/[escript]/trunk/doc/cookbook/einsteinETA.tex
ViewVC logotype

Annotation of /trunk/doc/cookbook/einsteinETA.tex

Parent Directory Parent Directory | Revision Log Revision Log


Revision 2632 - (hide annotations)
Wed Aug 26 22:18:19 2009 UTC (9 years, 10 months ago) by ahallam
File MIME type: application/x-tex
File size: 1768 byte(s)
Regigger of cookbook directory structure. Examlples->examples/cookbook TEXT->doc/cookbook Figures-> doc/cookbook/figures
1 ahallam 2411
2     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3     %
4     % Copyright (c) 2003-2009 by University of Queensland
5     % Earth Systems Science Computational Center (ESSCC)
6     % http://www.uq.edu.au/esscc
7     %
8     % Primary Business: Queensland, Australia
9     % Licensed under the Open Software License version 3.0
10     % http://www.opensource.org/licenses/osl-3.0.php
11     %
12     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13    
14 gross 2477 \section{The Einstein Summation Convention}
15 ahallam 2411
16 ahallam 2495 The Einstein Summation Convention (ESC) is a notational convention that is prefered by the \ESCRIPT 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.
17     So we have;
18     \begin{equation}
19     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}
20     \end{equation}
21    
22     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:
23     \begin{equation}
24     \mathbf{u}=\sum_{i}u_ie^i = u_ie^i
25     \end{equation}
26     \begin{equation}
27     \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
28     \end{equation}
29     \begin{equation}
30     div(\mathbf{u}) = \mathbf{\nabla}.\mathbf{u} = \sum_{i}\frac{\partial u_i}{\partial x_i} = \partial_i u_i = u_{i,i}
31     \end{equation}
32     \begin{equation}
33     div(\mathbf{grad}(f)) = \nabla^2 f = \Delta f = \sum_{i}\frac{\partial^2 f}{\partial x_i^2} = f_{,ii}
34     \end{equation}

  ViewVC Help
Powered by ViewVC 1.1.26