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revision 3330 by jfenwick, Fri Oct 22 01:56:02 2010 UTC revision 3331 by caltinay, Mon Nov 1 05:21:18 2010 UTC
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11  %  %
12  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13    
   
14  \section{Einstein Notation}  \section{Einstein Notation}
15  \label{EINSTEIN NOTATION}  \label{EINSTEIN NOTATION}
16    
17  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.  Compact notation is used in equations such continuum mechanics and linear
18    algebra; it is known as Einstein notation or the Einstein summation convention.
19  There are two rules which make up the convention:  It makes the conventional notation of equations involving tensors more compact
20    by shortening and simplifying them.
21  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.  
22    There are two rules which make up the convention.
23  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:  Firstly, the rank of a tensor is represented by an index.
24    For example, $a$ is a scalar, $b_{i}$ represents a vector, and $c_{ij}$
25    represents a matrix.
26    Secondly, if an expression contains repeated subscripted variables, they are
27    assumed to be summed over all possible values, from $0$ to $n$.
28    For example, the expression
29  \begin{equation}  \begin{equation}
30  y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n}  y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n}
31  \label{NOTATION1}  \label{NOTATION1}
32  \end{equation}  \end{equation}
33    can be represented as
 can be represented as:  
   
34  \begin{equation}  \begin{equation}
35  y = \sum_{i=0}^n  a_{i}b_{i}  y = \sum_{i=0}^n  a_{i}b_{i}
36  \label{NOTATION2}  \label{NOTATION2}
37  \end{equation}  \end{equation}
38    then in Einstein notation:
 then in Einstein notion:  
   
39  \begin{equation}  \begin{equation}
40  y = a_{i}b_{i}  y = a_{i}b_{i}
41  \label{NOTATION3}  \label{NOTATION3}
42  \end{equation}  \end{equation}
43    %
44  Another example:  Another example:
   
45  \begin{equation}  \begin{equation}
46  \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}  \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}
47  \label{NOTATION4}  \label{NOTATION4}
48  \end{equation}  \end{equation}
49    can be expressed in Einstein notation as
 can be expressed in Einstein notation as:  
   
50  \begin{equation}  \begin{equation}
51  \nabla p = p,_{i}  \nabla p = p,_{i}
52  \label{NOTATION5}  \label{NOTATION5}
53  \end{equation}  \end{equation}
54    where the comma ',' in the subscript indicates the partial derivative.
55    
56  where the comma ',' indicates the partial derivative.  \noindent For a tensor:
   
 For a tensor:  
   
57  \begin{equation}  \begin{equation}
58  \sigma _{ij}=  \sigma _{ij}=
59  \left[ \begin{array}{ccc}  \left[ \begin{array}{ccc}
# Line 72  For a tensor: Line 64  For a tensor:
64  \label{NOTATION6}  \label{NOTATION6}
65  \end{equation}  \end{equation}
66    
67    The $\delta_{ij}$ is the Kronecker $\delta$-symbol, which is a matrix with ones
68  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 ($i \neq j$).  in its diagonal entries ($i = j$) and zeros in the remaining entries
69    ($i \neq j$).
70    
71  \begin{equation}  \begin{equation}
72  \delta _{ij} =  \delta _{ij} =
# Line 84  The $\delta_{ij}$ is the Kronecker $\del Line 77  The $\delta_{ij}$ is the Kronecker $\del
77  \right.  \right.
78  \label{KRONECKER}  \label{KRONECKER}
79  \end{equation}  \end{equation}
80    
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