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If the idea of tensors ever intrigued you and you wanted to get to the real physical understanding of them, this article might help.
Tensors in physics and mathematics have two different but related interpretations - as physical entities and as transformation mapping. Let's look at the former interpretation below. But before that, if you want to quickly glance at some interesting ways to look at tensors, click on the buttons below.
This description is based on the aphorism - "a tensor is something that behaves like a tensor"!
From a physical entity point of view, a tensor can be interpreted as something that brings together different components of the same entity together without adding them together in a scalar or vector sense of addition. E.g.
$\chi$ above is an example of a tensor. Though we may not be able to see $\chi$ as a single perceivable thing, it can be used to fetch or understand perfectly comprehensible entities, e.g. for a given area $\vec{s}$, we can get the total number of pixels emitting light perpendicular to it by the product:
$$\begin{bmatrix}\chi_x&\chi_y&\chi_z
\end{bmatrix}\cdot
\begin{bmatrix}s_x\\s_y\\s_z
\end{bmatrix}$$
Change the monochromatic pixels in this example to RGB pixels, and we get something very similar to the stress tensor (a tensor of rank 2), and we can get the traction vector (force per unit area for a given unit area n) by the equation:
$$\begin{align}
\textbf{T}^{(\textbf{n})}
& = \begin{bmatrix} T_x\\T_y\\T_z \end{bmatrix}^{(n)}
= \textbf{n} \cdot \boldsymbol{\sigma}\\
& =
\begin{bmatrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\
\sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\
\sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\
\end{bmatrix}
\begin{bmatrix}n_x\\n_y\\n_z \end{bmatrix}
\end{align}$$
Though it's difficult to visualize the stress tensor in totality, each of its components tells us something very discrete, e.g. $\sigma_{xx}$ tells us how much force in x-direction is being experienced by a unit surface area that is perpendicular to the x-direction (at a given point in a solid). The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction, we get the traction vector from the stress tensor, or, I do not mean literally though, the stress tensor collapses to the traction vector.
Note that the possibility of interpreting tensors as a single physical entity or something that makes sense visually is not zero. E.g., vectors are tensors and we can visualize most of them (e.g. velocity, electromagnetic field).
Also note that in all the three cases - scalars, vectors and tensors, the resulting entity - the collection or the grouping of the individual components, whether or not it makes any sense to you intuitionally or visually (e.g. the total amount of Calcium, the net velocity, or the pixel density above), remains invariant. If we change the bases, we get different values of the components of these entities in different coordinate systems, but the resulting entities remain the same, and it is possible to find the values of the components of an entity in another system by knowing it in one - by virtue of having transformation rules between the two coordinate systems.
This invariance, forced by the transformation rules, is the meaning of the phrase "behaving like a Tensor".
We know how a physical entity can have a magnitude, or a magnitude and a direction. But what doesn't seem natural to intuition, but is nevertheless possible, is the fact that entities can have a magnitude and two (or more) directions interwoven together.
The phenomenon being referred here is not of addition or multiplication of directions. Adding is possible only when the directions have the same dimensions, e.g. adding two velocities, and that doesn't give us any new entity. Multiplication of directions is possible and that gives us a vector.
But other operations are possible between dimensions having directions (or vectors), e.g. when we define $\text{stress} = {\text{force} \over \text{area}}$, both the terms on the right have directions in addition to having magnitude, and these directions (each having three components) interweave together to give an entity called the stress tensor having 9 components. The traction, or total force per unit area (for a given unit area) at any point is given by
\[\begin{align}
\textbf{T}^{(\textbf{n})}
& = \begin{bmatrix} T_x\\T_y\\T_z \end{bmatrix}^{(n)}
= \textbf{n} \cdot \boldsymbol{\sigma}\\
& =
\begin{bmatrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\
\sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\
\sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\
\end{bmatrix}
\begin{bmatrix}n_x\\n_y\\n_z \end{bmatrix}
\end{align}\]
The stress tensor ($\sigma$), in its entirety, may not make intuitional sense to us, but its 9 components can manifest this entity in a way that we can get to know the direction of force on a surface of a given area and direction. These two directions (or vectors) - of force and of the surface are lying hidden in the stress tensor ($\sigma$) that come to surface as and when needed.
Though it's difficult to visualize the stress tensor in totality, each of its components tells us something very discrete, e.g. $\sigma_{xx}$ tell us how much force in x-direction is being experienced by a unit surface area that is perpendicular to the x-direction (at a given point in a solid). The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction of the surface, we get the traction vector, or, I do not mean verbally though, the stress tensor collapses to the traction vector.
Further, if we fix the direction of the force as well, i.e. we are only interested in force in a specific direction, we get a scalar - force in that specific direction on a surface facing a given direction. The traction vector has now collapsed to a scalar!
Every dimension of a physical entity can be comprised of different components, e.g. the three components of the velocity dimension $\vec{v} = v_x\vec{i}+v_j\vec{j}+v_z\vec{k}$. When we combine physical entities that are comprised of different dimensions in order to define new physical entities, the different components of the individual entities can interplay in different ways, e.g.
To say that tensors are physical entities is to say that there are physical entities out there that are represented by tensors, just like there are physical entities that represented by scalars or vectors. So, what is actually a tensor physical entity? In order to understand tensors as physical entities, we must have a fair understanding of the following. These ideas or phenomena are behind any and every physical entity.
The gist is that depending upon the context (or the reference frame) and our familiarity, a physical entity can make sense to us either in its own right, or in terms of its components (flavors and dimensions), or in both. If you have understood this correctly, tensors are going to be a piece of cake now.
