Insights What Is a Tensor? The mathematical point of view

[FactChecker]

This cut from Wikipedia shows a motive of using tensors:

"Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of basis. The basis independence of a tensor then takes the form of a https://www.physicsforums.com/x-dictionary:r:'Covariant_transformation?lang=en&signature=com.apple.DictionaryApp.Wikipedia' that relates the array computed in one basis to that computed in another one. "

I believe this might be one of the most important characteristics of tensors for differential geometry and general relativity. (both essentially over my head)

Thanks for taking the time and effort to write this article.

I agree that this is the key feature of a tensor. It is an entity that is defined in such a way that its representations in different coordinate systems satisfy the covariant/contravariant transformation rules. Mathematically, a tensor can be considered an equivalence class of coordinate system representations that satisfy the covariant/contravariant transformation rules. This gives tensors the great advantage of being coordinate system agnostic.

 

[fresh_42]

This gives tensors the great advantage of being coordinate system agnostic.

Yes, but this is similar difficult as to why vectors and linear transformations, although usually represented by an array of numbers or a matrix, are not those arrays but rather entities on their own, as you said agnostic to coordinates. A tensor is in the end merely a continuation of scalar $\rightarrow$ vector $\rightarrow$ matrix to simply higher dimensions. They often appear as if they were something special, if we speak of stress-energy tensors or curvature tensors. This is as if we associated automatically a force with a vector, or a rotation with a matrix. I often get the impression that physicists use the term tensor but mean a certain example. It's just a multilinear array or transformation - any multilinear transformation, which are already two interpretations of the same thing. The entire co-contra-variant stuff is also an interpretation, and - in my mind - sometimes a bit deliberate.

 

[Thuring]

Sooo, are you hinting that perhaps the biggest advantage to using "tensors" is the notation?

 

[Thuring]

Perhaps the biggest difference between vectors, matrices, and linear algebra with "tensors" is the attitude or conception of the users. My simple minded pragmatic definition of a tensor essentially is a matrix using the tensor notation. Matrices are easily visualized, they have a shape and size. Tensors notation considers only one
coefficient at a time, but a lot of them.

(Butt then, eye am knot a reel physicist)

 

[fresh_42]

Sooo, are you hinting that perhaps the biggest advantage to using "tensors" is the notation?

To some extend, yes. Tensors on the other hand are quite variable, same as matrices are. E.g. scalars, vectors and matrices are also tensors. And they build a tensor algebra with a universal property, i.e. many algebras can be realized as quotient algebras of the tensor algebra. So it is the same as with vectors: it all depends on what we use it for. In the end they are simply an arrow, an arrow that serves many, many applications.

 

[FactChecker]

Yes, but this is similar difficult as to why vectors and linear transformations, although usually represented by an array of numbers or a matrix, are not those arrays but rather entities on their own, as you said agnostic to coordinates.

Yes, if they are defined in a way that is agnostic to coordinate systems. It is possible to define tuples that can not have any physical or geometric meaning. I can define the tuple (1,0) in all coordinate systems, but it does not transform at all -- it is (1,0) in any coordinate system. It is not a tensor. Tensors can have a physical or geometric meaning that is independent of the choice of coordinate system. The tuple (1,0) defined that way regardless of coordinate system can not have a physical or geometric meaning. There are similar examples for matrices and they can be found throughout mathematics.

 

[fresh_42]

Of course, because you defined $(0,1)$ as a tuple plus the absence of meaning and then reasoned that it has no meaning. That's a tautology. $(0,1)$ has a meaning, as soon as it is associated with a point in a coordinate system, namely the vector from the origin to this point, even over $\mathbb{Z}_2$. It transforms, at the very least by permutation of the axis to $(1,0)$ and is of course a tensor, as all vectors are. Examples of matrices which aren't a representative of a linear transformation in some coordinate system, need to have a meaning attached to them, which excludes such a transformation. Maybe a matrix of pixels in an image. But even then each entry has a RGB coordinate and is again in some sense a tensor. Whether this tensor as a multilinear object makes any sense is another question.

 

[FactChecker]

Of course, because you defined $(0,1)$ as a tuple plus the absence of meaning and then reasoned that it has no meaning.

It has no physical or geometric meaning, but can have mathematical meaning and properties. Those concepts are common in mathematics.