what the fuck is a tensor. i have no fucking clue. I work with tensors daily. I have solved literally hundreds of tensor problems. yet I cannot for the live of me figure what the fuck one is. literally every person I ask gives me a different answer. what the fuck is a tensor
pic unrelated
only video I understood
youtube.com
Personally I'm convinced that tensors are a spook. You're better off just accepting what you know about them and moving forward with your head down.
>spends half the video explaining what basis vectors are
>Last two minutes:
>Here's a rank 3 tensor. It's basically just a logical extension of a rank 2 tensor.
You understood something from this?
Tensors bend dimensions nigga
just a higher dimensional matrix
If you properly learn linear algebra. It's just a multilinear map, or more precisly, the representation of that multilinear map as a higher dimensional array in the same wat a matrix is a representation of a linear function given a basis. More care needs to be taken to construct that array, but the general concept is not that hard. Physicists just teach it in the most backwards and convoluted way.
t. Mad physics student.
OP, have you taken a proper linear algebra course?
a tensor is an object that satisfies the tensor transformation law
Shit definition.
You don't like THE ONLY definition of a tensor? You definitely won't like the definition of a vector then.
Nigger, the definition of a vector is independent of it's transformation laws and its representation as a basis. Have you ever read a linear algebra books above memegineer level? The same goes with tensors.
If you have another definition, buddy, state it.
>hair magically covering nipples
the future sucks
Look tensors as multilinear maps.
...
Nice comback faggot.
That's not a definition. It's not even a complete sentence. Are you implying all multiliear maps are tensors? If so, that is fucking wrong as fuck. If not, what separates the general case of a multi-linear map, buddy, from the tensor case?
HINT: The definition of a tensor is what separates the general case from the tensor case.
>Are you implying all multiliear maps are tensors?
not him, but this is true
The transformation laws come when you try to make a change of basis for your representation of the multilinear function. When someone tells you "not all matrices are tensors" is just saying that a matrix can represent different objects such as a linear transformation or a bilineal form which are different in how you defined them abstractly given your vector space.
rather than admit I was wrong I will cite the thing I just found about that said you can have non-tensorial multilinear maps in infinite dimensional vectors spaces, and it is only true that they are always tensors in finite dimensional vector spaces.
Well in infinite dimensional VS, not all linear maps are continuous, so that's a different story.
ok, well I am glad to know that all multilinear maps are not tensors.
Has anyone fucking ever?
I took the intro one but then I graduate with a physics degree and went to a graduate school where they wanted you t immediately become a device technician without taking courses for breadth. I would really like to go back and get at least a BS in mathematics. There is a lot I don't know, and a lot of jargon I don't know too.
Me
Just read friedberg or linear algebra done right.
I'd like to "just go to college" but unfortunately that is too expensive for me in my destitution
So a scaler is a tensor of rank 0 because it just has a magnitude and no directional index. A vector is a tensor of rank 1 because it will give you a magnitude and a direction. Tensors of rank 2 will have magnitude and two directional indexes.
You keep building on from there to higher and higher rank. Use matrices to define them as a nice way of organizing.
I can understand that a vector can have 1 direction.
But how a Rank 2 Tensor can have two directions?
Can you give me there any real life example? Rotation would be one kind of direction?
Ok, you can still read those books on your own time. Learning math is much easier than physics and you can do it as an autists in your cave.
I wish I had a cave, I only have a hovel.
Tensors are like matrices but better
No
>what the fuck is a tensor.
This question doesn't have a single good answer, because there isn't a universally agreed upon definition of "tensor" in mathematics. In particular:
Tensors are sometimes defined as multidimensional arrays, in the same way that a matrix is a two-dimensional array. From this point of view, a matrix is certainly a special case of a tensor.
In differential geometry and physics, "tensor" refers to a certain kind of object that can be described at a point on a manifold (though the word "tensor" is often used to refer to a tensor field, in which one tensor is chosen for every point). From this point of view, a matrix can be used to describe a rank-two tensor in local coordinates, but a rank-two tensor is not itself a matrix.
In linear algebra, "tensor" sometimes refers to an element of a tensor product, and sometimes refers to a certain kind of multilinear map. Again, neither of these is a generalization of "matrix", though you can get a matrix from a rank-two tensor if you choose a basis for your vector space.
You run into the same problem if you ask a question like "Is a vector just a tuple of numbers?" Sometimes a vector is defined as a tuple of numbers, in which case the answer is yes. However, in differential geometry and physics, the word "vector" refers to an element of the tangent space to a manifold, while in linear algebra, a "vector" may be any element of a vector space.
On a basic level, the statement "a vector is a rank 1 tensor, and a matrix is a rank 2 tensor" is roughly correct. This is certainly the simplest way of thinking about tensors, and is reflected in the Einstein notation. However, it is important to appreciate the subtleties of this identification, and to realize that "tensor" often means something slightly different and more abstract than a multidimensional array.
The main gestalt.
It's a multilinear function (i.e. linear in each variable) that takes vectors and covectors (also called functionals in basic linear algebra texts) and outputs elements of the associated field (like R or C).
>Are you implying all multiliear maps are tensors?
They are. Follows directly from the definition of the tensor product.
>because there isn't a universally agreed upon definition of "tensor" in mathematics
Yes there is.
>an element of a tensor product, and sometimes refers to a certain kind of multilinear map
Multilinear maps on MxN and elements of M⊗N are in bijection by definition of the tensor product. There is no distinction.
>"tensor" refers to a certain kind of object that can be described at a point on a manifold
Just people being lazy. Anyone talking about tensor fields know they are not actually tensors. Tensor fields are defined on vector bundles, while tensors are defined on vector spaces. A tensor field is fiberwise a tensor.