Before bringing in tensors, a last piece of analogy. Let's say we have cube that has LED pixels embedded all through its volume. We can create an entity called pixel density, or pixels per unit volume to get a rough idea of how bright or how luminous this cube could be at different points inside it. Let's denote it by $\chi$. We'll be using the term luminosity for brightness (which, in turn, will be equal to the number of pixels) - do not confuse it with the traditional definition of luminosity. In the video below, we are looking at a unit cube inside our pixel cube:
But, now we realize that since these pixels have to be connected with wires and some gluing mechanism, they do not emit light in all directions. In fact, each pixel can only illuminate the volume in front of it. So, our physical entity (pixels/m3) is not doing justice in understanding the brightness of different points of our cube. Because, at any moment, we can look at only one surface inside the cube, that too only from a specific direction. Depending upon how the pixels have been glued at any given location, the brightness in a specific direction can be different from any other.
What we have in reality is something like this:
So, let's create another entity - pixel density by area, or pixels per unit area. Here, unit area is supposed to have a direction that is perpendicular to the unit area in question, and the pixels counted will be the ones that emit light in that direction. Because the unit areas can be facing different directions, we'll need to have not one but three values for our new entity pixels-per-unit-area.
Let's assume these pixel densities are constant (they do not change at different points of the cube). And, the values to be $\chi_x = 150$, $\chi_y = 23$ and $\chi_x = 85$. So now, we can get the total luminosity ($L^{(s)}$) perpendicular to any surface ($\vec{s}$) within the cube:
\[
L^{(s)}
=
\begin{bmatrix}150&23&85
\end{bmatrix}
\begin{bmatrix}s_x\\s_y\\s_z
\end{bmatrix}
\]
Or, for a unit surface ($\vec{n}$):
\[
L^{(n)}
=
\begin{bmatrix}150&23&85
\end{bmatrix}
\begin{bmatrix}n_x\\n_y\\n_z
\end{bmatrix}
\]
The things to note here are:
Let's get back to the physical entity we defined - pixels per unit area ($\chi$). Here, we have combined two entities - pixels (p) and area (s) - to generate a new entity. But, it so happens that one of our underlying entities - area - comes in three flavors - $s_x$, $s_y$ and $s_z$. So, in order to have any meaning of our entity pixels per unit area, we need to break it into three flavors - pixels per unit-area-perpendicular-to-the-x-direction, and so on. These individual flavors, in spite of being scalars by themselves, and not even having a direction, cannot be just added together to give the total pixels by unit area at a point!
This entity of ours - pixel density per unit area per direction- is neither a scalar, nor a vector. It has got flavors each of which has a magnitude but not a direction associated to it, and these flavors separately do make sense to us. So, we just write these flavors or components together and understand this thing to represent the total luminosity per unit area at a point (for all directions!). We call this combined thing a tensor. It is something that brings together different components of the same entity together without adding them together in a scalar or vector sense of addition. It exists by definition, but we can (mostly) make sense of only its individual components.
So, what we have seen here - $\begin{bmatrix}150&23&85\end{bmatrix}$, or more generally $\begin{bmatrix}\chi_x&\chi_y&\chi_z\end{bmatrix}$ - is a tensor that represents $\chi$ or pixel density per unit area per direction in our pixel-cube.
Now, let's upscale. Instead of having just one color pixels, let's have RGB pixels. What we are now going to have for $\chi$ is:
\[
\chi
=
\begin{bmatrix}\chi_{xr}&\chi_{yr}&\chi_{zr}\\
\chi_{xg}&\chi_{yg}&\chi_{zg}\\
\chi_{xb}&\chi_{yb}&\chi_{zb}\\
\end{bmatrix}
\]
Now, our total luminosity per unit area (for a given unit area $\vec{n}$) itself is going to have these three flavors - red, green and blue, and we can get it as:
\[
L^{(n)}
=
\begin{bmatrix} L_x\\L_y\\L_z \end{bmatrix}^{(n)}
=
\begin{bmatrix}\chi_{xr}&\chi_{yr}&\chi_{zr}\\
\chi_{xg}&\chi_{yg}&\chi_{zg}\\
\chi_{xb}&\chi_{yb}&\chi_{zb}\\
\end{bmatrix}
\begin{bmatrix}n_x\\n_y\\n_z
\end{bmatrix}
\]
This total luminosity per unit area for a given unit area $\vec{n}$, $L^{(n)}$, can be understood either as the total brightness of all the three colors combined, or as just a convenient way of writing together the different luminosities (of its different colors together). If we were looking at it ourselves in real world, we'll see it more as the total brightness of the combined color (that is, a composite of the three colors or flavors).
And, each component of the tensor $\chi$ can be understood as the number of pixels of a specific color emitting light in a specific direction (x, y or z) at a given point.
The flavors or the components, may not always add or combine together at all, or at least, as seamlessly as colors do. One example where do they add together in a sense that we can interpret their addition in a physical way, is directions.
In the above example, just replace the total luminosity by force, the three colors by the three directions force can have, and the number of pixels by 9 proportionality constants, and we get the total force per unit area (for a given unit area) at a given point in a solid, which is nothing else but the traction vector. And, the 9 constants together, denote the stress tensor (see TL;DR 2 above)! Each of the 9 components of the stress tensor represents the force in a specific direction (x, y or z) for a given unit area facing a specific direction (x, y, or z). By having this information contained in it,the stress tensor can tell us the magnitude and net direction of the force vector (traction vector) given any surface vector in such a way that the direction of the traction vector is not constrained to be parallel or perpendicular to that of the surface area (which would happen in vector scaling or multiplication).
Though it's difficult to visualize the stress tensor in totality, each of its component tells us something very discrete, e.g. $\sigma_{xx}$ tell us how much force in x-direction is being experienced by a unit surface area that is perpendicular to the x-direction (at a given point in a solid). The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction, we get the traction vector, or, I do not mean literally though, the stress tensor collapses to the traction vector.
