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\chapter{Einstein Notation} 
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\label{EINSTEIN NOTATION} 
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Compact notation is used in equations such continuum mechanics and linear 
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algebra; it is known as Einstein notation or the Einstein summation convention. 
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It makes the conventional notation of equations involving tensors more compact 
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by shortening and simplifying them. 
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There are two rules which make up the convention. 
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Firstly, the rank of a tensor is represented by an index. 
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For example, $a$ is a scalar, $b_{i}$ represents a vector, and $c_{ij}$ 
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represents a matrix. 
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Secondly, if an expression contains repeated subscripted variables, they are 
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assumed to be summed over all possible values, from $0$ to $n$. 
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For example, the expression 
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\begin{equation} 
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y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n} 
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\label{NOTATION1} 
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\end{equation} 
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can be represented as 
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\begin{equation} 
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y = \sum_{i=0}^n a_{i}b_{i} 
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\label{NOTATION2} 
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\end{equation} 
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then in Einstein notation: 
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\begin{equation} 
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y = a_{i}b_{i} 
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\label{NOTATION3} 
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\end{equation} 
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% 
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Another example: 
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\begin{equation} 
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\nabla p = \frac{\partial p}{\partial x_{0}}\textbf{i} + \frac{\partial p}{\partial x_{1}}\textbf{j} + \frac{\partial p}{\partial x_{2}}\textbf{k} 
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\label{NOTATION4} 
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\end{equation} 
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can be expressed in Einstein notation as 
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\begin{equation} 
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\nabla p = p,_{i} 
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\label{NOTATION5} 
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\end{equation} 
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where the comma ',' in the subscript indicates the partial derivative. 
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\noindent For a tensor: 
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\begin{equation} 
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\sigma _{ij}= 
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\left[ \begin{array}{ccc} 
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\sigma_{00} & \sigma_{01} & \sigma_{02} \\ 
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\sigma_{10} & \sigma_{11} & \sigma_{12} \\ 
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\sigma_{20} & \sigma_{21} & \sigma_{22} \\ 
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\end{array} \right] 
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\label{NOTATION6} 
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\end{equation} 
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The $\delta_{ij}$ is the Kronecker $\delta$symbol, which is a matrix with ones 
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in its diagonal entries ($i = j$) and zeros in the remaining entries 
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($i \neq j$). 
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\begin{equation} 
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\delta _{ij} = 
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\left \{ \begin{array}{cc} 
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1, & \mbox{if $i = j$} \\ 
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0, & \mbox{if $i \neq j$} \\ 
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\end{array} 
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\right. 
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\label{KRONECKER} 
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\end{equation} 
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