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1
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 % Copyright (c) 2003-2018 by The University of Queensland
4 % http://www.uq.edu.au
5 %
6 % Primary Business: Queensland, Australia
7 % Licensed under the Apache License, version 2.0
8 % http://www.apache.org/licenses/LICENSE-2.0
9 %
10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC)
11 % Development 2012-2013 by School of Earth Sciences
12 % Development from 2014 by Centre for Geoscience Computing (GeoComp)
13 %
14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
15
16 \chapter{Einstein Notation}
17 \label{EINSTEIN NOTATION}
18
19 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
24 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 \begin{equation}
32 y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n}
33 \label{NOTATION1}
34 \end{equation}
35 can be represented as
36 \begin{equation}
37 y = \sum_{i=0}^n a_{i}b_{i}
38 \label{NOTATION2}
39 \end{equation}
40 then in Einstein notation:
41 \begin{equation}
42 y = a_{i}b_{i}
43 \label{NOTATION3}
44 \end{equation}
45 %
46 Another example:
47 \begin{equation}
48 \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 \label{NOTATION4}
50 \end{equation}
51 can be expressed in Einstein notation as
52 \begin{equation}
53 \nabla p = p,_{i}
54 \label{NOTATION5}
55 \end{equation}
56 where the comma ',' in the subscript indicates the partial derivative.
57
58 \noindent For a tensor:
59 \begin{equation}
60 \sigma _{ij}=
61 \left[ \begin{array}{ccc}
62 \sigma_{00} & \sigma_{01} & \sigma_{02} \\
63 \sigma_{10} & \sigma_{11} & \sigma_{12} \\
64 \sigma_{20} & \sigma_{21} & \sigma_{22} \\
65 \end{array} \right]
66 \label{NOTATION6}
67 \end{equation}
68
69 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
73 \begin{equation}
74 \delta _{ij} =
75 \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

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