doc/user/notation.tex


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Copyright (c) 2003-2018 by The University of Queensland
% http://www.uq.edu.au
%
% Primary Business: Queensland, Australia
% Licensed under the Apache License, version 2.0
% http://www.apache.org/licenses/LICENSE-2.0
%
% Development until 2012 by Earth Systems Science Computational Center (ESSCC)
% Development 2012-2013 by School of Earth Sciences
% Development from 2014 by Centre for Geoscience Computing (GeoComp)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{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 involving tensors more compact
by shortening and simplifying them.

There are two rules which make up the convention.
Firstly, the rank of a 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, the 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 notation:
\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 ',' in the subscript indicates the partial derivative.

\noindent 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
in its diagonal entries ($i = j$) and zeros in 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}

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