/[escript]/trunk/doc/user/notation.tex
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revision 2923 by jfenwick, Thu Feb 4 04:05:36 2010 UTC revision 3295 by jfenwick, Fri Oct 22 01:56:02 2010 UTC
# Line 19  Compact notation is used in equations su Line 19  Compact notation is used in equations su
19    
20  There are two rules which make up the convention:  There are two rules which make up the convention:
21    
22  firstly, the rank of the tensor is represented by an index. For example, $a$ is a scalar; $b\hackscore{i}$ represents a vector; and $c\hackscore{ij}$ represents a matrix.  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    
24  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:  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:
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26    
27    
28  \begin{equation}  \begin{equation}
29  y = a\hackscore{0}b\hackscore{0} + a\hackscore{1}b\hackscore{1} + \ldots + a\hackscore{n}b\hackscore{n}  y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n}
30  \label{NOTATION1}  \label{NOTATION1}
31  \end{equation}  \end{equation}
32    
33  can be represented as:  can be represented as:
34    
35  \begin{equation}  \begin{equation}
36  y = \sum\hackscore{i=0}^n  a\hackscore{i}b\hackscore{i}  y = \sum_{i=0}^n  a_{i}b_{i}
37  \label{NOTATION2}  \label{NOTATION2}
38  \end{equation}  \end{equation}
39    
40  then in Einstein notion:  then in Einstein notion:
41    
42  \begin{equation}  \begin{equation}
43  y = a\hackscore{i}b\hackscore{i}  y = a_{i}b_{i}
44  \label{NOTATION3}  \label{NOTATION3}
45  \end{equation}  \end{equation}
46    
47  Another example:  Another example:
48    
49  \begin{equation}  \begin{equation}
50  \nabla p = \frac{\partial p}{\partial x\hackscore{0}}\textbf{i} + \frac{\partial p}{\partial x\hackscore{1}}\textbf{j} + \frac{\partial p}{\partial x\hackscore{2}}\textbf{k}  \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  \label{NOTATION4}  \label{NOTATION4}
52  \end{equation}  \end{equation}
53    
54  can be expressed in Einstein notation as:  can be expressed in Einstein notation as:
55    
56  \begin{equation}  \begin{equation}
57  \nabla p = p,\hackscore{i}  \nabla p = p,_{i}
58  \label{NOTATION5}  \label{NOTATION5}
59  \end{equation}  \end{equation}
60    
# Line 63  where the comma ',' indicates the partia Line 63  where the comma ',' indicates the partia
63  For a tensor:  For a tensor:
64    
65  \begin{equation}  \begin{equation}
66  \sigma \hackscore{ij}=  \sigma _{ij}=
67  \left[ \begin{array}{ccc}  \left[ \begin{array}{ccc}
68  \sigma\hackscore{00} & \sigma\hackscore{01} & \sigma\hackscore{02} \\  \sigma_{00} & \sigma_{01} & \sigma_{02} \\
69  \sigma\hackscore{10} & \sigma\hackscore{11} & \sigma\hackscore{12} \\  \sigma_{10} & \sigma_{11} & \sigma_{12} \\
70  \sigma\hackscore{20} & \sigma\hackscore{21} & \sigma\hackscore{22} \\  \sigma_{20} & \sigma_{21} & \sigma_{22} \\
71  \end{array} \right]  \end{array} \right]
72  \label{NOTATION6}  \label{NOTATION6}
73  \end{equation}  \end{equation}
74    
75    
76  The $\delta\hackscore{ij}$ is the Kronecker $\delta$-symbol, which is a matrix with ones for its diagonal entries ($i = j$) and zeros for the remaining entries ($i \neq j$).  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    
78  \begin{equation}  \begin{equation}
79  \delta \hackscore{ij} =  \delta _{ij} =
80  \left \{ \begin{array}{cc}  \left \{ \begin{array}{cc}
81  1, & \mbox{if $i = j$} \\  1, & \mbox{if $i = j$} \\
82  0, & \mbox{if $i \neq j$} \\  0, & \mbox{if $i \neq j$} \\

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