Indexes of tensors and simple physical applications in high school

Question

In this picture I have colored the cubes of a tensor of rank $3$ or $3-$way tensor $\mathcal A_{\mathrm{klm}}$.

My questions are:

1. If the red cube has like subscripts $\mathrm{klm}$ the yellow cube has subscript $\mathrm{(k+1)l(m+1)}$ or is it wrong? In general the subscripts of each cube, how should they be indicated, starting from the red one for example?
2. Remaining in the field of the tensors is there any simple example in physics that could attract high school students?

Answer 1

To me considering a tensor as a cube is a particularly bad way of thinking about things, because a tensor is so much more than an array/cube/hypercube of data. One possible definition of an $(r,s)$ tensor over a (real, finite-dimensional) vector space $V$ is a multilinear map $T: (V^*)^r \times V^s \to \Bbb{R}$.

Once you choose a basis for the vector space $V$, then of course, everything about the tensor is contained in its components $T^{i_1\dots i_r}_{j_1\dots j_s}$ relative to that choice of basis. And in this case, sure, if you consider a $(0,3)$ tensor $A$, once you fix a basis, you only have to think about its components $A_{klm}$. The answer to which indices are for the yellow cube depend on how you're labelling your indices: it could be either $A_{k,l,m+1}$ or $A_{k,l+1,m}$ or $A_{k+1,l,m}$ (once again, depending on how you define things).

Finally, there are several examples of tensors (and tensor fields) which are very relevant for math and physics. Let me just give a few "simple" examples:

1. The standard dot/inner product on a finite-dimensional real vector space $V$ (for example $V= \Bbb{R}^n$) is a $(0,2)$-tensor. i.e $\langle\cdot, \cdot\rangle:V \times V \to \Bbb{R}$ is bilinear (and also symmetric and positive-definite). This allows you to define the geometry of the space, and gives rise to notions of angles, lengths. If you go from a single vector space to the collection of all tangent spaces to a manifold, you get the notion of a metric tensor field $g$ (of course, you need certain technical smoothness conditions also), and this is what's used in Riemannian geometry. Finally, if you replace the positive-definiteness condition to being non-degenerate with Lorentzian signature, you get the idea of a Lorentzian metric, and this is heavily used in Einstein's theory of Relativity (both Special and General).

2. The determinant of a matrix, thought of as an operation on the columns, $\det: \underbrace{\Bbb{R}^n \times \dots \times \Bbb{R}^n}_{\text{$n$ times}} \to \Bbb{R}$ is a $(0,n)$ tensor (and this is VERY VERY important, because it is also alternating, and so it is very closely related to the notion of volumes). For example, given $a,b,c \in \Bbb{R}^3$, the number $\det(a,b,c) \in \Bbb{R}$ represents the signed volume of the parallelepiped spanned by the vectors $a,b,c$.

3. Another example of a tensor, more on the physical side is the moment of inertia tensor $I$ associated to a rigid body (this is a $(0,2)$ tensor, or a $(1,1)$ tensor depending on how you define things). This roughly captures the information of "how hard" it is to rotate about various axes. Based on memory, I remember learning in highschool that the moment of inertia is a single number ($I = \dfrac{1}{2}mr^2$ for a point particle, $I = ml^2/12$ for a thin rod about the center), but of course this is only part of the story, and perhaps this is a good time to introduce students to the idea that there is more to the story.

4. A very commonly encountered tensor field is the exterior derivative of a function: $df$, which in local coordinates reads $df = \dfrac{\partial f}{\partial x^i}dx^i$ (I'm not sure this is understandable for highschoolers, but perhaps the one-dimensional version $df = f' dx$ should be possible to explain assuming they have learnt a little differential calculus).

5. Electromagnetism is full of tensor (fields), from the electromagnetic field strength tensor $F_{\alpha\beta}$, to the stress-energy tensor $T_{\alpha\beta}$. But the physical significance of this is I think pretty hard to explain at a High-school level.

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I think out of the examples I presented, $1,2,3$ are probably the easiest to explain to high school students, and $3$ is perhaps the simplest example of a tensor whose "physical interpretation" can be explained at a high-school level (compared to $5$ which talks about the EM field strength tensor and stress-energy tensor, whose significance is I think much harder to explain).

Answer 2

If you view a 3-way tensor as a cube, the indices of an entry are simply the coordinates of the corresponding small cube. So here if the red cube has indices $(k,l,m)$, the yellow cube has indices $(k+1,l,m)$ or $(k-1,l,m)$ since you have only moved one step in one direction from the red cube (if you assume the direction you moved along corresponds to the first axis).

Concerning applications, I know continuum mechanics make an extensive use of tensors but it might be a bit hard for high school students...