What do the symbols $\delta_{ij}$ and $\epsilon_{ijk}$ mean?

Question

I am reading David Tong's notes on vector calculus (which are amazing), and the symbols $\delta_{ij}$ and $\epsilon_{ijk}$ keep coming up in the notes. I do not know what these actually mean. I think they might represent some sort of unit vector combination? I am not sure. He defines the symbol $\delta_{ij}$ for one sentence in the notes, saying that it is "The orthonormality $\mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}$". He also uses the same style of symbol to express the equation $\nabla \cdot \mathbf{r} = \frac{\partial x^i}{\partial x^i} = \delta_{ii} = n$, where he is using summation notation for the $x^i$. If someone would be kind enough to tell me what these symbols mean, I would greatly appreciate it (Google does not have any results for "delta calc 3 symbol"). Thank you.

Answer 1

The symbol $\delta$ in this context is known as the Kronecker delta. It denotes a function $\mathbb Z\times\mathbb Z\to\{0,1\}$ whose value at $(i,j)$ is written as $\delta_{ij}$. This function is defined by the following rule: $\delta_{ij}=1$ if $i=j$, and $0$ otherwise.

The symbol $\varepsilon$ is known as the Levi-Civita symbol. It denotes the function $\{1,2,3\}\times\{1,2,3\}\times\{1,2,3\}\to\{-1,0,1\}$ given by $$ \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is } (1,2,3), (2,3,1), \text{ or } (3,1,2), \\ -1 & \text{if } (i,j,k) \text{ is } (3,2,1), (1,3,2), \text{ or } (2,1,3), \\ 0 & \text{if } i = j, \text{ or } j = k, \text{ or } k = i. \end{cases} $$ Put differently, $\varepsilon_{ijk}$ equals $0$ when there are any repeated entries, equals $1$ for even permutations of $(1,2,3)$, and equals $-1$ for odd permutations of $(1,2,3)$. The Levi-Civita symbol can also be defined more generally for $n$ indices $i_1i_2\dots i_n$, so that $\varepsilon_{i_1i_2\dots i_n}$ equals $0$ when there are any repeated entries, equals $1$ for even permutations of $(1,\dots,n)$, and equals $-1$ for odd permutations.

Both of these notations are often used in conjunction with the summation convention, also known as Einstein notation. For instance, if $A$ is a $3\times 3$ matrix, we can express its determinant as $$ \det(A)=\varepsilon_{ijk}a_{1i}a_{2j}a_{3k} \, , $$ where the double appearance of the indices $i,j,k$ indicates that the above is a shorthand for $$ \det(A)=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3}\varepsilon_{ijk}a_{1i}a_{2j}a_{3k} \, . $$