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MathGroup Archive 2005

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Einstein Summation with Utilities`Notation`

  • To: mathgroup at smc.vnet.net
  • Subject: [mg61003] Einstein Summation with Utilities`Notation`
  • From: "Steven T. Hatton" <hattons at globalsymmetry.com>
  • Date: Fri, 7 Oct 2005 03:37:47 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Has anybody worked out an implementation of Einstein summation notation
using Mathematica's Utilities`Notation` package?  I started toying with the
idea and realized it could become rather tedious to write all the possible
combinations of raised and lowered indices for tensors of higher order.  It
is also mathematical blasphemy to write this out explicitly for each
instance.

It appears to me that the ideal way of producing notations is to use vi or
better (a.k.a. (X)Emacs).  That is, the form produced using the Mathematica
front end is virtually unintelligible for modestly complex notations.  I'm
confident that creating notations with a traditional text editor can be
done using NotationBoxTag, though I haven't worked out the general
approach.

This is the general idea of what I've been attempting as it appears in
Notebookese: 

\!\(\*
  RowBox[{
    RowBox[{"Symbolize", "[", 
      TagBox[\(e\&^\),
        NotationBoxTag,
        TagStyle->"NotationTemplateStyle"], "]"}], "\[IndentingNewLine]", 
    RowBox[{"Notation", "[", 
      RowBox[{
        TagBox[\(\(e_\&^\)\_i_\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], " ", 
        "\[DoubleLongLeftRightArrow]", " ", 
        TagBox[\(\(e_\&^\)\[LeftDoubleBracket]i_\[RightDoubleBracket]\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"]}], "]"}],
"\[IndentingNewLine]", 
    
    RowBox[{"Notation", "[", 
      RowBox[{
        TagBox[\(\(e_\&^\)\_\(i_, j_\)\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], " ", 
        "\[DoubleLongLeftRightArrow]", " ", 
        TagBox[\(\(e_\&^\)\[LeftDoubleBracket]i_, j_\[RightDoubleBracket]\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"]}], "]"}], 
    "\[IndentingNewLine]", \(Bases[x_:  e1, y_:  e2, z_:  e3] := {x, y, 
        z}\)}]\)
-- 
"Philosophy is written in this grand book, The Universe. ... But the book
cannot be understood unless one first learns to comprehend the language...
in which it is written. It is written in the language of mathematics, ...;
without which wanders about in a dark labyrinth."   The Lion of Gaul


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