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

Revision 3382 - (show annotations)
Thu Nov 25 00:43:18 2010 UTC (10 years ago) by jfenwick
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I've added better credits for dev team.
There is a new page just before the contents page with current dev team
and a more detailed list in the user guide appendix.

Please check this and let me know if you think there are errors or
omissions.

Also fixed a typo.
Removed reference to python style files we no longer use from the

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