Levi-Civita Properties
Prove the following
a) Let $(\hat e_1,\hat e_2. \hat e_3)$ be the unit vectors of a right handed, orthogonal coordinate system. Demonstrate that Levi-Civita symbol satisfies $\epsilon_{ijk} =\hat e_i(\hat e_j \times \hat e_k)$
b) Prove that $\epsilon_{ijk}\epsilon_{ist}=\delta_{js}\delta_{kt}-\delta_{jt}\delta_{ks}$
c) Prove that $a \times b=\epsilon_{ijk}\hat e_i a_j b_k$
a)
$ϵ_{ijk}$ is defined as
$det\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{vmatrix}=∑_{i=1}^3 ∑_{j=1}^3 ∑_{k=1}^3 ϵ_{ijk} a_{1i} a_{2j} a_{3k}=ϵ_{ijk} a_{1i} a_{2j} a_{3k}$
Taking
$a_i=|\hat e_i |$ $a_j=|\hat e_j|$, $a_k=|\hat e_k|$ we obtain $ϵ_{ijk}=det |\hat e_i,\hat e_j ,\hat e_k |=(\hat e_i )(\hat e_j × \hat e_k)$
b)
In general
$ϵ_{ijk} ϵ_{lmn}=δ_{il} δ_{jm} δ_{kn}+δ_{im} δ_{jl} δ_{kl}+δ_{in} δ_{jl} δ_{km}-δ_{im} δ_{jl} δ_{kn}-δ_{il} δ_{jn} δ_{km}-δ_{in} δ_{jm} δ_{kl}$
Taking
$i=l$, $m=s$ and $n=t$ one has
$ϵ_{ijk} ϵ_{ist}=δ_{ii} δ_{js} δ_{kt}-δ_{ii} δ_{jt} δ_{ks}=δ_{js} δ_{kt}-δ_{jt} δ_{ks}$
c)
$(a×b)=det \begin {vmatrix}\hat e_1&\hat e_2&\hat e_3 \\a_1&a_2&a_3\\b_1&b_2&b_3 \end{vmatrix}=ϵ_{ijk} (\hat e_i ) a_j b_k$ and thus $(a×b)_i=ϵ_{ijk} a_j b_k$
Evaluate the following expressions
a) $\delta_{ii}$
b) $\delta_{ij}\epsilon_{ijk}$
c) $\epsilon_{ijk}\epsilon_{ljk}$
d) $\epsilon{ijk}\epsilon_{ijk}$
a)
$δ_{ii}=1$
b)
$δ_{ij} ϵ_{ijk}=ϵ_{iik}=0$ because $i=j$
c)
$ϵ_{ijk} ϵ_{lmn}=δ_{il} δ_{jm} δ_{kn}+δ_{im} δ_{jl} δ_{kl}+δ_{in} δ_{jl} δ_{km}-δ_{im} δ_{jl} δ_{kn}-δ_{il} δ_{jn} δ_{km}-δ_{in} δ_{jm} δ_{kl}$
therefore
$ϵ_{ijk} ϵ_{ljk}=δ_{il} δ_{jj} δ_{kk}+δ_{ij} δ_{jl} δ_{kk}+δ_{ik} δ_{jl} δ_{kj}-δ_{ij} δ_{jl} δ_{kk}-δ_{il} δ_{jk} δ_{kj}-δ_{ik} δ_{jj} δ_{ki}$
$ϵ_{ijk} ϵ_{ljk}=1+δ_{ik} δ_{jl} δ_{kj}-δ_{ij} δ_{jk} δ_{kj}-δ_{ik}^2$
d)
$ϵ_{ijk} ϵ_{ijk}=δ_{ii} δ_{jj} δ_{kk}+δ_{ij} δ_{ji} δ_{kk}+δ_{ij} δ_{jk} δ_{kk}-δ_{ij} δ_{jk} δ_{kk}-δ_{ii} δ_{jk} δ_{kj}-δ_{ik} δ_{jj} δ_{ki}$
$ϵ_{ikj} ϵ_{ijk}=1+δ_{ij}^2+δ_{ij} δ_{jk}-δ_{ik} δ_{jk}-δ_{jk}^2-δ_{ik}^2=1+δ_{ij}^2-δ_{jk}^2-δ_{ik}^2$
