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User:Jon Awbrey/ATLAS (view source)

Revision as of 15:00, 25 August 2007

23,594 bytes added ,  15:00, 25 August 2007
add image stub
Line 1,141: Line 1,141:  
Figure 12.  The Anchor
 
Figure 12.  The Anchor
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 12.  The Anchor'''</font></center></p>
    
===Figure 13.  The Tiller===
 
===Figure 13.  The Tiller===
Line 1,174: Line 1,178:  
Figure 13.  The Tiller
 
Figure 13.  The Tiller
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 13.  The Tiller'''</font></center></p>
    
===Table 14.  Differential Propositions===
 
===Table 14.  Differential Propositions===
Line 1,667: Line 1,675:  
|}
 
|}
 
</font><br>
 
</font><br>
 +
 +
===Figure 16.  A Couple of Fourth Gear Orbits===
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 16.  A Couple of Fourth Gear Orbits'''</font></center></p>
    
===Figure 16-a.  A Couple of Fourth Gear Orbits:  1===
 
===Figure 16-a.  A Couple of Fourth Gear Orbits:  1===
Line 2,064: Line 2,078:  
Figure 18-a.  Extension from 1 to 2 Dimensions:  Areal
 
Figure 18-a.  Extension from 1 to 2 Dimensions:  Areal
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 18-a.  Extension from 1 to 2 Dimensions:  Areal'''</font></center></p>
    
===Figure 18-b.  Extension from 1 to 2 Dimensions:  Bundle===
 
===Figure 18-b.  Extension from 1 to 2 Dimensions:  Bundle===
Line 2,093: Line 2,111:  
Figure 18-b.  Extension from 1 to 2 Dimensions:  Bundle
 
Figure 18-b.  Extension from 1 to 2 Dimensions:  Bundle
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 18-b.  Extension from 1 to 2 Dimensions:  Bundle'''</font></center></p>
    
===Figure 18-c.  Extension from 1 to 2 Dimensions:  Compact===
 
===Figure 18-c.  Extension from 1 to 2 Dimensions:  Compact===
Line 2,124: Line 2,146:  
Figure 18-c.  Extension from 1 to 2 Dimensions:  Compact
 
Figure 18-c.  Extension from 1 to 2 Dimensions:  Compact
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 18-c.  Extension from 1 to 2 Dimensions:  Compact'''</font></center></p>
    
===Figure 18-d.  Extension from 1 to 2 Dimensions:  Digraph===
 
===Figure 18-d.  Extension from 1 to 2 Dimensions:  Digraph===
Line 2,143: Line 2,169:  
Figure 18-d.  Extension from 1 to 2 Dimensions:  Digraph
 
Figure 18-d.  Extension from 1 to 2 Dimensions:  Digraph
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 18-d.  Extension from 1 to 2 Dimensions:  Digraph'''</font></center></p>
    
===Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal===
 
===Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal===
Line 2,186: Line 2,216:  
Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal
 
Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal'''</font></center></p>
    
===Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle===
 
===Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle===
Line 2,247: Line 2,281:  
Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle
 
Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle'''</font></center></p>
    
===Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact===
 
===Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact===
Line 2,287: Line 2,325:  
Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact
 
Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact'''</font></center></p>
    
===Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph===
 
===Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph===
Line 2,330: Line 2,372:  
Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph
 
Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph'''</font></center></p>
    
===Figure 20-i.  Thematization of Conjunction (Stage 1)===
 
===Figure 20-i.  Thematization of Conjunction (Stage 1)===
Line 2,360: Line 2,406:  
Figure 20-i.  Thematization of Conjunction (Stage 1)
 
Figure 20-i.  Thematization of Conjunction (Stage 1)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 20-i.  Thematization of Conjunction (Stage 1)'''</font></center></p>
    
===Figure 20-ii.  Thematization of Conjunction (Stage 2)===
 
===Figure 20-ii.  Thematization of Conjunction (Stage 2)===
Line 2,407: Line 2,457:  
Figure 20-ii.  Thematization of Conjunction (Stage 2)
 
Figure 20-ii.  Thematization of Conjunction (Stage 2)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 20-ii.  Thematization of Conjunction (Stage 2)'''</font></center></p>
    
===Figure 20-iii.  Thematization of Conjunction (Stage 3)===
 
===Figure 20-iii.  Thematization of Conjunction (Stage 3)===
Line 2,450: Line 2,504:  
Figure 20-iii.  Thematization of Conjunction (Stage 3)
 
Figure 20-iii.  Thematization of Conjunction (Stage 3)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 20-iii.  Thematization of Conjunction (Stage 3)'''</font></center></p>
    
