% this tex file was generated by the Physics Derivation Graph 
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage[dvipdfmx,colorlinks=true,pdfkeywords={physics derivation graph}]{hyperref}
\usepackage{graphicx} % for including PNG files
% inference rules as newcommand for use in the body
\newcommand\declareidentity[1]{Eq.~\ref{eq:#1} is an identity.}
\newcommand\replacecurlwithLeviCevitasummationcontravariant[2]{Replace curl in Eq.~\ref{eq:#1} with Levi-Cevita contravariant; yields Eq.~\ref{eq:#2}.}
\newcommand\substituteRHSofexprintoexpr[3]{Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.}
\newcommand\simplify[2]{Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.}
\newcommand\replacesummationnotationwithvectornotation[2]{Replace summation notation in Eq.~\ref{eq:#1} with vector notation; yields Eq.~\ref{eq:#2}.}
\newcommand\claimLHSequalsRHS[1]{Thus we see that LHS of Eq.~\ref{eq:#1} is equal to RHS.}
\title{curl curl identity}
\date{\today}
%\author{a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}
\begin{document}
\maketitle
\begin{abstract}
Generated by the \href{http://allofphysics.com/}{Physics Derivation Graph}.
\end{abstract}

% step ID = 0003948572
\declareidentity{557e0500d6bdafec0d1c1c80da47a04a5c9fa526ed234342c0300418d2ed8a95}
\begin{equation}
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
\label{eq:557e0500d6bdafec0d1c1c80da47a04a5c9fa526ed234342c0300418d2ed8a95}
\end{equation}

% step ID = 0002339482
\replacecurlwithLeviCevitasummationcontravariant{557e0500d6bdafec0d1c1c80da47a04a5c9fa526ed234342c0300418d2ed8a95}{07d000c06849fed374eb2954327952468a414687a27d4e11e48090bfa2ce6def}
\begin{equation}
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
\label{eq:07d000c06849fed374eb2954327952468a414687a27d4e11e48090bfa2ce6def}
\end{equation}

% step ID = 0003948552
\replacecurlwithLeviCevitasummationcontravariant{07d000c06849fed374eb2954327952468a414687a27d4e11e48090bfa2ce6def}{317a953ea3c40fe5855319d1ba819732e56e1699f047314537f1c45aec556a58}
\begin{equation}
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
\label{eq:317a953ea3c40fe5855319d1ba819732e56e1699f047314537f1c45aec556a58}
\end{equation}

% step ID = 0004295822
\declareidentity{2a343fef4a1d0b32da87dbdd08c2314bc838bb85edbd7e1c8afca76d2e9f784e}
\begin{equation}
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
\label{eq:2a343fef4a1d0b32da87dbdd08c2314bc838bb85edbd7e1c8afca76d2e9f784e}
\end{equation}

% step ID = 0002930454
\substituteRHSofexprintoexpr{317a953ea3c40fe5855319d1ba819732e56e1699f047314537f1c45aec556a58}{2a343fef4a1d0b32da87dbdd08c2314bc838bb85edbd7e1c8afca76d2e9f784e}{5421e8e4c07d7a26d623c0af8dc63bbfd2bfe2fee38ead6b603dca9d2167ac4f}
\begin{equation}
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
\label{eq:5421e8e4c07d7a26d623c0af8dc63bbfd2bfe2fee38ead6b603dca9d2167ac4f}
\end{equation}

% step ID = 0003848292
\simplify{5421e8e4c07d7a26d623c0af8dc63bbfd2bfe2fee38ead6b603dca9d2167ac4f}{4c31b6c87b4f69572fbeaba68b9172116a349d5a2329377ed1069fb05b24f5f8}
\begin{equation}
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
\label{eq:4c31b6c87b4f69572fbeaba68b9172116a349d5a2329377ed1069fb05b24f5f8}
\end{equation}

% step ID = 0003838233
\simplify{4c31b6c87b4f69572fbeaba68b9172116a349d5a2329377ed1069fb05b24f5f8}{f9398ae2f53749b1a524f1093a2d60074a31c892267cbe842b93b7cb47fea501}
\begin{equation}
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
\label{eq:f9398ae2f53749b1a524f1093a2d60074a31c892267cbe842b93b7cb47fea501}
\end{equation}

% step ID = 0001393411
\replacesummationnotationwithvectornotation{f9398ae2f53749b1a524f1093a2d60074a31c892267cbe842b93b7cb47fea501}{a9a04bc9d22bdd2f91bec55c2a5752aa2227ce8baa64ff1ce12b9c32b5f51cd4}
\begin{equation}
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
\label{eq:a9a04bc9d22bdd2f91bec55c2a5752aa2227ce8baa64ff1ce12b9c32b5f51cd4}
\end{equation}

% step ID = 0003949211
\claimLHSequalsRHS{a9a04bc9d22bdd2f91bec55c2a5752aa2227ce8baa64ff1ce12b9c32b5f51cd4}
\bibliographystyle{plain}
\bibliography{pdg_derivation_citations.bib}
\end{document}
% EOF