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\section{Einstein Notation} 
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\label{EINSTEIN NOTATION} 
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Compact notation is used in equations such continuum mechanics and linear algebra; it is known as Einstein notation or the Einstein summation convention. It makes the conventional notation of equations involing tensors more compact, 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 the tensor is represented by an index. For example, $a$ is a scalar; $b\hackscore{i}$ represents a vector; and $c\hackscore{ij}$ represents a matrix. 
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Secondly, if an expression contains subscripted variables, they are assumed to be summed over all possible values, from $0$ to $n$. For example, for the following expression: 
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\begin{equation} 
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y = a\hackscore{0}b\hackscore{0} + a\hackscore{1}b\hackscore{1} + \ldots + a\hackscore{n}b\hackscore{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\hackscore{i=0}^n a\hackscore{i}b\hackscore{i} 
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\label{NOTATION2} 
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\end{equation} 
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then in Einstein notion: 
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\begin{equation} 
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y = a\hackscore{i}b\hackscore{i} 
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\label{NOTATION3} 
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\end{equation} 
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Another example: 
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\begin{equation} 
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\nabla p = \frac{\partial p}{\partial x\hackscore{0}}\textbf{i} + \frac{\partial p}{\partial x\hackscore{1}}\textbf{j} + \frac{\partial p}{\partial x\hackscore{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,\hackscore{i} 
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\label{NOTATION5} 
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\end{equation} 
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where the comma ',' indicates the partial derivative. 
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For a tensor: 
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\begin{equation} 
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\sigma \hackscore{ij}= 
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\left[ \begin{array}{ccc} 
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\sigma\hackscore{00} & \sigma\hackscore{01} & \sigma\hackscore{02} \\ 
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\sigma\hackscore{10} & \sigma\hackscore{11} & \sigma\hackscore{12} \\ 
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\sigma\hackscore{20} & \sigma\hackscore{21} & \sigma\hackscore{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 $\delta\hackscore{ij}$ is the Kronecker $\delta$symbol, which is a matrix with ones for its diagonal entries ($i = j$) and zeros for the remaining entries ($i \neq j$). 
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\begin{equation} 
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\delta \hackscore{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} 