===Figure 21.  Thematization of Disjunction and Equality===
 
===Figure 21.  Thematization of Disjunction and Equality===
Line 2,516: Line 2,574:  
Figure 21.  Thematization of Disjunction and Equality
 
Figure 21.  Thematization of Disjunction and Equality
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 21.  Thematization of Disjunction and Equality'''</font></center></p>
    
===Table 22.  Disjunction ''f'' and Equality ''g''===
 
===Table 22.  Disjunction ''f'' and Equality ''g''===
Line 3,673: Line 3,735:  
Figure 30.  Generic Frame of a Logical Transformation
 
Figure 30.  Generic Frame of a Logical Transformation
 
</pre>
 
</pre>
 +
 +
'''Note.'''  The following image was corrupted in transit between software platforms.
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 30 -- Generic Frame of a Logical Transformation.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 30.  Generic Frame of a Logical Transformation'''</font></center></p>
    
===Formula Display 3===
 
===Formula Display 3===
Line 3,729: Line 3,797:  
Figure 31.  Operator Diagram (1)
 
Figure 31.  Operator Diagram (1)
 
</pre>
 
</pre>
 +
 +
'''Note.'''  The following image was corrupted in transit between software platforms.
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 31 -- Operator Diagram (1).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 31.  Operator Diagram (1)'''</font></center></p>
    
===Figure 32.  Operator Diagram (2)===
 
===Figure 32.  Operator Diagram (2)===
Line 3,754: Line 3,828:  
Figure 32.  Operator Diagram (2)
 
Figure 32.  Operator Diagram (2)
 
</pre>
 
</pre>
 +
 +
'''Note.'''  The following image was corrupted in transit between software platforms.
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 32 -- Operator Diagram (2).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 32.  Operator Diagram (2)'''</font></center></p>
    
===Figure 33-i.  Analytic Diagram (1)===
 
===Figure 33-i.  Analytic Diagram (1)===
Line 3,774: Line 3,854:  
Figure 33-i.  Analytic Diagram (1)
 
Figure 33-i.  Analytic Diagram (1)
 
</pre>
 
</pre>
 +
 +
'''Note.'''  The following image was corrupted in transit between software platforms.
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 33-i -- Analytic Diagram (1).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 33-i.  Analytic Diagram (1)'''</font></center></p>
    
===Figure 33-ii.  Analytic Diagram (2)===
 
===Figure 33-ii.  Analytic Diagram (2)===
Line 3,794: Line 3,880:  
Figure 33-ii.  Analytic Diagram (2)
 
Figure 33-ii.  Analytic Diagram (2)
 
</pre>
 
</pre>
 +
 +
'''Note.'''  The following image was corrupted in transit between software platforms.
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 33-ii -- Analytic Diagram (2).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 33-ii.  Analytic Diagram (2)'''</font></center></p>
    
===Formula Display 4===
 
===Formula Display 4===
Line 4,012: Line 4,104:  
Figure 34.  Tangent Functor Diagram
 
Figure 34.  Tangent Functor Diagram
 
</pre>
 
</pre>
 +
 +
'''Note.'''  The following image was corrupted in transit between software platforms.
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 34 -- Tangent Functor Diagram.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 34.  Tangent Functor Diagram'''</font></center></p>
    
===Figure 35.  Conjunction as Transformation===
 
===Figure 35.  Conjunction as Transformation===
Line 4,067: Line 4,165:  
Figure 35.  Conjunction as Transformation
 
Figure 35.  Conjunction as Transformation
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 35.  Conjunction as Transformation'''</font></center></p>
    
===Table 36.  Computation of !e!J===
 
===Table 36.  Computation of !e!J===
Line 4,140: Line 4,242:  
</font><br>
 
</font><br>
   −
===Figure 37-a.  Tacit Extension of J (Areal)===
+
===Figure 37-a.  Tacit Extension of ''J''&nbsp;&nbsp;(Areal)===
    
<pre>
 
<pre>
Line 4,183: Line 4,285:  
</pre>
 
</pre>
   −
===Figure 37-b.  Tacit Extension of J (Bundle)===
+
<br>
 +
<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>
 +
 
 +
===Figure 37-b.  Tacit Extension of ''J''&nbsp;&nbsp;(Bundle)===
    
<pre>
 
<pre>
Line 4,252: Line 4,358:  
</pre>
 
</pre>
   −
===Figure 37-c.  Tacit Extension of J (Compact)===
+
<br>
 +
<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-c.  Tacit Extension of ''J''&nbsp;&nbsp;(Compact)===
    
