- #1

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## Main Question or Discussion Point

There is a transformation:

[tex]\vec{E}\times\left(\nabla\times\vec{E}\right)=

\vec{k}_{i}\epsilon_{ijk}E_j\left(\nabla\times\vec{E}\right)_k=

\vec{k}_{i}\epsilon_{jk}E_j\epsilon_{klm}\nabla_l E_m=

\vec{k}_i\epsilon_{ijk}\epsilon_{klm}E_j\nabla_l E_m

=\vec{k}_i\left(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}\right)E_j\nabla_l E_m= [/tex]

[tex]=\vec{k}_i\delta_{il}\delta_{jm}E_j\nabla_l E_m-\vec{k}_i\delta_{im}\delta_{jl}E_j\nabla_l E_m=\vec{k}_i jE_j\nabla_i E_j-\vec{k}_i E_j\nabla_j E_i={{1}\over{2}}\nabla E^2-\left(\vec{E}\nabla\right)\vec{E} [/tex]

I think it is connected somehow to matrixes, but I do not get it at all. The way how it is done looks quite nice so maybe it is worth to learn this method. Can someone just help me with understanding it?

PS. Sorry if I posted it in a wrong forum, i just do not know where I am supposed to write it...

[tex]\vec{E}\times\left(\nabla\times\vec{E}\right)=

\vec{k}_{i}\epsilon_{ijk}E_j\left(\nabla\times\vec{E}\right)_k=

\vec{k}_{i}\epsilon_{jk}E_j\epsilon_{klm}\nabla_l E_m=

\vec{k}_i\epsilon_{ijk}\epsilon_{klm}E_j\nabla_l E_m

=\vec{k}_i\left(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}\right)E_j\nabla_l E_m= [/tex]

[tex]=\vec{k}_i\delta_{il}\delta_{jm}E_j\nabla_l E_m-\vec{k}_i\delta_{im}\delta_{jl}E_j\nabla_l E_m=\vec{k}_i jE_j\nabla_i E_j-\vec{k}_i E_j\nabla_j E_i={{1}\over{2}}\nabla E^2-\left(\vec{E}\nabla\right)\vec{E} [/tex]

I think it is connected somehow to matrixes, but I do not get it at all. The way how it is done looks quite nice so maybe it is worth to learn this method. Can someone just help me with understanding it?

PS. Sorry if I posted it in a wrong forum, i just do not know where I am supposed to write it...