\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

Examples of the Einstein Summation Convention

\[a_i \sim \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix} = a\] \[a_{ij} \sim \begin{bmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & A_{22} & A_{23}\\ A_{31} & A_{32} & A_{33} \end{bmatrix} =A\]

\begin{equation} R_{i}y_i= \sum_{i=1}^{n} x_iy_i =x_1y_1+x_2y_2+x_3y_3 \end{equation} \begin{equation} x_jy_j \end{equation} \begin{equation} xkyj \end{equation}

\begin{equation} m_iv^2 = \sum_{i=1}^{n} m_iv^i = m_1v^1 + m_2v^2 + m_3v^3 \end{equation}

\begin{equation} a_{ij}x_iy_j \Rightarrow a_{i1}x_iy_i + a_{i2}x_iy_2 + a_{i3}x_iy_3
= a_{11}x_1y_1 +a_{21}x_2y_1 + a_{31}x_3y_1 +
a_{12}x_1y_2 + a_{22}x_2y_2 + a_{32}x_3y_2 +
a_{13}x_1y_3 + a_{23}x_2y_3 + a_{33}x_3y_3 \end{equation}

\begin{equation}x_ia_jx_ja_i = \sum_{i=1}^{3} \sum_{j=1}^{3} (x_ia_jx_ja_i)
\end{equation} = \begin{equation} \sum_{i=1}^{3}x_ia_i(\sum_{j=1}^{3} x_ja_j) \end{equation} = \begin{equation} \sum_{i=1}^{3}x_ia_j \end{equation} = \begin{equation} (\sum_{i=1}^{3}x_ia_i)(\sum_{j=1}^{3}x_ja_j) \end{equation} = \begin{equation} (x_1a_1 + x_2a_2 + x_3a_3)(x_1a_1 + x_2a_2 + x_3a_3) \end{equation} =\begin{equation} (a_1)^2 (x_1)^2 + 2a_1a_2x_1x_2 + 2a_1a_3x_1x_3 + (a_2)^2(x_2)^2 +2 a_2a_3 x_2x_3 + (a_3)^2(x_3)^2 \end{equation}

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