<pre>
 
<pre>
Line 4,292: Line 4,402:  
</pre>
 
</pre>
   −
===Figure 37-d.  Tacit Extension of J (Digraph)===
+
<br>
 +
<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>
 +
 
 +
===Figure 37-d.  Tacit Extension of ''J''&nbsp;&nbsp;(Digraph)===
    
<pre>
 
<pre>
Line 4,333: Line 4,447:  
Figure 37-d.  Tacit Extension of J (Digraph)
 
Figure 37-d.  Tacit Extension of J (Digraph)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<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>
    
===Table 38.  Computation of EJ (Method 1)===
 
===Table 38.  Computation of EJ (Method 1)===
Line 4,504: Line 4,622:  
</font><br>
 
</font><br>
   −
===Figure 40-a.  Enlargement of J (Areal)===
+
===Figure 40-a.  Enlargement of ''J''&nbsp;&nbsp;(Areal)===
    
<pre>
 
<pre>
Line 4,547: Line 4,665:  
</pre>
 
</pre>
   −
===Figure 40-b.  Enlargement of J (Bundle)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 40-a.  Enlargement of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p>
 +
 
 +
===Figure 40-b.  Enlargement of ''J''&nbsp;&nbsp;(Bundle)===
    
<pre>
 
<pre>
Line 4,616: Line 4,738:  
</pre>
 
</pre>
   −
===Figure 40-c.  Enlargement of J (Compact)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 40-b.  Enlargement of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>
 +
 
 +
===Figure 40-c.  Enlargement of ''J''&nbsp;&nbsp;(Compact)===
    
<pre>
 
<pre>
Line 4,656: Line 4,782:  
</pre>
 
</pre>
   −
===Figure 40-d.  Enlargement of J (Digraph)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 40-c.  Enlargement of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p>
 +
 
 +
===Figure 40-d.  Enlargement of ''J''&nbsp;&nbsp;(Digraph)===
    
<pre>
 
<pre>
Line 4,697: Line 4,827:  
Figure 40-d.  Enlargement of J (Digraph)
 
Figure 40-d.  Enlargement of J (Digraph)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 40-d.  Enlargement of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>
    
===Table 41.  Computation of DJ (Method 1)===
 
===Table 41.  Computation of DJ (Method 1)===
Line 4,964: Line 5,098:  
</font><br>
 
</font><br>
   −
===Figure 44-a.  Difference Map of J (Areal)===
+
===Figure 44-a.  Difference Map of ''J''&nbsp;&nbsp;(Areal)===
    
<pre>
 
<pre>
Line 5,007: Line 5,141:  
</pre>
 
</pre>
   −
===Figure 44-b.  Difference Map of J (Bundle)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 44-a.  Difference Map of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p>
 +
 
 +
===Figure 44-b.  Difference Map of ''J''&nbsp;&nbsp;(Bundle)===
    
<pre>
 
<pre>
Line 5,076: Line 5,214:  
</pre>
 
</pre>
   −
===Figure 44-c.  Difference Map of J (Compact)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 44-b.  Difference Map of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>
 +
 
 +
===Figure 44-c.  Difference Map of ''J''&nbsp;&nbsp;(Compact)===
    
<pre>
 
<pre>
Line 5,117: Line 5,259:  
</pre>
 
</pre>
   −
===Figure 44-d.  Difference Map of J (Digraph)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 44-c.  Difference Map of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p>
 +
 
 +
===Figure 44-d.  Difference Map of ''J''&nbsp;&nbsp;(Digraph)===
    
<pre>
 
<pre>
Line 5,155: Line 5,301:  
Figure 44-d.  Difference Map of J (Digraph)
 
Figure 44-d.  Difference Map of J (Digraph)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 44-d.  Difference Map of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>
    
===Table 45.  Computation of dJ===
 
===Table 45.  Computation of dJ===
Line 5,193: Line 5,343:  
</font><br>
 
</font><br>
   −
===Figure 46-a.  Differential of J (Areal)===
+
===Figure 46-a.  Differential of ''J''&nbsp;&nbsp;(Areal)===
    
<pre>
 
<pre>
Line 5,236: Line 5,386:  
</pre>
 
</pre>
   −
===Figure 46-b.  Differential of J (Bundle)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 46-a.  Differential of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p>
 +
 
 +
===Figure 46-b.  Differential of ''J''&nbsp;&nbsp;(Bundle)===
    
<pre>
 
<pre>
Line 5,305: Line 5,459:  
</pre>
 
</pre>
   −
===Figure 46-c.  Differential of J (Compact)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 46-b.  Differential of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>
 +
 
