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<a name="Functions-and-Variables-for-atensor"></a>
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<p>
Previous: <a href="maxima_136.html#Introduction-to-atensor" accesskey="p" rel="previous">Introduction to 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="Functions-and-Variables-for-atensor-1"></a>
<h3 class="section">27.2 Functions and Variables for atensor</h3>
<a name="init_005fatensor"></a><a name="Item_003a-Atensor_002fdeffn_002finit_005fatensor"></a><dl>
<dt><a name="index-init_005fatensor"></a>Function: <strong>init_atensor</strong> <em><br> <tt>init_atensor</tt> (<var>alg_type</var>, <var>opt_dims</var>) <br> <tt>init_atensor</tt> (<var>alg_type</var>)</em></dt>
<dd>
<p>Initializes the <code>atensor</code> package with the specified algebra type. <var>alg_type</var>
can be one of the following:
</p>
<p><code>universal</code>: The universal algebra has no commutation rules.
</p>
<p><code>grassmann</code>: The Grassman algebra is defined by the commutation
relation <code>u.v+v.u=0</code>.
</p>
<p><code>clifford</code>: The Clifford algebra is defined by the commutation
relation <code>u.v+v.u=-2*sf(u,v)</code> where <code>sf</code> is a symmetric
scalar-valued function. For this algebra, <var>opt_dims</var> can be up
to three nonnegative integers, representing the number of positive,
degenerate, and negative dimensions of the algebra, respectively. If
any <var>opt_dims</var> values are supplied, <code>atensor</code> will configure the
values of <code>adim</code> and <code>aform</code> appropriately. Otherwise,
<code>adim</code> will default to 0 and <code>aform</code> will not be defined.
</p>
<p><code>symmetric</code>: The symmetric algebra is defined by the commutation
relation <code>u.v-v.u=0</code>.
</p>
<p><code>symplectic</code>: The symplectic algebra is defined by the commutation
relation <code>u.v-v.u=2*af(u,v)</code> where <code>af</code> is an antisymmetric
scalar-valued function. For the symplectic algebra, <var>opt_dims</var> can
be up to two nonnegative integers, representing the nondegenerate and
degenerate dimensions, respectively. If any <var>opt_dims</var> values are
supplied, <code>atensor</code> will configure the values of <code>adim</code> and <code>aform</code>
appropriately. Otherwise, <code>adim</code> will default to 0 and <code>aform</code>
will not be defined.
</p>
<p><code>lie_envelop</code>: The algebra of the Lie envelope is defined by the
commutation relation <code>u.v-v.u=2*av(u,v)</code> where <code>av</code> is
an antisymmetric function.
</p>
<p>The <code>init_atensor</code> function also recognizes several predefined
algebra types:
</p>
<p><code>complex</code> implements the algebra of complex numbers as the
Clifford algebra Cl(0,1). The call <code>init_atensor(complex)</code> is
equivalent to <code>init_atensor(clifford,0,0,1)</code>.
</p>
<p><code>quaternion</code> implements the algebra of quaternions. The call
<code>init_atensor (quaternion)</code> is equivalent to
<code>init_atensor (clifford,0,0,2)</code>.
</p>
<p><code>pauli</code> implements the algebra of Pauli-spinors as the Clifford-algebra
Cl(3,0). A call to <code>init_atensor(pauli)</code> is equivalent to
<code>init_atensor(clifford,3)</code>.
