>12076905
it's literally the "tensor is something which transforms like tensor" definition but precise