--- trunk/doc/user/notation.tex 2010/10/22 01:15:56 3294 +++ trunk/doc/user/notation.tex 2010/10/22 01:56:02 3295 @@ -19,42 +19,42 @@ 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; $b\hackscore{i}$ 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. 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: $$-y = a\hackscore{0}b\hackscore{0} + a\hackscore{1}b\hackscore{1} + \ldots + a\hackscore{n}b\hackscore{n} +y = a_{0}b_{0} + a_{1}b_{1} + \ldots + a_{n}b_{n} \label{NOTATION1}$$ can be represented as: $$-y = \sum\hackscore{i=0}^n a\hackscore{i}b\hackscore{i} +y = \sum_{i=0}^n a_{i}b_{i} \label{NOTATION2}$$ then in Einstein notion: $$-y = a\hackscore{i}b\hackscore{i} +y = a_{i}b_{i} \label{NOTATION3}$$ Another example: $$-\nabla p = \frac{\partial p}{\partial x\hackscore{0}}\textbf{i} + \frac{\partial p}{\partial x\hackscore{1}}\textbf{j} + \frac{\partial p}{\partial x\hackscore{2}}\textbf{k} +\nabla p = \frac{\partial p}{\partial x_{0}}\textbf{i} + \frac{\partial p}{\partial x_{1}}\textbf{j} + \frac{\partial p}{\partial x_{2}}\textbf{k} \label{NOTATION4}$$ can be expressed in Einstein notation as: $$-\nabla p = p,\hackscore{i} +\nabla p = p,_{i} \label{NOTATION5}$$ @@ -63,20 +63,20 @@ For a tensor: $$-\sigma \hackscore{ij}= +\sigma _{ij}= \left[ \begin{array}{ccc} -\sigma\hackscore{00} & \sigma\hackscore{01} & \sigma\hackscore{02} \\ -\sigma\hackscore{10} & \sigma\hackscore{11} & \sigma\hackscore{12} \\ -\sigma\hackscore{20} & \sigma\hackscore{21} & \sigma\hackscore{22} \\ +\sigma_{00} & \sigma_{01} & \sigma_{02} \\ +\sigma_{10} & \sigma_{11} & \sigma_{12} \\ +\sigma_{20} & \sigma_{21} & \sigma_{22} \\ \end{array} \right] \label{NOTATION6}$$ -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 ($i \neq j$). -\delta \hackscore{ij} = +\delta _{ij} = \left \{ \begin{array}{cc} 1, & \mbox{if $i = j$} \\ 0, & \mbox{if $i \neq j$} \\