Equations of fluid dynamics E Fluid dynamics in one dimension G Jacobian determinant

Appendix F Levi-Civita-Symbol

The symbol $\epsilon_{ijkl\ldots}$ is called Levi-Civita symbol and defined as follows:

\displaystyle\epsilon_{ijkl\ldots}=\begin{cases}1&\text{if $i,j,k,l\ldots$ is % an even permutation},\\ -1&\text{if $i,j,k,l\ldots$ is an odd permutation},\\ 0&\text{otherwise (two or more labels are the same)}.\end{cases}

(F.1)

Therefore the Levi-Civita-Symbol will change its sign, if two labels are exchanged

\displaystyle\epsilon_{ijkl\ldots u\ldots v}=-\epsilon_{ijkl\ldots v\ldots u}.

The Levi-Civita-Symbol is not a tensor, but a pseudotensor, because it transforms like a tensor under rotation, but not under reflection (Pope, 2000).²⁸²⁸Is it possible to fix up the Levi-Civita-Symbol so it becomes a real tensor?

F.1 Levi-Civita-Symbol in 3D

In 3D only 6 of the 27 components of the Levi-Civita-Symbolare are unequal zero

	$\displaystyle\epsilon_{123}=\epsilon_{312}=\epsilon_{231}$	$\displaystyle=1$
	$\displaystyle\epsilon_{321}=\epsilon_{132}=\epsilon_{213}$	$\displaystyle=-1$

The Levi-Civita-Symbol in 3D is most often used to express components of a cross product of vectors in cartesian tensor notation

	$\displaystyle{\left[\boldsymbol{u}\times\boldsymbol{v}\right]}_{i}=\epsilon_{% ijk}u_{j}v_{k}=$	$\displaystyle\ \epsilon_{i11}u_{1}v_{1}+\epsilon_{i12}u_{1}v_{2}+\epsilon_{i13% }u_{1}v_{3}$
		$\displaystyle+\epsilon_{i21}u_{2}v_{1}+\epsilon_{i22}u_{2}v_{2}+\epsilon_{i23}% u_{2}v_{3}$
		$\displaystyle+\epsilon_{i31}u_{3}v_{1}+\epsilon_{i32}u_{3}v_{2}+\epsilon_{i32}% u_{3}v_{3}$
	$\displaystyle=$	$\displaystyle\ \delta_{i3}u_{1}v_{2}-\delta_{i2}u_{1}v_{3}-\delta_{i3}u_{2}v_{1}$
		$\displaystyle+\delta_{i1}u_{2}v_{3}+\delta_{i2}u_{3}v_{1}-\delta_{i1}u_{3}v_{2}$
	$\displaystyle=$	$\displaystyle\ \delta_{i1}{\left(u_{2}v_{3}-u_{3}v_{2}\right)}$
		$\displaystyle+\delta_{i2}{\left(u_{3}v_{1}-u_{1}v_{3}\right)}$
		$\displaystyle+\delta_{i3}{\left(u_{1}v_{2}-u_{2}v_{1}\right)}$

or the components of the curl of a vector field

\displaystyle{\left[\nabla\times\boldsymbol{v}\right]}_{i}=\epsilon_{ijk}\frac% {\partial}{\partial r_{j}}v_{k}

To express double cross product other more complicated expressions we need the following important relation between the Kronecker Delta and the Levi-Civita-Symbol

\displaystyle\begin{split}\displaystyle\epsilon_{ijk}\epsilon_{lmn}=&% \displaystyle\begin{vmatrix}\delta_{il}&\delta_{im}&\delta_{in}\\ \delta_{jl}&\delta_{jm}&\delta_{jn}\\ \delta_{kl}&\delta_{km}&\delta_{kn}\end{vmatrix}\\ \displaystyle=&\displaystyle\ \delta_{il}\delta_{jm}\delta_{kn}+\delta_{im}% \delta_{jn}\delta_{kl}+\delta_{in}\delta_{jl}\delta_{km}\\ &\displaystyle-\delta_{in}\delta_{jm}\delta_{kl}-\delta_{il}\delta_{jn}\delta_% {km}-\delta_{im}\delta_{jl}\delta_{kn}\end{split}

(F.2)

From this relation we can derive the following

	$\displaystyle\begin{split}\displaystyle\epsilon_{ijk}\epsilon_{imn}=&% \displaystyle\ \delta_{ii}\delta_{jm}\delta_{kn}+\delta_{im}\delta_{jn}\delta_% {ki}+\delta_{in}\delta_{ji}\delta_{km}\\ &\displaystyle-\delta_{in}\delta_{jm}\delta_{ki}-\delta_{ii}\delta_{jn}\delta_% {km}-\delta_{im}\delta_{ji}\delta_{kn}\\ \displaystyle=&\displaystyle\ 3\delta_{jm}\delta_{kn}+\delta_{km}\delta_{jn}+% \delta_{jn}\delta_{km}\\ &\displaystyle-\delta_{kn}\delta_{jm}-3\delta_{jn}\delta_{km}-\delta_{jm}% \delta_{kn}\\ \displaystyle=&\displaystyle\ \delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}% \end{split}$		(F.3)
	$\displaystyle\begin{split}\displaystyle\epsilon_{ijk}\epsilon_{ijn}=&% \displaystyle\ \delta_{jj}\delta_{kn}-\delta_{jn}\delta_{kj}\\ \displaystyle=&\displaystyle\ 3\delta_{kn}-\delta_{kn}\\ \displaystyle=&\displaystyle\ 2\delta_{kn}\end{split}$		(F.4)
	$\displaystyle\begin{split}\displaystyle\epsilon_{ijk}\epsilon_{ijk}=&% \displaystyle\ 2\delta_{kk}\\ \displaystyle=&\displaystyle\ 6\end{split}$		(F.5)