Home > LaTeX, Mathematics > Einstein, Feynman, Kronecker and Levi-Civita

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}

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