# Annotation of /trunk/doc/user/notation.tex

Revision 3295 - (hide annotations)
Fri Oct 22 01:56:02 2010 UTC (10 years, 2 months ago) by jfenwick
File MIME type: application/x-tex
File size: 2393 byte(s)

 1 ksteube 1811 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 lgraham 1709 % 4 jfenwick 2881 % Copyright (c) 2003-2010 by University of Queensland 5 ksteube 1811 % Earth Systems Science Computational Center (ESSCC) 6 7 lgraham 1709 % 8 ksteube 1811 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 lgraham 1709 % 12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 lgraham 1709 14 ksteube 1811 15 lgraham 1709 \section{Einstein Notation} 16 \label{EINSTEIN NOTATION} 17 18 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. 19 20 There are two rules which make up the convention: 21 22 jfenwick 3295 firstly, the rank of the tensor is represented by an index. For example, $a$ is a scalar; $b_{i}$ represents a vector; and $c_{ij}$ represents a matrix. 23 lgraham 1709 24 jfenwick 2923 Secondly, if an expression contains repeated subscripted variables, they are assumed to be summed over all possible values, from $0$ to $n$. For example, for the following expression: 25 lgraham 1709 26 27 28 \begin{equation} 29 jfenwick 3295 y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n} 30 lgraham 1709 \label{NOTATION1} 31 \end{equation} 32 33 can be represented as: 34 35 \begin{equation} 36 jfenwick 3295 y = \sum_{i=0}^n a_{i}b_{i} 37 lgraham 1709 \label{NOTATION2} 38 \end{equation} 39 40 then in Einstein notion: 41 42 \begin{equation} 43 jfenwick 3295 y = a_{i}b_{i} 44 lgraham 1709 \label{NOTATION3} 45 \end{equation} 46 47 Another example: 48 49 \begin{equation} 50 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} 51 lgraham 1709 \label{NOTATION4} 52 \end{equation} 53 54 can be expressed in Einstein notation as: 55 56 \begin{equation} 57 jfenwick 3295 \nabla p = p,_{i} 58 lgraham 1709 \label{NOTATION5} 59 \end{equation} 60 61 where the comma ',' indicates the partial derivative. 62 63 For a tensor: 64 65 \begin{equation} 66 jfenwick 3295 \sigma _{ij}= 67 lgraham 1709 \left[ \begin{array}{ccc} 68 jfenwick 3295 \sigma_{00} & \sigma_{01} & \sigma_{02} \\ 69 \sigma_{10} & \sigma_{11} & \sigma_{12} \\ 70 \sigma_{20} & \sigma_{21} & \sigma_{22} \\ 71 lgraham 1709 \end{array} \right] 72 \label{NOTATION6} 73 \end{equation} 74 75 76 jfenwick 3295 The $\delta_{ij}$ is the Kronecker $\delta$-symbol, which is a matrix with ones for its diagonal entries ($i = j$) and zeros for the remaining entries ($i \neq j$). 77 lgraham 1709 78 \begin{equation} 79 jfenwick 3295 \delta _{ij} = 80 lgraham 1709 \left \{ \begin{array}{cc} 81 1, & \mbox{if $i = j$} \\ 82 0, & \mbox{if $i \neq j$} \\ 83 \end{array} 84 \right. 85 \label{KRONECKER} 86 \end{equation}