Changes

Figures&nbsp;37-a through 37-d present several pictures of the proposition ''J'' and its tacit extension <math>\epsilon</math>''J''.  Notice in these Figures how <math>\epsilon</math>''J'' in E''U''^&nbsp;&bull; visibly extends ''J'' in ''U''^&nbsp;&bull;, by annexing to the indicated cells of ''J'' all of the arcs that exit from or flow out of them.  In effect, this extension attaches to these cells all of the dispositions that spring from them, in other words, it attributes to these cells all of the conceivable changes that are their issue.

Figures&nbsp;37-a through 37-d present several pictures of the proposition ''J'' and its tacit extension <math>\epsilon</math>''J''.  Notice in these Figures how <math>\epsilon</math>''J'' in E''U''^&nbsp;&bull; visibly extends ''J'' in ''U''^&nbsp;&bull;, by annexing to the indicated cells of ''J'' all of the arcs that exit from or flow out of them.  In effect, this extension attaches to these cells all of the dispositions that spring from them, in other words, it attributes to these cells all of the conceivable changes that are their issue.

<pre>

<br>

o---------------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]</p>

|                                       |

<p><center><font size="+1">'''Figure 37-a.  Tacit Extension of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p>

|                  /%\                  |

|  o---+---o      o      o---+---o  |

Figure 37-a.  Tacit Extension of J (Areal)

</pre>

<pre>

<br>

                                                  o-----------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]</p>

                                                  |                             |

<p><center><font size="+1">'''Figure 37-b.  Tacit Extension of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>

                                                  |    /       \ /      \    |

Figure 37-b.  Tacit Extension of J (Bundle)

</pre>

<pre>

<br>

o---------------------------------------------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]</p>

|                                                                     |

<p><center><font size="+1">'''Figure 37-c.  Tacit Extension of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p>

|          /                     \ /                    \          |

|    |                      |  -->--  |                      |    |

|    |      u      <---------------@--------------->     v      |    |

Figure 37-c.  Tacit Extension of J (Compact)

</pre>

<pre>

<br>

o-----------------------------------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]</p>

<p><center><font size="+1">'''Figure 37-d.  Tacit Extension of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>

|                        (du).(dv)                        |

|                          --->---                          |

Figure 37-d.  Tacit Extension of J (Digraph)

</pre>

The computational scheme that was shown in Table&nbsp;36 treated ''J'' as a proposition in ''U''^&nbsp;&bull; and formed <math>\epsilon</math>''J'' as a proposition in E''U''^&nbsp;&bull;.  When ''J'' is regarded as a mapping ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull; then <math>\epsilon</math>''J'' must be obtained as a mapping <math>\epsilon</math>''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;.  By default, the tacit extension of the map ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''] is naturally taken to be a particular map, of the following form:

The computational scheme that was shown in Table&nbsp;36 treated ''J'' as a proposition in ''U''^&nbsp;&bull; and formed <math>\epsilon</math>''J'' as a proposition in E''U''^&nbsp;&bull;.  When ''J'' is regarded as a mapping ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull; then <math>\epsilon</math>''J'' must be obtained as a mapping <math>\epsilon</math>''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;.  By default, the tacit extension of the map ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''] is naturally taken to be a particular map, of the following form: