doc/user/notation.tex


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\section{Einstein Notation}
\label{EINSTEIN NOTATION}

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.

There are two rules which make up the convention:

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.

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:


\begin{equation}
y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n}
\label{NOTATION1}
\end{equation}

can be represented as:

\begin{equation}
y = \sum_{i=0}^n  a_{i}b_{i}
\label{NOTATION2}
\end{equation}

then in Einstein notion:

\begin{equation}
y = a_{i}b_{i}
\label{NOTATION3}
\end{equation}

Another example:

\begin{equation}
\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}
\label{NOTATION4}
\end{equation}

can be expressed in Einstein notation as:

\begin{equation}
\nabla p = p,_{i}
\label{NOTATION5}
\end{equation}

where the comma ',' indicates the partial derivative.

For a tensor:

\begin{equation}
\sigma _{ij}= 
\left[ \begin{array}{ccc}
\sigma_{00} & \sigma_{01} & \sigma_{02} \\
\sigma_{10} & \sigma_{11} & \sigma_{12} \\
\sigma_{20} & \sigma_{21} & \sigma_{22} \\
\end{array} \right]
\label{NOTATION6}
\end{equation}


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 ($i \neq j$).

\begin{equation}
\delta _{ij} = 
\left \{ \begin{array}{cc}
1, & \mbox{if $i = j$} \\
0, & \mbox{if $i \neq j$} \\
\end{array}
\right.
\label{KRONECKER}
\end{equation}
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3	lgraham	1709	%
4	jfenwick	2881	% Copyright (c) 2003-2010 by University of Queensland
5	ksteube	1811	% Earth Systems Science Computational Center (ESSCC)
6			% http://www.uq.edu.au/esscc
7	lgraham	1709	%
8	ksteube	1811	% Primary Business: Queensland, Australia
9			% Licensed under the Open Software License version 3.0
10			% http://www.opensource.org/licenses/osl-3.0.php
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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}