 +
===Figure 46-c.  Differential of ''J''&nbsp;&nbsp;(Compact)===
    
<pre>
 
<pre>
Line 5,342: Line 5,500:  
</pre>
 
</pre>
   −
===Figure 46-d.  Differential of J (Digraph)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 46-c.  Differential of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p>
 +
 
 +
===Figure 46-d.  Differential of ''J''&nbsp;&nbsp;(Digraph)===
    
<pre>
 
<pre>
Line 5,378: Line 5,540:  
Figure 46-d.  Differential of J (Digraph)
 
Figure 46-d.  Differential of J (Digraph)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 46-d.  Differential of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>
    
===Table 47.  Computation of rJ===
 
===Table 47.  Computation of rJ===
Line 5,439: Line 5,605:  
</font><br>
 
</font><br>
   −
===Figure 48-a.  Remainder of J (Areal)===
+
===Figure 48-a.  Remainder of ''J''&nbsp;&nbsp;(Areal)===
    
<pre>
 
<pre>
Line 5,482: Line 5,648:  
</pre>
 
</pre>
   −
===Figure 48-b.  Remainder of J (Bundle)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 48-a.  Remainder of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p>
 +
 
 +
===Figure 48-b.  Remainder of ''J''&nbsp;&nbsp;(Bundle)===
    
<pre>
 
<pre>
Line 5,551: Line 5,721:  
</pre>
 
</pre>
   −
===Figure 48-c.  Remainder of J (Compact)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 48-b.  Remainder of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>
 +
 
 +
===Figure 48-c.  Remainder of ''J''&nbsp;&nbsp;(Compact)===
    
<pre>
 
<pre>
Line 5,591: Line 5,765:  
</pre>
 
</pre>
   −
===Figure 48-d.  Remainder of J (Digraph)===
+
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 48-c.  Remainder of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p>
 +
 
 +
===Figure 48-d.  Remainder of ''J''&nbsp;&nbsp;(Digraph)===
    
<pre>
 
<pre>
Line 5,627: Line 5,805:  
Figure 48-d.  Remainder of J (Digraph)
 
Figure 48-d.  Remainder of J (Digraph)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 48-d.  Remainder of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>
    
===Table 49.  Computation Summary for J===
 
===Table 49.  Computation Summary for J===
Line 6,228: Line 6,410:  
Figure 52.  Decomposition of the Enlarged Conjunction EJ = (J, DJ)
 
Figure 52.  Decomposition of the Enlarged Conjunction EJ = (J, DJ)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 52.  Decomposition of E''J'''''</font></center></p>
    
===Figure 53.  Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===
 
===Figure 53.  Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===
Line 6,279: Line 6,465:  
Figure 53.  Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
 
Figure 53.  Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 53.  Decomposition of D''J'''''</font></center></p>
    
===Table 54.  Cast of Characters:  Expansive Subtypes of Objects and Operators===
 
===Table 54.  Cast of Characters:  Expansive Subtypes of Objects and Operators===
Line 6,981: Line 7,171:  
Figure 56-a1.  Radius Map of the Conjunction J = uv
 
Figure 56-a1.  Radius Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-a1.  Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 56-a2.  Secant Map of the Conjunction J = uv===
 
===Figure 56-a2.  Secant Map of the Conjunction J = uv===
Line 7,049: Line 7,243:  
Figure 56-a2.  Secant Map of the Conjunction J = uv
 
Figure 56-a2.  Secant Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-a2.  Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 56-a3.  Chord Map of the Conjunction J = uv===
 
===Figure 56-a3.  Chord Map of the Conjunction J = uv===
Line 7,117: Line 7,315:  
Figure 56-a3.  Chord Map of the Conjunction J = uv
 
Figure 56-a3.  Chord Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-a3.  Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 56-a4.  Tangent Map of the Conjunction J = uv===
 
===Figure 56-a4.  Tangent Map of the Conjunction J = uv===
Line 7,185: Line 7,387:  
Figure 56-a4.  Tangent Map of the Conjunction J = uv
 
Figure 56-a4.  Tangent Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-a4.  Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 56-b1.  Radius Map of the Conjunction J = uv===
 
===Figure 56-b1.  Radius Map of the Conjunction J = uv===
Line 7,285: Line 7,491:  
Figure 56-b1.  Radius Map of the Conjunction J = uv
 
Figure 56-b1.  Radius Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-b1.  Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 56-b2.  Secant Map of the Conjunction J = uv===
 
===Figure 56-b2.  Secant Map of the Conjunction J = uv===
Line 7,385: Line 7,595:  
Figure 56-b2.  Secant Map of the Conjunction J = uv
 
Figure 56-b2.  Secant Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-b2.  Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 56-b3.  Chord Map of the Conjunction J = uv===
 
