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<a name="Introduction-to-atensor"></a>
<div class="header">
<p>
Next: <a href="maxima_137.html#Functions-and-Variables-for-atensor" accesskey="n" rel="next">Functions and Variables for atensor</a>, Previous: <a href="maxima_135.html#atensor" accesskey="p" rel="previous">atensor</a>, Up: <a href="maxima_135.html#atensor" accesskey="u" rel="up">atensor</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Introduction-to-atensor-1"></a>
<h3 class="section">27.1 Introduction to atensor</h3>
<p><code>atensor</code> is an algebraic tensor manipulation package. To use <code>atensor</code>,
type <code>load("atensor")</code>, followed by a call to the <code>init_atensor</code>
function.
</p>
<p>The essence of <code>atensor</code> is a set of simplification rules for the
noncommutative (dot) product operator ("<code>.</code>"). <code>atensor</code> recognizes
several algebra types; the corresponding simplification rules are put
into effect when the <code>init_atensor</code> function is called.
</p>
<p>The capabilities of <code>atensor</code> can be demonstrated by defining the
algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis
vectors. The three quaternionic imaginary units are then the two
basis vectors and their product, i.e.:
</p>
<div class="example">
<pre class="example"> i = v j = v k = v . v
1 2 1 2
</pre></div>
<p>Although the <code>atensor</code> package has a built-in definition for the
quaternion algebra, it is not used in this example, in which we
endeavour to build the quaternion multiplication table as a matrix:
</p>
<div class="example">
<pre class="example">(%i1) load("atensor");
(%o1) /share/tensor/atensor.mac
(%i2) init_atensor(clifford,0,0,2);
(%o2) done
(%i3) atensimp(v[1].v[1]);
(%o3) - 1
(%i4) atensimp((v[1].v[2]).(v[1].v[2]));
(%o4) - 1
(%i5) q:zeromatrix(4,4);
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
(%o5) [ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
(%i6) q[1,1]:1;
(%o6) 1
(%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
(%o7) done
(%i8) q[1,4]:q[4,1]:v[1].v[2];
(%o8) v . v
1 2
(%i9) for i from 2 thru 4 do for j from 2 thru 4 do
q[i,j]:atensimp(q[i,1].q[1,j]);
(%o9) done
(%i10) q;
</pre><pre class="example"> [ 1 v v v . v ]
[ 1 2 1 2 ]
[ ]
[ v - 1 v . v - v ]
[ 1 1 2 2 ]
(%o10) [ ]
[ v - v . v - 1 v ]
[ 2 1 2 1 ]
[ ]
[ v . v v - v - 1 ]
[ 1 2 2 1 ]
</pre></div>
<p><code>atensor</code> recognizes as base vectors indexed symbols, where the symbol
is that stored in <code>asymbol</code> and the index runs between 1 and <code>adim</code>.
For indexed symbols, and indexed symbols only, the bilinear forms
<code>sf</code>, <code>af</code>, and <code>av</code> are evaluated. The evaluation
substitutes the value of <code>aform[i,j]</code> in place of <code>fun(v[i],v[j])</code>
where <code>v</code> represents the value of <code>asymbol</code> and <code>fun</code> is
either <code>af</code> or <code>sf</code>; or, it substitutes <code>v[aform[i,j]]</code>
in place of <code>av(v[i],v[j])</code>.
</p>
<p>Needless to say, the functions <code>sf</code>, <code>af</code> and <code>av</code>
can be redefined.
</p>
<p>When the <code>atensor</code> package is loaded, the following flags are set:
</p>
<div class="example">
<pre class="example">dotscrules:true;
dotdistrib:true;
dotexptsimp:false;
</pre></div>
<p>If you wish to experiment with a nonassociative algebra, you may also
consider setting <code>dotassoc</code> to <code>false</code>. In this case, however,
<code>atensimp</code> will not always be able to obtain the desired
simplifications.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Tensors">Tensors</a>
·<a href="maxima_424.html#Category_003a-Share-packages">Share packages</a>
·<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·</div>
<a name="Item_003a-Atensor_002fnode_002fFunctions-and-Variables-for-atensor"></a><hr>
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<p>
Next: <a href="maxima_137.html#Functions-and-Variables-for-atensor" accesskey="n" rel="next">Functions and Variables for atensor</a>, Previous: <a href="maxima_135.html#atensor" accesskey="p" rel="previous">atensor</a>, Up: <a href="maxima_135.html#atensor" accesskey="u" rel="up">atensor</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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