1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html401/loose.dtd">
<html>
<!-- Created on September, 20 2006 by texi2html 1.76 -->
<!--
Written by: Lionel Cons <Lionel.Cons@cern.ch> (original author)
Karl Berry <karl@freefriends.org>
Olaf Bachmann <obachman@mathematik.uni-kl.de>
and many others.
Maintained by: Many creative people <dev@texi2html.cvshome.org>
Send bugs and suggestions to <users@texi2html.cvshome.org>
-->
<head>
<title>Maxima Manual: 30. atensor</title>
<meta name="description" content="Maxima Manual: 30. atensor">
<meta name="keywords" content="Maxima Manual: 30. atensor">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="texi2html 1.76">
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
pre.display {font-family: serif}
pre.format {font-family: serif}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: serif; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: serif; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.sansserif {font-family:sans-serif; font-weight:normal;}
ul.toc {list-style: none}
body
{
color: black;
background: white;
margin-left: 8%;
margin-right: 13%;
}
h1
{
margin-left: +8%;
font-size: 150%;
font-family: sans-serif
}
h2
{
font-size: 125%;
font-family: sans-serif
}
h3
{
font-size: 100%;
font-family: sans-serif
}
h2,h3,h4,h5,h6 { margin-left: +4%; }
div.textbox
{
border: solid;
border-width: thin;
/* width: 100%; */
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em
}
div.titlebox
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em;
background: rgb(200,255,255);
font-family: sans-serif
}
div.synopsisbox
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em;
background: rgb(255,220,255);
/*background: rgb(200,255,255); */
/* font-family: fixed */
}
pre.example
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 1em;
padding-right: 1em;
background: rgb(247,242,180); /* kind of sandy */
/* background: rgb(200,255,255); */ /* sky blue */
font-family: "Lucida Console", monospace
}
div.spacerbox
{
border: none;
padding-top: 2em;
padding-bottom: 2em
}
div.image
{
margin: 0;
padding: 1em;
text-align: center;
}
-->
</style>
<link rel="icon" href="http://maxima.sourceforge.net/favicon.ico"/>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="atensor"></a>
<a name="SEC122"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="maxima_29.html#SEC121" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#SEC123" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima_29.html#SEC108" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_31.html#SEC125" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h1 class="chapter"> 30. atensor </h1>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC123">30.1 Introduction to atensor</a></td><td> </td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC124">30.2 Definitions for atensor</a></td><td> </td><td align="left" valign="top">
</td></tr>
</table>
<hr size="6">
<a name="Introduction-to-atensor"></a>
<a name="SEC123"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC122" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#SEC124" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="#SEC122" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="#SEC122" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_31.html#SEC125" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 30.1 Introduction to atensor </h2>
<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>
<table><tr><td> </td><td><pre class="example"> i = v j = v k = v . v
1 2 1 2
</pre></td></tr></table>
<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>
<table><tr><td> </td><td><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;
[ 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></td></tr></table>
<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>
<table><tr><td> </td><td><pre class="example">dotscrules:true;
dotdistrib:true;
dotexptsimp:false;
</pre></td></tr></table>
<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>
<hr size="6">
<a name="Definitions-for-atensor"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC123" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="maxima_31.html#SEC125" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="#SEC122" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="#SEC122" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_31.html#SEC125" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<a name="SEC124"></a>
<h2 class="section"> 30.2 Definitions for atensor </h2>
<dl>
<dt><u>Function:</u> <b>init_atensor</b><i> (<var>alg_type</var>, <var>opt_dims</var>)</i>
<a name="IDX1008"></a>
</dt>
<dt><u>Function:</u> <b>init_atensor</b><i> (<var>alg_type</var>)</i>
<a name="IDX1009"></a>
</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>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>atensimp</b><i> (<var>expr</var>)</i>
<a name="IDX1010"></a>
</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>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>alg_type</b>
<a name="IDX1011"></a>
</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>
</dd></dl>
<dl>
<dt><u>Variable:</u> <b>adim</b>
<a name="IDX1012"></a>
</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>
</dd></dl>
<dl>
<dt><u>Variable:</u> <b>aform</b>
<a name="IDX1013"></a>
</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>
</dd></dl>
<dl>
<dt><u>Variable:</u> <b>asymbol</b>
<a name="IDX1014"></a>
</dt>
<dd><p>Default value: <code>v</code>
</p>
<p>The symbol for base vectors..
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>sf</b><i> (<var>u</var>, <var>v</var>)</i>
<a name="IDX1015"></a>
</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>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>af</b><i> (<var>u</var>, <var>v</var>)</i>
<a name="IDX1016"></a>
</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>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>av</b><i> (<var>u</var>, <var>v</var>)</i>
<a name="IDX1017"></a>
</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>
<table><tr><td> </td><td><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></td></tr></table>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>abasep</b><i> (<var>v</var>)</i>
<a name="IDX1018"></a>
</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>
</dd></dl>
<hr size="6">
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC122" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="maxima_31.html#SEC125" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<p>
<font size="-1">
This document was generated by <em>Robert Dodier</em> on <em>September, 20 2006</em> using <a href="http://texi2html.cvshome.org/"><em>texi2html 1.76</em></a>.
</font>
<br>
</p>
</body>
</html>
|
