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Direct tensor algebra

• To: mathgroup at smc.vnet.net
• Subject: [mg37385] Direct tensor algebra
• From: Alexey Skoblikov <alexey.skoblikov at compusoft.no>
• Date: Sat, 26 Oct 2002 02:03:08 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

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

• Follow-Ups:
  • Re: Direct tensor algebra
    • From: John Browne <jbrowne@swin.edu.au>