Understanding Tensors: A Simplified Approach for Beginners

[stabu]

Hi,

I'm trying to close in on a more intuitive way of understanding tensors. For some reason, they've always held an aura of mystique for me, may be also their similarity to the word "tense" has meant that I've never really warmed to the many defintions and explanations available. So, in many ways, I already have a mental block over them.

Recently, I (re-) watched Dan Fleisch's video at
and the tone is very much what I was looking for, but I now feel he spends too much time on vectors (which I have no problems with), and glosses over the rank 1 tensor (a vector) to rank 2 tensor (tensor "proper" say) move, which actually is the part I find hardest.

I have had conceptual problems like this in the past, and found afterwards that I've made too much of a big deal about them. In other words, that in many of the big, supposedly difficult concepts, there is a smaller, much simpler idea trying to get through. For example, complex numbers. The word "complex" can form quite a barrier, but in fact the underlying idea is actually pretty simple if you give it time.

So I was revisiting tensors in that light. My only way of explaining (say, rank 2) to myself is to imagine a vector represented by velocity (perhaps), suddenly thrust from free space into a field of some sort, so that for every basis vector of velocity, a full set of basis vectors for the field are required. So in 3D, this means we get nine components. However, I'm not entirely happy with this. It's what I call "situational", i.e. depends on a specific situation.

Another way is just to examine how a scalar becomes a vector, and then decide to generalize up to tensor in terms of what I may call imprecisely "directional agents". A scalar has no such thing, it's only a quantity, so it's rank 0. If we are say in 3D, moving up a rank, means we add a "directional agent" which by necessity must have three components. That's rank 1. Now if we add a second "directional agent" it must correspond the vector in the three components available in the current context (3D space), and that correspondance must be three times for each of the vectors components, so that means nine (i.e. not 3 or 6).

Any help or insights welcome, thanks for reading! Cheers.

 

[rolotomassi]

I too have had the same problem intuitively understanding them. I think there not intuitively understood is the problem and perhaps working with them over time will clear it up.

I remember one maths lecturer would always emphasise for example that a tensor has 'components' which are represented by a matrix. The tensor itself is independent of the co-ordinate system, while of course the matrix is not.
One example I recently had in a lecture was that in non homogeneous materials the magnetic succeptibility is represented by a tensor because its different in all directions. Similarly for dielectric constant (or refractive index), the wavevector is effected differently in different directions, this is birefringence i believe. So if you want a 3D description of the material and how light travels through it or whatever situation you have, tensors facilitate this.

I think you are on the right lines with the idea a vector ( also tensor ) transforms a scalar. A matrix ( represents a tensor) transforms a vector etc.
This is about as much as I know and I would like to know what you think about these ideas I am not 100% they're all correct either just my thoughts.

 

[rolotomassi]

To link what I've said to your example. If i think about a velocity vector in a non homogeneous material so that the index varies throughout. Then the x component of the velocity needs 3 components to desribe the effect the index has on it. So initial face of the material it might see the refractive index ( n_x) then in the middle (n_y) and the the last face (n_z) So 9 components needed ?

 

[stabu]

Hi rolotomassi,

Thanks for the replies! I had not thought of a non-homogeneous material/field, and it helps enormously, thanks! I can see more fully how the tensor components could be needed in that situation.

Let me say that yes, intuition develops, so to speak, and with usage, as you say, it's possible to become comfortable with tensors, sure.

I see matrices as more general myself, and would probably associate tensors more with the kronecker product operation.

I must find now an easy tensor problem that can be worked on ... to begin working with them, as you say.

Cheers!

 

[Chestermiller]

Hi rolotomassi,

Thanks for the replies! I had not thought of a non-homogeneous material/field, and it helps enormously, thanks! I can see more fully how the tensor components could be needed in that situation.

Let me say that yes, intuition develops, so to speak, and with usage, as you say, it's possible to become comfortable with tensors, sure.

I see matrices as more general myself, and would probably associate tensors more with the kronecker product operation.

I must find now an easy tensor problem that can be worked on ... to begin working with them, as you say.

Cheers!

Have you had any exposure to the subject of tensors using so-called dyadic tensor representation (where a 2nd order tensor, for example, is represented as the sum of components times two coordinate basis vectors placed in juxtaposition)?

Chet

 

[Chandra Prayaga]

Physical situations and concrete examples help reduce your tension. But, ultimately, we have to face the fact that a tensor (even a vector or a scalar) is a mathematical entity, and needs to be defined as such.
Incidentally, it is not necessary to think of an inhomogenous material to need tensors. What you need is an anisotropic material (many crystalline materials). The properties of such a material depend on the direction. Here is an example physical property which might make you feel better:

In an isotropic dielectric (such as water), the polarization of the material is proportional to, and in the same direction as the applied electric field. The proportionality constant is the susceptibility, and is a scalar. In this case, if you apply an electric field in the x-direction, you will have a polarization, also in the x-direction. If you apply the same magnitude of the electric field in the y-direction, you will get the same magnitude of the polarization, in the y-direction. The proportionality constant (susceptibility) is the same in all directions. That is why the susceptibility is a scalar.

In a general anisotropic dielectric, if you apply an electric field in the x-direction, you can get polarization in all three directions, with three different susceptibilities. So for the three components of the electric field, there are nine susceptibilities ( components of a tensor of rank two)

 

[rolotomassi]

Nice description.
When you apply a E-field in the x direction say, how is it that this causes a polarization in perpendicular directions? Is this due to some internal effects within the material or because the crystal structure does not allow 'free movement' of bulk of the material or something?

Also could you explain how tensors are used more generally, I mean how would you describe them mathematically.

 

[BobG]

Hi,

I'm trying to close in on a more intuitive way of understanding tensors. For some reason, they've always held an aura of mystique for me, may be also their similarity to the word "tense" has meant that I've never really warmed to the many defintions and explanations available. So, in many ways, I already have a mental block over them.

...

Any help or insights welcome, thanks for reading! Cheers.

Why not start with coordinate transformations - transforming the components of a vector from one coordinate system to the other. Not quite the same, but very close. How much difference is there between rotating a vector through a stationary coordinate system and rotating a coordinate system around a stationary vector? It gives a basic grasp of the idea at least.

By the way, I couldn't listen to the sound on that video and so chose the CC option. What horrible closed captioning! I had to start laughing a couple of minutes in as I translated from the closed captioning version to what the professor must actually be saying. The entire discussion was about bassists and cancers instead of tensors. If tensors make you tense, cancers will surely terrify you!

 

[Chandra Prayaga]

Nice description.
When you apply a E-field in the x direction say, how is it that this causes a polarization in perpendicular directions? Is this due to some internal effects within the material or because the crystal structure does not allow 'free movement' of bulk of the material or something?

Also could you explain how tensors are used more generally, I mean how would you describe them mathematically.

Nice description.
When you apply a E-field in the x direction say, how is it that this causes a polarization in perpendicular directions? Is this due to some internal effects within the material or because the crystal structure does not allow 'free movement' of bulk of the material or something?

Also could you explain how tensors are used more generally, I mean how would you describe them mathematically.

I would suggest any standard book on Mathematical Methods in Physics, such as the one by Boas.