[Maxima-commits] CVS: maximabook/share/tensor tensor.bbl,NONE,1.1 tensor.tex,NONE,1.1

[Cliff Y. <sta...@us...>]

Update of /cvsroot/maxima/maximabook/share/tensor
In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv20010

Added Files:
	tensor.bbl tensor.tex 
Log Message:
Adding Viktor T. Toth's tensor paper.  Needs to be edited and integrated 
with maximabook - at this point I'm just getting it into cvs in case I 
wipe out my computer.


--- NEW FILE: tensor.bbl ---
\begin{thebibliography}{10}

\bibitem{D1994}
Geoffrey~M. Dixon.
\newblock {\em Division Algebras: Octonions, Quaternions, Complex Numbers and
  the Algebraic Design of Physics}.
\newblock Kluwer Academic Publishers, 1994.

\bibitem{F2004}
Theodore Frankel.
\newblock {\em The Geometry of Physics}.
\newblock Cambridge University Press, second edition, 2004.

\bibitem{GS1989}
M.~G{\"o}ckeler and T.~Sch{\"u}cker.
\newblock {\em Differential geometry, gauge theories, and gravity}.
\newblock Cambridge University Press, 1989.

\bibitem{M1996}
Macsyma Inc.
\newblock {\em Macsyma Mathematics and System Reference Manual}.
\newblock Macsyma, Inc., 1996.

\bibitem{LL1975}
L.~D. Landau and E.~M. Lifshitz.
\newblock {\em Theoretical Physics}, volume II: Field Theory.
\newblock Nauka, 1975.

\bibitem{LR1989}
David Lovelock and Hanno Rund.
\newblock {\em Tensors, Differential Forms, and Variational Principles}.
\newblock Dover Publications, 1989.

\bibitem{PR1990}
Roger Penrose and Wolfgang Rindler.
\newblock {\em Spinors and space-time}, volume~1.
\newblock Cambridge University Press, 1990.

\bibitem{PSD2000}
D.~Pollney, J.~E.~F. Skea, and R.~A. d'Inverno.
\newblock Classifying geometries in general relativity: Iii. classification in
  practice.
\newblock {\em Class. Quant. Grav.}, 17:2885--2902, 2000.

\bibitem{SKMHH2003}
Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and
  Eduard Herlt.
\newblock {\em Exact Solutions to Einstein's Field Equations, Second Edition}.
\newblock Cambridge University Press, 2003.

\bibitem{W1984}
Robert~M. Wald.
\newblock {\em General Relativity}.
\newblock The University of Chicago Press, 1984.

\bibitem{JAW1984}
Joseph~A. Wolf.
\newblock {\em Spaces of Constant Curvature}.
\newblock Publish or Perish, 1984.

\end{thebibliography}

--- NEW FILE: tensor.tex ---
\documentclass[10pt]{article} \usepackage[papersize={8.5in,11in}]{geometry} \usepackage{bm} \usepackage{graphicx} \usepackage{longtable}
\geometry{left=0.75in,right=0.75in,top=0.75in,bottom=1.25in}
\renewcommand{\baselinestretch}{1}
\newcommand{\boxedeqn}[1]{%
  \[\fbox{%
    \addtolength{\linewidth}{-2\fboxsep}%
    \addtolength{\linewidth}{-2\fboxrule}%
    \begin{minipage}{\linewidth}%
    \begin{equation}#1\end{equation}%
    \end{minipage}%
  }\]%
}
\renewcommand{\vec}[1]{%
  \bm{\mathrm{#1}}%
}
\newenvironment{changemargin}[2]{%
 \begin{list}{}{%
  \setlength{\topsep}{0pt}%
  \setlength{\leftmargin}{#1}%
[...1126 lines suppressed...]
For a Clifford algebra, the matrix {\tt aform} is initialized as ${\rm diag}(1\ldots 1,0\ldots 0,-1\ldots-1)$, where the number of 1's, 0's, and $-1$'s correspond with the number of positive, degenerate, and negative dimensions, respectively.
\par
For a symplectic algebra of (regular) dimension $n$, {\tt aform} is initialized to an $n\times n$ matrix with its off-diagonal values set to 1 or $-1$ depending on whether they represent an odd or even index permutation. If the algebra also has degenerate dimenions, the appropriate number of null columns and rows are added.
\par
Lastly, for Lie enveloping algebras {\tt aform} is initialized to an antisymmetric matrix whose elements are defined as $a_{ij}=\left({\rm mod}_n(2n+2-i-j)+1\right){\rm perm}(i,j)$
where ${\rm perm}(i,j)$ is a permutator function that gives $+1$ if $(i,j,1\ldots i-1,i+1,\ldots j-1,j+1,\ldots n)$ is an even permutation of the sequence $(1\ldots n)$, and $-1$ otherwise. For instance, the initialization function call {\tt init\_atensor(lie\_envelop,3)} produces the following matrix:
$$
\pmatrix{0&3&-2\cr-3&0&1\cr2&-1&0}
$$
\par
It is known (see, for instance, \cite{D1994}) that the Clifford algebra of 0 positive and $-2$ negative dimensions corresponds with the algebra of quaternions. The ATENSOR package can correctly reproduce the quaternionic multiplication table as
$$
\pmatrix{1&v_1&v_2&v_1\cdot v_2\cr v_1&-1&v_1\cdot v_2&-v_2\cr v_2&-v_1\cdot v_2&-1&v_1\cr v_1\cdot v_2&v_2&-v_1&-1}
$$
where the base vectors $v_1$, $v_2$, and their product $v_1\cdot v_2$ serve as the three quaternionic imaginary units. This was one of the tests used to ascertain that the ATENSOR package produces mathematically correct results.
\section{Conclusions}
The Maxima tensor packages remain a work in progress. It is the author's desire to maintain these packages in good working order so long as GPL Maxima itself remains actively maintained. It is hoped that the tensor packages will again be accepted by a user community, and as they are actively ``field tested'', it will be possible to make them more robust and mathematically accurate.
\bibliography{../refs}
\bibliographystyle{plain}
\end{document}