===Figure 56-b3.  Chord Map of the Conjunction J = uv===
Line 7,485: Line 7,699:  
Figure 56-b3.  Chord Map of the Conjunction J = uv
 
Figure 56-b3.  Chord Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-b3.  Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 56-b4.  Tangent Map of the Conjunction J = uv===
 
===Figure 56-b4.  Tangent Map of the Conjunction J = uv===
Line 7,585: Line 7,803:  
Figure 56-b4.  Tangent Map of the Conjunction J = uv
 
Figure 56-b4.  Tangent Map of the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 56-b4.  Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 57-1.  Radius Operator Diagram for the Conjunction J = uv===
 
===Figure 57-1.  Radius Operator Diagram for the Conjunction J = uv===
Line 7,655: Line 7,877:  
Figure 57-1.  Radius Operator Diagram for the Conjunction J = uv
 
Figure 57-1.  Radius Operator Diagram for the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 57-1.  Radius Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 57-2.  Secant Operator Diagram for the Conjunction J = uv===
 
===Figure 57-2.  Secant Operator Diagram for the Conjunction J = uv===
Line 7,725: Line 7,951:  
Figure 57-2.  Secant Operator Diagram for the Conjunction J = uv
 
Figure 57-2.  Secant Operator Diagram for the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 57-2.  Secant Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 57-3.  Chord Operator Diagram for the Conjunction J = uv===
 
===Figure 57-3.  Chord Operator Diagram for the Conjunction J = uv===
Line 7,795: Line 8,025:  
Figure 57-3.  Chord Operator Diagram for the Conjunction J = uv
 
Figure 57-3.  Chord Operator Diagram for the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 57-3.  Chord Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Figure 57-4.  Tangent Functor Diagram for the Conjunction J = uv===
 
===Figure 57-4.  Tangent Functor Diagram for the Conjunction J = uv===
Line 7,865: Line 8,099:  
Figure 57-4.  Tangent Functor Diagram for the Conjunction J = uv
 
Figure 57-4.  Tangent Functor Diagram for the Conjunction J = uv
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 57-4.  Tangent Functor Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
    
===Formula Display 11===
 
===Formula Display 11===
Line 8,328: Line 8,566:  
</pre>
 
</pre>
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−
|+ '''Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
+
|+ '''Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes'''
−
'''
   
|- style="background:paleturquoise"
 
|- style="background:paleturquoise"
 
| &nbsp;
 
| &nbsp;
Line 8,337: Line 8,574:  
| align="center" | '''Transformation<br>or<br>Mapping'''
 
| align="center" | '''Transformation<br>or<br>Mapping'''
 
|-
 
|-
−
| valign="top" | Operand
+
| Operand
 
| valign="top" |
 
| valign="top" |
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
Line 8,367: Line 8,604:  
| <math>\epsilon</math> :
 
| <math>\epsilon</math> :
 
|-
 
|-
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 
|-
 
|-
−
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>)
+
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)&nbsp;&rarr;&nbsp;(E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)
 
|}
 
|}
 
|
 
|
Line 8,404: Line 8,641:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| <math>\eta</math>''J'' :
+
| <math>\eta</math>''F''<sub>''i''</sub> :
 
|-
 
|-
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
|-
 
|-
−
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 
|}
 
|}
 
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| <math>\eta</math>''J'' :
+
| <math>\eta</math>''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,435: Line 8,672:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| E''J'' :
+
| E''F''<sub>''i''</sub> :
 
|-
 
|-
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
|-
 
|-
−
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 
|}
 
|}
 
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| E''J'' :
+
| E''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,466: Line 8,703:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| D''J'' :
+
| D''F''<sub>''i''</sub> :
 
|-
 
|-
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
|-
 
|-
−
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 
|}
 
|}
 
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| D''J'' :
+
| D''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,497: Line 8,734:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| d''J'' :
+
| d''F''<sub>''i''</sub> :
 
|-
 
|-
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
|-
 
|-
−
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 
|}
 
|}
 
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| d''J'' :
+
| d''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,528: Line 8,765:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| r''J'' :
+
| r''F''<sub>''i''</sub> :
 
|-
 
|-
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
|-
 
|-
−
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 
|}
 
|}
 
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| r''J'' :
+
| r''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,567: Line 8,804:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| <font face=georgia>'''e'''</font>''J'' :
+
| <font face=georgia>'''e'''</font>''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,598: Line 8,835:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| <font face=georgia>'''E'''</font>''J'' :
+
| <font face=georgia>'''E'''</font>''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,629: Line 8,866:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| <font face=georgia>'''D'''</font>''J'' :
+
| <font face=georgia>'''D'''</font>''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|-
 
