What Are Tensors and How Can They Be Easily Understood by High School Students?

[Amrutha.phy]

what is meant by a tensor?

 

[Drakkith]

Have you read this article?
http://en.wikipedia.org/wiki/Tensor

 

[Mark M]

That's a very general question, and you're going to need to look into some resources if you want to learn tensor algebra/calculus.

Essentially, a tensor is a function that maps vectors and one-forms to the real numbers (or a new vector). A tensor that is a function of m one-forms and n vectors is called a tensor of rank [m, n]. An example of a tensor of rank one is a vector - it is a function of a basis vector. For a vector A, we can write it as a tensor as A^ae_a. The e refers to a basis vector, for which we define the components of the vector. When you see an upper index match the lower index of the next term, a sum is implied. If we write out this sum, we'll see that the above expression is the normal expression of a vector through it's components. A tensor of higher rank has more than one basis vector for each dimension ( a vector, a rank one tensor, needs only one basis vector for each orthogonal axis).

Like I said, it's a thorough subject, and you'll need to find a good text to get familiar with it.

 

[Muphrid]

There's an easy way to understand tensors and some of the differences between them.

There are tensors that represent generalizations of vectors. If a vector represents a directed line, some tensors represent directed planes (i.e. where if you turn the plane upside-down, the result is negated), directed volumes, and so on. Tensors give you a way of talking about the algebra of these objects in a way similar to the algebra of ordinary vectors.

There are also tensors that represent linear operators--for instance, a rotation matrix transforms a coordinate system from one basis to another and would be considered a tensor. As linear operators can act on vectors, they can also act on tensors of the first type that I described above.

Unfortunately, both these kinds of objects are called tensors, which I think is very misleading. Linear operators are not the same things as higher-dimensional geometric objects, but the term tensor has stuck and can generally mean either kind of object. I think keeping them separate in your mind will make the geometric picture clearer, however.

 

[Khashishi]

It might help to have a less rigorous, less general, but more intuitive definition. A rank 0 tensor is a scalar. A rank 1 tensor is a vector. A rank 2 tensor is essentially a vector of vectors, and can be represented as a matrix. For example, energy is a scalar value, a rank 0 tensor. Momentum is a rank 1 tensor. Flux density is defined as the amount of something that is flowing along some direction. So the flux density has a rank one higher than the stuff that is carried. Energy flux density has rank 1, because it has a direction and magnitude. But momentum flux density has rank 2 because momentum already has a direction, and flux adds another direction to it. This is perhaps the most natural way of multiplying two vectors together.

So the rank 2 tensor has 2 kinds of direction in it. You could express a 3D vector in components a*i + b*j + c*k. Likewise, you might express a 3D rank 2 tensor as a*i1*i2 + b*i1*j2 + c*i1*k2 + d*j1*i2 + e*j1*j2 + f*j1*k2 + g*k1*i2 + h*k1*j2 + i*kk1*k2, or alternatively as a 3x3 matrix. Note that in general the two sets of directions aren't interchangeable, which is why I added numbers to the dimension place holders. So when you do operations on tensors, you need to be explicit in the notation which dimensions are being operated on. People use index notation to do this.

 

[Amrutha.phy]

Can the explanation be more simpler such that XI standard student can understand?

 

[Muphrid]

With all due respect, you've received four different explanations. Which of them (or which parts of them) do you find too complex?

 

[mikeph]

I always just thought of them as scalars, vectors, matrices, and then 3D, 4D,... matrices. So a tensor of rank N is an N-dimensional matrix, and to find a specific element in that matrix, you need to specify where along each of the dimensions you have to look.

eg. A_ijkl, where _ denotes "subscript", might be a rank 4 tensor, and it holds data along four dimensions, and each dimension is spanned by an index, in this case, i, j, k and l. For a rank 2 tensor (a matrix), i and j are the traditional row and column numbers where you would find the specific element you're looking for, but for a rank 4 tensor there are now two more indices, l and k.

