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1 ahallam 2411
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15 ahallam 2411
16 ahallam 2975 \chapter{The Einstein Summation Convention}
17 ahallam 2411
18 caltinay 4286 The Einstein Summation Convention (ESC) is a notational convention that is preferred by the \esc developers. It is a condensed and practical way to deal with multi-dimensional and convoluted PDEs. By suppressing the need to write out the many terms of each problem it is possible to increase efficiency and reduce the number of errors created through poor working. According to the convention, when an index variable appears twice in a single term, it implies that we are summing over all of its possible values.
19 ahallam 2495 So we have;
20     \begin{equation}
21 jfenwick 3308 a_{1}\frac{\partial^2 f}{\partial x_{1}^2} + a_{2}\frac{\partial^2 f}{\partial x_{2}^2} = a_{i}\frac{\partial^2 f}{\partial x_{i}^2}
22 ahallam 2495 \end{equation}
23    
24 jfenwick 3308 For a scalar function $f(x_{1},x_{2},..x_{i})$ and a vector $\mathbf{u}(u_{1},u_{2},..u_{i})$ with $u_{i}(x_{1},x_{2},..x_{i})$, we have the following notation:
25 ahallam 2495 \begin{equation}
26 jfenwick 3308 \mathbf{u}=\sum_{i}u_{i}e^i = u_{i}e^i
27 ahallam 2495 \end{equation}
28     \begin{equation}
29 jfenwick 3308 \mathbf{grad}(f) = \mathbf{\nabla}(f) = \sum_{i}\frac{\partial f}{\partial x_{i}}e^i = (\partial_{i} f)e^i = f_{,i}e^i
30 ahallam 2495 \end{equation}
31     \begin{equation}
32 jfenwick 3308 div(\mathbf{u}) = \mathbf{\nabla}.\mathbf{u} = \sum_{i}\frac{\partial u_{i}}{\partial x_{i}} = \partial_{i} u_{i} = u_{i,i}
33 ahallam 2495 \end{equation}
34     \begin{equation}
35 jfenwick 3308 div(\mathbf{grad}(f)) = \nabla^2 f = \Delta f = \sum_{i}\frac{\partial^2 f}{\partial x_{i}^2} = f_{,ii}
36 ahallam 2495 \end{equation}

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