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## Einstein, Feynman, Kronecker and Levi-Civita

A number of notations exist to facilitate concise documentation and manipulation of vector identities. In this post I look at the vector identity:

$\vec{a}\times(\nabla \times \vec{b}) = \nabla_{b}(\vec{a}\cdot\vec{b})-(\vec{a}\cdot\nabla)\vec{b}$

$\nabla_b$ is the Feynman notation meaning the operator $\nabla$ acts on $b$.

Let $\vec{c} =\vec{a}\times(\nabla\times \vec{b})$.  The component $c_i$ of the $\vec{c}$ is :

$\begin{array}{lll}c_i&=&\epsilon_{ijk} a_j\epsilon_{klm}\frac{\partial b_m}{\partial x_l}\\&=&\epsilon_{ijk}\epsilon_{lmk}a_j\frac{\partial b_m}{\partial x_l}\\&=&(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})a_j\frac{\partial b_m}{\partial x_l}\\&=& a_j\frac{\partial b_j}{\partial x_i}-a_j\frac{\partial b_i}{\partial x_j}\\&=&\vec{a}\cdot (\frac{\partial \vec{b}}{\partial x_i}-\nabla b_i)\end{array}$

where the Einstein summation convention, the Kronecker delta and Levi-Civita symbols have been used.

The Feynman notation, therefore, means:

$\nabla_b (\vec{a}\cdot\vec{b})_i =a_j \frac{\partial b_j}{\partial x_i} =\vec{a}\cdot\frac{\partial \vec{b}}{\partial x_i}$