So I came across this proof and was unsure about the notation they used. I did the proof by hand and similar to them used product rule to proof the identity.
My question:
What do the epsiplon ijks represent?
Proof:
Notation Inquiry.
1
$\begingroup$
notation
vector-analysis
1 Answers
2
$\epsilon$ is the Levi-Civita symbol. $\epsilon_{ijk}$ is equal to $0$ unless the $i,j,k$ are all different; it's $1$ if $(i,j,k)$ is some rotation of $(1,2,3)$; and it's $-1$ if $(i,j,k)$ is some rotation of $(1,3,2)$.
A rotation of $(a,b,c)$ is $(a,b,c)$, $(b,c,a)$, or $(c,a,b)$.
In general, for $\sigma \in S_n$ the symmetric group on $n$ elements (above, we have $n=3$), we define $\epsilon_{\sigma}$ to be equal to $\mathrm{sgn}(\sigma)$; and $\epsilon_{e}$ is defined to be $0$ if $e$ is some expression not in $S_n$. We write $\epsilon_{ijk}$ for $\epsilon_{(i,j,k)}$ when $\epsilon$ is defined in this way. For example, $\epsilon_{112} = 0$ because $(1,1,2)$ is not in $S_n$.