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Fwd: Re: Direct tensor algebra

  • To: mathgroup at smc.vnet.net
  • Subject: [mg37519] Fwd: [mg37423] Re: [mg37385] Direct tensor algebra
  • From: Garry Helzer <gah at math.umd.edu>
  • Date: Sat, 2 Nov 2002 03:31:50 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Alexey,

You probably want some sort of rule based implementation, though these 
can be slow. John Browne mentions Grassmann algebra below but his code 
is not yet available. You can find a rules based implementation of 
Grassman algebra--in the G.-C. Rota, et al version referred to as Peano 
algebra--in the package Peano.m that accompanies my Geometry and 
Computer Graphics lecture notes at

http://www.math.umd.edu/~gah/Pages/Math431Lecturesf02.html

This implementation is not "really" coordinate free since the 
operations are defined on basis vectors and extended by rules to be 
bilinear, but it shows how to use such rules to define functions.

Note: The notation is different in the Peano algera context. The 
exterior product is denoted by \[Vee], instead of \[Wedge] (which 
computes intersections) and the Clifford product is \[CircleDot]



Begin forwarded message:

> From: John Browne <jbrowne at swin.edu.au>
To: mathgroup at smc.vnet.net
> Date: Mon Oct 28, 2002  3:40:41  AM US/Eastern
> To: mathgroup at smc.vnet.net
> Subject: [mg37519] [mg37423] Re: [mg37385] Direct tensor algebra
> Reply-To: jbrowne at swin.edu.au
>
> Alexey,
>
> Below is a link to the draft of a book on Grassmann algebra. Although 
> not
> the full tensor algebra, it might give you some ideas for what can be
> achieved in Mathematica. I think you'll find Mathematica is ideal for
> encoding mathematical systems. The actual Grassmann algebra code is 
> still
> being finalized, but should be available early next year.
>
> http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/index.htm
>
> John
>
>
> Alexey Skoblikov wrote:
>
>> Dear colleagues.
>>
>> Can anyone help me with making direct tensor algebra in Mathematica?
>>
>> In direct tensor algebra tensors are not components. Tensors are 
>> special
>> objects, that could be presented in the component form in sonme basis,
>> but even then they dont appear as S_{mn}, but as
>> S_{mn}r^m r^n
>> where r^m and r^n - are vectors of reciprocal basis and S_{mn} -
>> covariant components. NB! Basis vectors, are not columns like {1,0,0},
>> but exaclty vectors i.e. "directed line segment" as is.
>>
>> Is it possible to make this kind of package in Mathematica, that could
>> deal with such objects and also go to the component form - on the 
>> lower
>> level of abstraction - on demand.
>>
>> In particular such system would calculate that
>> a . b x a = 0 (mixed product - cross and dot) WITHOUT making the
>> constructs like
>> E^{mnk}b_m a_n a_k, where E^{mnk} - Levi-Chivitta symbols.
>>
>> The example is on leshakk.chat.ru - file tensor.pdf
>
> -- _________________________________
> John Browne
> School of Engineering and Science
> Swinburne University of Technology
> John Street, Hawthorn, Victoria, Australia
> Quantica phone: +613 9431 4007
> Quantica fax: +613 9431 0940
> Email: jbrowne at swin.edu.au
>
>
>
Garry Helzer
Department of  Mathematics
University of Maryland
1303 Math Bldg
College Park, MD 20742-4015



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