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.

Every dimension of a physical entity can be comprised of different components, e.g. $\vec{v} = v_x\vec{i}+v_j\vec{j}+v_z\vec{k}+$. 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.

- Each component can be
**scaled**by a constant amount to give the corresponding component of the resulting entity, e.g. momentum, $\vec{p} = m\vec{v}$. - The individual components can
**multiply and add**(or divide and get subtracted) with one another and**result in components that can still look like components of a vector**. E.g. force in magnetic field $\vec{F_{magnetic}} = q(\vec{v} \times \vec{B})$. - The individual components can combine with one another in a way that the
**resulting entity doesn't look like a scalar or vector**!**This is where tensors come in**. They represent such entities. What happens mostly is that, we are able to interpret the resulting entity manifested in specific ways, or the individual components themselves. But we may not be able to visualize the resulting physical entity in its entirety.

Nevertheless, we can, and do,**write the resulting entity in its entirety, and call them tensors**. E.g. electric**polarization of a crystal**in an electric field is given by:

\[

P =

\begin{bmatrix} P_x\\P_y\\P_z \end{bmatrix} =

\begin{bmatrix}\alpha_{xx}&\alpha_{xy}&\alpha_{xz}\\

\alpha_{yx}&\alpha_{yy}&\alpha_{yz}\\

\alpha_{zx}&\alpha_{zy}&\alpha_{zz}\\

\end{bmatrix}

\begin{bmatrix}E_x\\E_y\\E_z

\end{bmatrix}

\]

While the polarization tensor $\alpha$ may not make much sense to us like a scalr or a vector, the way it results in the different components of $\vec{P}$, $\begin{bmatrix} P_x\\P_y\\P_z \end{bmatrix}$ for a given $\vec{E}$ does, and we keep using the tensor representation of $\alpha$.

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

\[

\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}

\]

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!

If I measure a $6m$ long pole with my ruler (basis) that is $2m$ in length, my measure will be $3$ units.

Another choice of basis (in another system) can be a $1m$ long ruler, and then the length of the pole will be $6$ units.

The actual length of the pole - the **scalar invariant** - remains the same, while its measure in different frames (using different bases) differs. And we can have transformation rules between the different frames, so that knowing the measure in one frame, we can *calculate* it in another without actually doing the measurement. Also note that the length of the pole in any frame is a **collection of its bases (defined by scalar addition)**.

Similarly, for vectors, we have not $1$ but $3$ bases. And the **vector is a collection of these** $3$ **bases**, however, this time, not defined by scalar but **both scalar and vector addition**. And while these bases can differ among different coordinate systems, the **vector remains invariant**. We have transformation rules between the different coordinate systems.

The three bases in a vector space give us $3$ components for any vector that we can visualize together applying vector addition. But the $3$ bases can also combine together in any number of ways giving more than $3$ components, with these components not adhering to the rules of vector addition. Additionally, we can have more than $3$ bases. **That's how we get tensors**.

As long as we are constrained to combine bases only according to scalar or vector addition, any number of bases will give us a vector, otherwise a tensor.

The entity denoted by the combination of the bases in different ways - the **tensor - remains invariant**, and this constraint gives us the transformation rules between different coordinate systems (having different bases).

Finally, any combination (or collection) of bases that doesn't follow scalar or vector addition cannot be visualized, and hence we cannot generally perceive tensors as a single coherent physical entity. But, both from physics and mathematics point of view, they are as much a single entity as scalars and vectors.

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.

- Dimension
- Magnitude
- Flavor (same same but different things)
- Combination of flavor and dimensions

- The
**dimension**of any physical entity is the thing that provides it its physics, its measurability. Examples are length, mass, time etc. Anything without a dimension is just a concept, a mathematical one, not physical. E.g. phrases like 'the temperature is 25', 'this is 4 of iron', 'the wire has 2 current', don't have any physical meaning until we add dimensions like 'the temperature is 25^{o}**Celsius**', 'this is 4**kg**of iron', 'the wire has 2**amperes**of current'.

There are 7 fundamental dimensions which is basically saying that there are 7 primary dimensions that are not a combination of other dimensions. - Then comes the
**magnitude**. This is the second mandatory requirement for the existence of a physical entity. I don't think I need to explain it. **Flavor**. This is my construct! But you'll see how natural and necessary it is in understanding physical entities. Let's say I've 5 apples which are identical in every aspect except one - 3 of them are red and 2 green.**Here 5 is the magnitude and apple-ness is the dimension**. Now, even though looking at my apples as apples will suffice in many scenarios, there will be situations in which it'll be helpful to differentiate between red and green apples. Here, red and green are the flavors of our entity (apples). Whether we want to look at them as just apples or red and green apples separately depends entirely upon the context.

Also, we may or may not be accustomed to looking at them one way or another. E.g. somebody gravely allergic to green apples will see them as hugely different things whereas somebody who is color blind will find them identical.

You may ask why not just see these as different entities (with different dimensions) altogether. The answer lies in the**measurability**- if we can measure an entity using the exact same unit, it makes sense to see them having the same dimension.

Once we move from scalar entities to vector ones, things become clearer.**In the world of vectors, flavor of an entity is its direction.**A velocity of 5m/s in x-direction is not the same as 5m/s in y-direction. But, the dimension for both is m/s. So, these are like, different flavors of the same physical entity - the velocity.- Now comes the last piece of the puzzle - the
**combination of flavors and dimensions**. The flavors of individual dimensions (entities), and different dimensions themselves can combine together to give rise to new physical entities.

And, we may or may not be used to understanding and comprehending the new physical entity in its own right, or conversely, we may or may not able to see the underlying combination. But, from a physical point of view, it always makes sense to able to describe the entities in terms of their components, even if its the combined entity that is more natural for us.

- Length and time - these two different dimensions can combine in a way to give rise to the physical entity called
**speed**, and we can very much comprehend it its own right. **Momentum**is comprised of two dimensions - mass and velocity, but we can quite fairly see and feel these two dimensions separately from each other, e.g. even with closed eyes, you can somewhat tell if a 5gm penny hit you at 20m/s or a 20gm coin hit you with 5m/s, even though both of these have the same momentum.- Red, green and blue - three flavors of the same dimension - luminosity - can combine together to form a
**composite color**. This time, there are two possibilities - we can see the composite color, or only one or two flavors (colors) in the composite color if there are any constraints like filters or color-blindness. - A
**velocity**comprised of two flavors - 5$\vec{i}$m/s and 4$\vec{j}$m/s, i.e. a velocity in the direction of (5$\vec{i}$ + 4$\vec{j}$)m/s is comprehensible to us as a single entity, and we do not see it as a composition of two individual velocities in real world. But, for a unidirectional being living on x-axis, only the 5$\vec{i}$m/s part of the velocity will be perspicuous.

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/m^{3}) 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:

- The luminosity increases if the size of the area increases (it's obvious)
- Luminosity per unit area at any point changes in value if ($\vec{n}$) changes direction

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 a single 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.

Ad

Ad

Ad