Reading the wikipedia totally confused me and I'm sure there's more to it, but for everything I ever needed in physics, the above understanding was sufficient. Most explanations I've ever received were either incorrect (in failing to distinguish between a tensor and a tensor field, and therefore giving me parts of explanations of two different things) or too complex, usually the latter coming from a mathematician.

 

[krd]

Reading the wikipedia totally confused me and I'm sure there's more to it, but for everything I ever needed in physics, the above understanding was sufficient.

Most explanations I've ever received were either incorrect (in failing to distinguish between a tensor and a tensor field, and therefore giving me parts of explanations of two different things) or too complex, usually the latter coming from a mathematician.

There's kind of a few problems with Wikipedia. If you're reading any article, don't take what you read as 100% correct. Look at the talk page. There's a lot more disagreement and messing going on with the pages.

Someone only asks you for an explanation of something if they don't understand it. Otherwise they don't ask. So, if you're ever explaining something to someone, you have to think through what you're explaining. You don't play Gotcha with someone - if they ask you about a maths topic you have think through it, and explain the crucial terms they may not know either. If that's too much for ya, keep your mouth shut.


Wikipedia has a nerd problem. Some of the editors, I've spotted a few on the maths topics, are going in and deliberately making things more obscure and harder to understand. And then putting notes in the talk like "I removed the explanation of such and such, because Wikipedia is not a maths book, or a substitute for a university education. If the reader doesn't understand such and such they shouldn't be looking at wikipedia.". The insane idea that if you don't know the subject already, you're not fit to be looking it up in an encyclopaedia.

Then you just get meaningless gibberish like "An eigenstate is the measured state of some object possessing quantifiable characteristics such as position, momentum, etc." From introduction to Eingenstates. A nerd will tell you that description is correct. It may be, but it tells you absolutely nothing.

If you can't explain something, it's a good sign that you don't understand it yourself. Nerds can do exam questions and get the correct results - it doesn't mean they understand topic, just they've rote learned the answers to the questions. Nerds will deliberately mystify because it gives them a kick.


If you see a bad explanation of something on Wikipedia. Do your bit - either edit it, or put a note in the talk page.

 

[mikeph]

I can't count the number of wikipedia articles that open with an example or feature of the relevant subject, in lieu of a concise definition. eg.

The rigid rotor is a mechanical model that is used to explain rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top.

A vague outline, an purpose, and an example are given.

Compare this to Wolfram:

Rigid body: a body R, regardless of shape and volume, with the characteristic that the relative distance between any two points of R remains constant, even if the body is acted upon by external forces.

95% people googling a scientific term want the exact meaning of the term before they want any peripheral information. It's a real problem.

 

[Mark M]

Can the explanation be more simpler such that XI standard student can understand?

I managed to find this video, probably the best explanation you can find:

 

[Andy Resnick]

Can the explanation be more simpler such that XI standard student can understand?

Your OP question does not have a simple answer, and there are different approaches to defining what a tensor is.

At one level, a tensor is defined by the way it transforms from one coordinate system to another- this approach is similar to defining broccoli by the way it tastes: if it tastes like broccoli, then it is broccoli. If a mathematical entity transforms a certain way, then it is a tensor.

Another approach is to define a tensor by what it does: for example, vector addition is defined according to the parallelogram law, tensors 'eat' vectors and spit out numbers. Quantities that add according to the parallelogram law are vectors, and quantities that eat vectors and give numbers are tensors.

Then there's the way to represent tensors: for example, a vector can be written without coordinates as a or in terms of components a_i: (a_x, a_y, a_z): similarly, a second-order tensor can be written as Q or as a matrix Q_ij.

Just to give some specifics, you may be used to thinking of some physical quantities as scalars (mass, temperature), or as vectors (velocity, force). Some tensor-valued quantities are stress, diffusion, and susceptibility.