Getting an Intuition of Tensors

[ashwinnarayan]

Homework Statement

This is not really a homework problem. I'm just trying to learn about tensors by myself. I'm new here (This is only my second post). From what I could gather from the forum rules this seems to be the place for my question.

I went through literally hundreds of websites for a simple explanation. The thing is I'm a very visual learner. If I can't visualize a concept I'm trying to learn I never understand it. If I do have a general picture however, I find I can understand new concepts in minutes.

So after my research I got this far: "A tensor is a multidimensional array of numbers."

Homework Equations

I already know that n x n matrices can be tensors.
My question is, if I take a Rubik's Cube and fill every cell with a number, do I get a tensor?
And does the analogy scale to higher dimensions?

I have a good understanding of basic linear algebra. I know about subspaces, linear transformations, eigenvalues and eigenvectors.

 

[HallsofIvy]

Homework Statement

This is not really a homework problem. I'm just trying to learn about tensors by myself. I'm new here (This is only my second post). From what I could gather from the forum rules this seems to be the place for my question.

I went through literally hundreds of websites for a simple explanation. The thing is I'm a very visual learner. If I can't visualize a concept I'm trying to learn I never understand it. If I do have a general picture however, I find I can understand new concepts in minutes.

So after my research I got this far: "A tensor is a multidimensional array of numbers."

Ouch! That's absolutely wrong. That's as wrong as to say a vector is a one-dimensional array of numbers. In a given coordinate system, a vector can be represented by a list of numbers but the important thing is how those numbers change if we change the coordinate system. I could, for example, write wind velocity, air pressure, and air temperature at a given point as a list of 5 numbers- the three components of the wind velocity, the air pressure and temperature. But if I change to a different coordinate systerm, only the components of the wind speed would change- they form a vector but the other numbers are not part of a vector.

Vectors and tensors are objects in some space that are independent of the coordinate system we happen to use. How we happen to represent them in a given coordinate system- the numbers we use, whether a single number (0 order tensor), a linear array of numbers (first order tensor- a vector), or a "multi-dimensioal array" (a general tensor) depend upon the coordinate system.

What we need is that the coordinates change "homogeneously". That is, the components of a tensor in one coordinate system are sums of given numbers (relating the two coordinate systems) times the components of the tensor in the other coordinate system. The crucial result of that is "If a tensor has all components equal to 0 in one coordinate system, it has all components equal to 0 in any coordinate system". That's because the new components will be a sum of numbers all multiplied by 0.

That may not seem like much but suppose we have an equation relating tensors that is true in one coordinate system: generally we can write it A= B where A and B are tensors. That is the same as saying that the tensor A-B= 0 in that coordinate system. It follows, then, that A- B= 0 in all coordinate systems or that A= B in all coordinate systems. If an equation in tensors is true in one coordinate system, it is true in all coodinate systems.

That, of course, is exactly what we want for physics! Coordinate systems are not "physical", we can impose any coordinate system we want on a physical situation. But any equation that is supposed to represent a "physical" situation must be true in any coordinate system. And we can guarantee that by writing our equations in terms of tensors.

Homework Equations

I already know that n x n matrices can be tensors.

n by n matrices represent tensors in the same sense that <a, b, c> can represent a vector. But don't confuse the representation of an object with the object.

My question is, if I take a Rubik's Cube and fill every cell with a number, do I get a tensor?
And does the analogy scale to higher dimensions?

No, it doesn't- although it might represent some tensor in a particular coordinate system.

I have a good understanding of basic linear algebra. I know about subspaces, linear transformations, eigenvalues and eigenvectors.

Be careful- "tensors" as used in linear algebra- "multi-linear" structures- are generalizations of the tensors used in physics.

 

[tiny-tim]

welcome to pf!

hi ashwinnarayan! welcome to pf!

a tensor is basically a linear rule for replacing one "input" vector by a different "output" vector

for example, the moment of inertia tensor converts the angular velocity vector to the angular momentum vector (usually in a different direction)

… if I take a Rubik's Cube and fill every cell with a number, do I get a tensor?

you (obviously) get a matrix

if you are told that it is the representation of a particular tensor in a particular basis (frame), then you can find the matrix which represents the same tensor in any other basis