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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 % http://www.uq.edu.au
5 lgraham 1709 %
6 ksteube 1811 % Primary Business: Queensland, Australia
7 jfenwick 6112 % Licensed under the Apache License, version 2.0
8     % http://www.apache.org/licenses/LICENSE-2.0
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

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