|-
Line 8,652: Line 8,889:  
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| d''J'' :
+
| d''F''<sub>''i''</sub> :
 
|-
 
|-
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 
|-
 
|-
−
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 
|}
 
|}
 
|
 
|
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| <font face=georgia>'''T'''</font>''J'' :
+
| <font face=georgia>'''T'''</font>''F'' :
 
|-
 
|-
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 
|-
 
|-
−
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 
|}
 
|}
 
|}<br>
 
|}<br>
 +
 +
===Formula Display 12===
    
<pre>
 
<pre>
−
o--------------o----------------------o--------------------o----------------------o
+
o-----------------------------------------------------------o
−
|              | Operator            | Proposition        | Transformation      |
+
|                                                          |
−
|              |    or                |    or              |    or                |
+
|        x  =  f(u, v)  =  ((u)(v))                    |
−
|              | Operand              | Component          | Mapping              |
+
|                                                          |
−
o--------------o----------------------o--------------------o----------------------o
+
|        y  =  g(u, v)  =  ((u, v))                    |
−
|              |                      |                    |                      |
+
|                                                          |
−
| Operand      | F = <F_1, F_2>      | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
+
o-----------------------------------------------------------o
−
|              |                      |                    |                      |
+
</pre>
−
|              | F = <f, g> : U -> X  | F_i : B^n -> B    | F : B^n -> B^k      |
+
 
−
|              |                      |                    |                      |
+
<br><font face="courier new">
−
o--------------o----------------------o--------------------o----------------------o
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−
|              |                      |                    |                      |
+
|
−
| Tacit        | !e! :                | !e!F_i :          | !e!F :              |
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y]  |
+
| &nbsp;
−
|              | (U%->X%)->(EU%->X%)  | B^n x D^n -> B    | [B^n x D^n]->[B^k]  |
+
| ''x''
−
|              |                      |                    |                      |
+
| =
−
o--------------o----------------------o--------------------o----------------------o
+
| ''f''‹''u'', ''v''›
−
|              |                      |                    |                      |
+
| =
−
| Trope        | !h! :                | !h!F_i :          | !h!F :              |
+
| ((''u'')(''v''))
−
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
+
| &nbsp;
−
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D    | [B^n x D^n]->[D^k]  |
+
|-
−
|              |                      |                    |                      |
+
| &nbsp;
−
o--------------o----------------------o--------------------o----------------------o
+
| ''y''
−
|              |                      |                    |                      |
+
| =
−
| Enlargement  | E :                  | EF_i :            | EF :                |
+
| ''g''‹''u'', ''v''›
−
| Operator    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
+
| =
−
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D    | [B^n x D^n]->[D^k]  |
+
| ((''u'', ''v''))
−
|              |                      |                    |                      |
+
| &nbsp;
−
o--------------o----------------------o--------------------o----------------------o
+
|}
−
|              |                      |                    |                      |
+
|}
−
| Difference  | D :                  | DF_i :            | DF :                |
+
</font><br>
−
| Operator    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
+
 
−
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D    | [B^n x D^n]->[D^k]  |
+
===Formula Display 13===
−
|              |                      |                    |                      |
+
 
−
o--------------o----------------------o--------------------o----------------------o
+
<pre>
−
|              |                      |                    |                      |
+
o-----------------------------------------------------------o
−
| Differential | d :                  | dF_i :            | dF :                |
+
|                                                          |
−
| Operator    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
+
|    <x, y>  =  F<u, v>  =  <((u)(v)), ((u, v))>        |
−
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D    | [B^n x D^n]->[D^k]  |
+
|                                                          |
−
|              |                      |                    |                      |
  −
o--------------o----------------------o--------------------o----------------------o
  −
|              |                      |                    |                      |
  −
| Remainder    | r :                  | rF_i :            | rF :                |
  −
| Operator    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
  −
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D    | [B^n x D^n]->[D^k]  |
  −
|              |                      |                    |                      |
  −
o--------------o----------------------o--------------------o----------------------o
  −
|              |                      |                    |                      |
  −
| Radius      | $e$ = <!e!, !h!> :  |                    | $e$F :              |
  −
| Operator    |                      |                    |                      |
  −
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
  −
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
  −
|              |                      |                    |                      |
  −
|              |                      |                    | [B^n x D^n] ->      |
  −
|              |                      |                    | [B^k x D^k]          |
  −
|              |                      |                    |                      |
  −
o--------------o----------------------o--------------------o----------------------o
  −
|              |                      |                    |                      |
  −
| Secant      | $E$ = <!e!, E> :    |                    | $E$F :              |
  −
| Operator    |                      |                    |                      |
  −
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
  −
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
  −
|              |                      |                    |                      |
  −
|              |                      |                    | [B^n x D^n] ->      |
  −
|              |                      |                    | [B^k x D^k]          |
  −
|              |                      |                    |                      |
  −
o--------------o----------------------o--------------------o----------------------o
  −
|              |                      |                    |                      |
  −
| Chord        | $D$ = <!e!, D> :    |                    | $D$F :              |
  −
| Operator    |                      |                    |                      |
  −
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
  −
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
  −
|              |                      |                    |                      |
  −
|              |                      |                    | [B^n x D^n] ->      |
  −
|              |                      |                    | [B^k x D^k]          |
  −
|              |                      |                    |                      |
  −
o--------------o----------------------o--------------------o----------------------o
  −
|              |                      |                    |                      |
  −
| Tangent      | $T$ = <!e!, d> :    | dF_i :            | $T$F :              |
  −
| Functor      |                      |                    |                      |
  −
|              | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u, v, du, dv] ->    |
  −
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
  −
|              |                      |                    |                      |
  −
|              |                      | B^n x D^n -> D    | [B^n x D^n] ->      |
  −
|              |                      |                    | [B^k x D^k]          |
  −
|              |                      |                    |                      |
  −
o--------------o----------------------o--------------------o----------------------o
  −
</pre>
  −
 
