In Physics, What is a Tensor
Overview
Without a doubt, this is the easiest explanation about tensors, so if you’re an undergraduate in physics who came here because you don’t know what a tensor is, I highly recommend reading this.
We’re not accepting corrections about mathematical inaccuracies. Teaching someone who hasn’t learned about negative numbers that ‘you can’t subtract a larger number from a smaller number’, or someone who hasn’t learned about complex numbers that ‘you can’t have a negative number under a square root’ is not considered teaching incorrect information. The purpose of this article is not to teach the exact definition of a tensor, but to prevent unnecessary time wastage in trying to understand tensors.
As you study physics and move up the grades, you will come across something called a tensor. The first tensor I remember seeing was the moment of inertia tensor in mechanics.
$$ \mathbf{I} =\overleftrightarrow{\mathbf{I}}= \begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} &I_{zz} \end{pmatrix} $$
At the time, as a sophomore, it was impossible for me to understand what this was. Despite searching through various books or online, it was difficult to grasp what a tensor was1. I forced myself to try and understand the concept of a tensor despite lacking the necessary knowledge to accurately grasp it. I was as interested in physics as I was in mathematics; hence, I couldn’t just use or accept the definition and meaning of a tensor without knowing what exactly it was. However, it’s clear that this isn’t a good attitude towards studying physics.
While it’s possible to accept and understand the mathematical definition of a tensor, doing so would require deep knowledge in linear algebra2. This goes far beyond the mathematics needed for undergraduate physics. Studying mathematics is certainly helpful for studying physics, but this case is an overkill. Trying to understand tensors mathematically could very well ruin your mechanics midterm. So, for an undergraduate in physics, it’s important to simply ‘feel’ what a tensor is. It’s enough to look at various examples to understand why it’s needed, how, and when it’s used. Much of physics uses mathematics loosely, but that’s because the mathematical rigor is guaranteed. Since tensors too have this rigor guaranteed, there’s no need for undergraduates studying physics to chase after this rigor.
The concept of a tensor arises because there are physical quantities that cannot be represented solely by scalars and vectors. In fact, tensors can represent all physical quantities, making them a general concept that includes scalars and vectors. So, you can just use tensors without mentioning scalars and vectors. However, that would be inefficient and not helpful at all when first learning physics, so tensors that are scalars and vectors are not specifically called tensors. In undergraduate physics, what is often referred to as a tensor is represented by a $3\times 3$ matrix. If you hate reading long articles or can’t understand them, thinking ‘a 3x3 matrix is considered a tensor, I guess,’ is fine. As far as undergraduate studies are concerned, this isn’t an incorrect explanation.
Classification of Tensors
Tensors are generally denoted as $(m,n)$-tensor or $\binom{m}{n}$tensor. Here, $m$ represents the dimension of space, and $n$ represents the number of subscripts attached to the components of the tensor. In this case, the number of components of a tensor is $m^{n}$. In most physics excluding relativity, space is always 3-dimensional, so it’s always $m=3$. Therefore, without specifically mentioning how many $m$s there are, tensors are classified into $0$order tensors, $1$order tensors, and so on, based on the value of $n$.
$0$Order Tensor=Scalar
$0$ order tensor signifies a $\binom{3}{0}$tensor, and this is equivalent to a scalar. A typical example of a $0$order tensor is mass. In 3-dimensional space, mass is simply represented by $m$, so there are 0 subscripts, making it a $0$order tensor. However, this is usually referred to as a scalar, not a tensor. The number of components is $3^0=1$.
$1$Order Tensor=Vector
$1$ order tensor signifies a $\binom{3}{1}$tensor, and this is equivalent to a vector. A typical example of a $1$order tensor is velocity. In 3-dimensional space, velocity is denoted by $\mathbf{v}=(v_{x},v_{y},v_{z})$, and since there’s one subscript attached to its components, it’s a $1$order tensor. However, this is usually referred to as a vector, not a tensor. The number of components is $3^1=3$.
$2$Order Tensor
A $2$ order tensor refers to a $\binom{3}{2}$tensor. Consider the following $3 \times 3$ matrix.
$$ A=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} $$
The components of the matrix are represented by $a_{ij}$, and since there are two subscripts, it’s a $2$order tensor. Usually, undergraduate physics textbooks refer to a $3 \times 3$ matrix as being a tensor, rather than specifically calling it a $2$order tensor. The Kronecker delta $\delta_{ij}$ also has two subscripts, hence it is a $2$order tensor. However, textbooks don’t typically call the Kronecker delta a tensor.
$$ \begin{align*} \delta_{ij}&=\begin{cases} 1 & \mathrm{if} \quad i=j \\ 0 & \mathrm{if} \quad i\ne j \end{cases} \\ &= \begin{pmatrix} \delta_{11} & \delta_{12} & \delta_{13} \\ \delta_{21} & \delta_{22} &\delta_{23} \\ \delta_{31} &\delta_{32} &\delta_{33} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \end{align*} $$
The number of components is $3^2=9$.
$3$Order Tensor
A notable example is the Levi-Civita symbol.
$$ \epsilon_{ijk}=\begin{cases} 1 & \mathrm{if} \quad ijk=123=231=312 \\ -1 & \mathrm{if} \quad ijk=132=213=321 \\ 0 & \mathrm{if} \quad i=j \ \mathrm{or}\ j=k\ \mathrm{or} \ k=i \end{cases} $$
Since there are three subscripts, it’s a $3$order tensor. However, textbooks usually don’t refer to the Levi-Civita symbol as a tensor. The number of components is $3^3=27$.
$4$Order Tensor
By now, you should understand that a $4$order tensor refers to a physical quantity with four subscripts, but tensors of $4$order and higher don’t appear in undergraduate physics.