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 1 ksteube 1811 2 jfenwick 3989 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 jfenwick 6651 % Copyright (c) 2003-2018 by The University of Queensland 4 jfenwick 3989 5 lgraham 1709 % 6 ksteube 1811 % Primary Business: Queensland, Australia 7 jfenwick 6112 % Licensed under the Apache License, version 2.0 8 9 lgraham 1709 % 10 jfenwick 3989 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 jfenwick 4657 % Development 2012-2013 by School of Earth Sciences 12 % Development from 2014 by Centre for Geoscience Computing (GeoComp) 13 jfenwick 3989 % 14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 15 lgraham 1709 16 jfenwick 3382 \chapter{Einstein Notation} 17 lgraham 1709 \label{EINSTEIN NOTATION} 18 19 caltinay 3331 Compact notation is used in equations such continuum mechanics and linear 20 algebra; it is known as Einstein notation or the Einstein summation convention. 21 It makes the conventional notation of equations involving tensors more compact 22 by shortening and simplifying them. 23 lgraham 1709 24 caltinay 3331 There are two rules which make up the convention. 25 Firstly, the rank of a tensor is represented by an index. 26 For example, $a$ is a scalar, $b_{i}$ represents a vector, and $c_{ij}$ 27 represents a matrix. 28 Secondly, if an expression contains repeated subscripted variables, they are 29 assumed to be summed over all possible values, from $0$ to $n$. 30 For example, the expression 31 lgraham 1709 \begin{equation} 32 jfenwick 3295 y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n} 33 lgraham 1709 \label{NOTATION1} 34 \end{equation} 35 caltinay 3331 can be represented as 36 lgraham 1709 \begin{equation} 37 jfenwick 3295 y = \sum_{i=0}^n a_{i}b_{i} 38 lgraham 1709 \label{NOTATION2} 39 \end{equation} 40 caltinay 3331 then in Einstein notation: 41 lgraham 1709 \begin{equation} 42 jfenwick 3295 y = a_{i}b_{i} 43 lgraham 1709 \label{NOTATION3} 44 \end{equation} 45 caltinay 3331 % 46 lgraham 1709 Another example: 47 \begin{equation} 48 jfenwick 3295 \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} 49 lgraham 1709 \label{NOTATION4} 50 \end{equation} 51 caltinay 3331 can be expressed in Einstein notation as 52 lgraham 1709 \begin{equation} 53 jfenwick 3295 \nabla p = p,_{i} 54 lgraham 1709 \label{NOTATION5} 55 \end{equation} 56 caltinay 3331 where the comma ',' in the subscript indicates the partial derivative. 57 lgraham 1709 58 caltinay 3331 \noindent For a tensor: 59 lgraham 1709 \begin{equation} 60 jfenwick 3295 \sigma _{ij}= 61 lgraham 1709 \left[ \begin{array}{ccc} 62 jfenwick 3295 \sigma_{00} & \sigma_{01} & \sigma_{02} \\ 63 \sigma_{10} & \sigma_{11} & \sigma_{12} \\ 64 \sigma_{20} & \sigma_{21} & \sigma_{22} \\ 65 lgraham 1709 \end{array} \right] 66 \label{NOTATION6} 67 \end{equation} 68 69 caltinay 3331 The $\delta_{ij}$ is the Kronecker $\delta$-symbol, which is a matrix with ones 70 in its diagonal entries ($i = j$) and zeros in the remaining entries 71 ($i \neq j$). 72 lgraham 1709 73 \begin{equation} 74 jfenwick 3295 \delta _{ij} = 75 lgraham 1709 \left \{ \begin{array}{cc} 76 1, & \mbox{if $i = j$} \\ 77 0, & \mbox{if $i \neq j$} \\ 78 \end{array} 79 \right. 80 \label{KRONECKER} 81 \end{equation} 82 caltinay 3331