  −
===Formula Display 12===
  −
 
  −
<pre>
  −
o-----------------------------------------------------------o
  −
|                                                          |
  −
|        x  =  f(u, v)  =  ((u)(v))                    |
  −
|                                                          |
  −
|        y  =  g(u, v)  =  ((u, v))                    |
  −
|                                                          |
  −
o-----------------------------------------------------------o
  −
</pre>
  −
 
  −
<br><font face="courier new">
  −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
  −
|
  −
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
  −
| &nbsp;
  −
| ''x''
  −
| =
  −
| ''f''‹''u'', ''v''›
  −
| =
  −
| ((''u'')(''v''))
  −
| &nbsp;
  −
|-
  −
| &nbsp;
  −
| ''y''
  −
| =
  −
| ''g''‹''u'', ''v''›
  −
| =
  −
| ((''u'', ''v''))
  −
| &nbsp;
  −
|}
  −
|}
  −
</font><br>
  −
 
  −
===Formula Display 13===
  −
 
  −
<pre>
  −
o-----------------------------------------------------------o
  −
|                                                          |
  −
|    <x, y>  =  F<u, v>  =  <((u)(v)), ((u, v))>        |
  −
|                                                          |
   
o-----------------------------------------------------------o
 
o-----------------------------------------------------------o
 
</pre>
 
</pre>
Line 8,852: Line 8,999:  
</pre>
 
</pre>
   −
===Figure 61.  Propositional Transformation===
+
<font face="courier new">
−
 
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−
<pre>
+
|+ '''Table 60.  Propositional Transformation'''
−
             o-----------------------------------------------------o
+
|- style="background:paleturquoise"
−
             | U                                                  |
+
| width="25%" | ''u''
−
             |                                                    |
+
| width="25%" | ''v''
−
             |            o-----------o  o-----------o            |
+
| width="25%" | ''f''
−
             |          /            \ /            \          |
+
| width="25%" | ''g''
−
             |          /              o              \          |
+
|-
−
             |        /              / \              \        |
+
| width="25%" |
−
             |        /              /  \              \        |
+
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−
             |      o              o    o              o      |
+
| 0
−
             |      |              |    |              |      |
+
|-
 +
| 0
 +
|-
 +
| 1
 +
|-
 +
| 1
 +
|}
 +
| width="25%" |
 +
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0
 +
|-
 +
| 1
 +
|-
 +
| 0
 +
|-
 +
| 1
 +
|}
 +
| width="25%" |
 +
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0
 +
|-
 +
| 1
 +
|-
 +
| 1
 +
|-
 +
| 1
 +
|}
 +
| width="25%" |
 +
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 1
 +
|}
 +
|-
 +
| width="25%" | &nbsp;
 +
| width="25%" | &nbsp;
 +
| width="25%" | ((''u'')(''v''))
 +
| width="25%" | ((''u'', ''v''))
 +
|}
 +
</font><br>
 +
 