</p>
<p><code>dirac</code> implements the algebra of Dirac-spinors as the Clifford-algebra
Cl(3,1). A call to <code>init_atensor(dirac)</code> is equivalent to
<code>init_atensor(clifford,3,0,1)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·</div></dd></dl>
<a name="atensimp"></a><a name="Item_003a-Atensor_002fdeffn_002fatensimp"></a><dl>
<dt><a name="index-atensimp"></a>Function: <strong>atensimp</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Simplifies an algebraic tensor expression <var>expr</var> according to the rules
configured by a call to <code>init_atensor</code>. Simplification includes
recursive application of commutation relations and resolving calls
to <code>sf</code>, <code>af</code>, and <code>av</code> where applicable. A
safeguard is used to ensure that the function always terminates, even
for complex expressions.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·<a href="maxima_424.html#Category_003a-Simplification-functions">Simplification functions</a>
·</div>
</dd></dl>
<a name="alg_005ftype"></a><a name="Item_003a-Atensor_002fdeffn_002falg_005ftype"></a><dl>
<dt><a name="index-alg_005ftype"></a>Function: <strong>alg_type</strong></dt>
<dd><p>The algebra type. Valid values are <code>universal</code>, <code>grassmann</code>,
<code>clifford</code>, <code>symmetric</code>, <code>symplectic</code> and <code>lie_envelop</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·</div>
</dd></dl>
<a name="adim"></a><a name="Item_003a-Atensor_002fdefvr_002fadim"></a><dl>
<dt><a name="index-adim"></a>Variable: <strong>adim</strong></dt>
<dd><p>Default value: 0
</p>
<p>The dimensionality of the algebra. <code>atensor</code> uses the value of <code>adim</code>
to determine if an indexed object is a valid base vector. See <code>abasep</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·<a href="maxima_424.html#Category_003a-Global-variables">Global variables</a>
·</div>
</dd></dl>
<a name="aform"></a><a name="Item_003a-Atensor_002fdefvr_002faform"></a><dl>
<dt><a name="index-aform"></a>Variable: <strong>aform</strong></dt>
<dd><p>Default value: <code>ident(3)</code>
</p>
<p>Default values for the bilinear forms <code>sf</code>, <code>af</code>, and
<code>av</code>. The default is the identity matrix <code>ident(3)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·<a href="maxima_424.html#Category_003a-Global-variables">Global variables</a>
·</div>
</dd></dl>
<a name="asymbol"></a><a name="Item_003a-Atensor_002fdefvr_002fasymbol"></a><dl>
<dt><a name="index-asymbol"></a>Variable: <strong>asymbol</strong></dt>
<dd><p>Default value: <code>v</code>
</p>
<p>The symbol for base vectors.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·<a href="maxima_424.html#Category_003a-Global-variables">Global variables</a>
·</div>
</dd></dl>
<a name="sf"></a><a name="Item_003a-Atensor_002fdeffn_002fsf"></a><dl>
<dt><a name="index-sf"></a>Function: <strong>sf</strong> <em>(<var>u</var>, <var>v</var>)</em></dt>
<dd>
<p>A symmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using <code>abasep</code> and if that is the case, substitutes the
corresponding value from the matrix <code>aform</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·</div>
</dd></dl>
<a name="af"></a><a name="Item_003a-Atensor_002fdeffn_002faf"></a><dl>
<dt><a name="index-af"></a>Function: <strong>af</strong> <em>(<var>u</var>, <var>v</var>)</em></dt>
<dd>
<p>An antisymmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using <code>abasep</code> and if that is the case, substitutes the
corresponding value from the matrix <code>aform</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·</div>
</dd></dl>
<a name="av"></a><a name="Item_003a-Atensor_002fdeffn_002fav"></a><dl>
<dt><a name="index-av"></a>Function: <strong>av</strong> <em>(<var>u</var>, <var>v</var>)</em></dt>
<dd>
<p>An antisymmetric function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using <code>abasep</code> and if that is the case, substitutes the
corresponding value from the matrix <code>aform</code>.
</p>
<p>For instance:
</p>
<div class="example">
<pre class="example">(%i1) load("atensor");
(%o1) /share/tensor/atensor.mac
(%i2) adim:3;
(%o2) 3
(%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
[ 0 3 - 2 ]
[ ]
(%o3) [ - 3 0 1 ]
[ ]
[ 2 - 1 0 ]
(%i4) asymbol:x;
(%o4) x
(%i5) av(x[1],x[2]);
(%o5) x
3
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·</div>
</dd></dl>
<a name="abasep"></a><a name="Item_003a-Atensor_002fdeffn_002fabasep"></a><dl>
<dt><a name="index-abasep"></a>Function: <strong>abasep</strong> <em>(<var>v</var>)</em></dt>
<dd>
<p>Checks if its argument is an <code>atensor</code> base vector. That is, if it is
an indexed symbol, with the symbol being the same as the value of
<code>asymbol</code>, and the index having a numeric value between 1
and <code>adim</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-atensor">Package atensor</a>
·<a href="maxima_424.html#Category_003a-Predicate-functions">Predicate functions</a>
·</div>
</dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="maxima_136.html#Introduction-to-atensor" accesskey="p" rel="previous">Introduction to 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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