 +
===Figure 61.  Propositional Transformation===
 +
 
 +
<pre>
 +
             o-----------------------------------------------------o
 +
             | U                                                  |
 +
             |                                                    |
 +
             |            o-----------o  o-----------o            |
 +
             |          /            \ /            \          |
 +
             |          /              o              \          |
 +
             |        /              / \              \        |
 +
             |        /              /  \              \        |
 +
             |      o              o    o              o      |
 +
             |      |              |    |              |      |
 
             |      |      u      |    |      v      |      |
 
             |      |      u      |    |      v      |      |
 
             |      |              |    |              |      |
 
             |      |              |    |              |      |
Line 8,936: Line 9,140:  
Figure 61.  Propositional Transformation
 
Figure 61.  Propositional Transformation
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 61.  Propositional Transformation'''</font></center></p>
    
===Figure 62.  Propositional Transformation (Short Form)===
 
===Figure 62.  Propositional Transformation (Short Form)===
Line 8,987: Line 9,195:  
Figure 62.  Propositional Transformation (Short Form)
 
Figure 62.  Propositional Transformation (Short Form)
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 62.  Propositional Transformation (Short Form)'''</font></center></p>
    
===Figure 63.  Transformation of Positions===
 
===Figure 63.  Transformation of Positions===
Line 9,064: Line 9,276:  
Figure 63.  Transformation of Positions
 
Figure 63.  Transformation of Positions
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 63.  Transformation of Positions'''</font></center></p>
    
===Table 64.  Transformation of Positions===
 
===Table 64.  Transformation of Positions===
Line 9,086: Line 9,302:  
</pre>
 
</pre>
   −
===Table 65.  Induced Transformation on Propositions===
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−
 
+
|+ '''Table 64.  Transformation of Positions'''
−
<pre>
+
|- style="background:paleturquoise"
−
Table 65.  Induced Transformation on Propositions
+
| ''u''&nbsp;&nbsp;''v''
−
o------------o---------------------------------o------------o
+
| ''x''
−
|    X%    |  <---  F  =  <f , g>  <---  |    U%    |
+
| ''y''
−
o------------o----------o-----------o----------o------------o
+
| ''x''&nbsp;''y''
−
|            |      u = |  1 1 0 0  | = u      |            |
+
| ''x''&nbsp;(''y'')
−
|            |      v = |  1 0 1 0  | = v      |            |
+
| (''x'')&nbsp;''y''
−
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
+
| (''x'')(''y'')
−
|            |      x = |  1 1 1 0  | = f<u,v> |            |
+
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y''&nbsp;]
−
|            |      y = |  1 0 0 1  | = g<u,v> |            |
+
|-
−
o------------o----------o-----------o----------o------------o
+
| width="12%" |
−
|            |          |          |          |            |
+
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−
|    f_0    |    ()    |  0 0 0 0  |    ()    |    f_0    |
+
| 0&nbsp;&nbsp;0
−
|            |          |          |          |            |
+
|-
−
|    f_1    |  (x)(y)  |  0 0 0 1  |    ()    |    f_0    |
+
| 0&nbsp;&nbsp;1
−
|            |          |          |          |            |
+
|-
−
|    f_2    |  (x) y  |  0 0 1 0  |  (u)(v)  |    f_1    |
+
| 1&nbsp;&nbsp;0
−
|            |          |          |          |            |
+
|-
−
|    f_3    |  (x)    |  0 0 1 1  |  (u)(v)  |    f_1    |
+
| 1&nbsp;&nbsp;1
−
|            |          |          |          |            |
+
|}
−
|    f_4    |  x (y)  |  0 1 0 0  |  (u, v)  |    f_6    |
+
| width="12%" |
−
|            |          |          |          |            |
+
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−
|    f_5    |    (y)  |  0 1 0 1  |  (u, v)  |    f_6    |
+
| 0
−
|            |          |          |          |            |
+
|-
−
|    f_6    |  (x, y)  |  0 1 1 0  |  (u  v)  |    f_7    |
+
| 1
−
|            |          |          |          |            |
+
|-
−
|    f_7    |  (x  y)  |  0 1 1 1  |  (u  v)  |    f_7    |
+
| 1
−
|            |          |          |          |            |
+
|-
−
o------------o----------o-----------o----------o------------o
+
| 1
−
|            |          |          |          |            |
+
|}
−
|    f_8    |  x  y  |  1 0 0 0  |  u  v  |    f_8    |
+
| width="12%" |
−
|            |          |          |          |            |
+
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−
|    f_9    | ((x, y)) |  1 0 0 1  |  u  v  |    f_8    |
+
| 1
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 1
 +
|}
 +
| width="12%" |
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 1
 +
|}
 +
| width="12%" |
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0
 +
|-
 +
| 1
 +
|-
 +
| 1
 +
|-
 +
| 0
 +
|}
 +
| width="12%" |
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|}
 +
| width="12%" |
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|}
 +
| width="12%" |
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| &uarr;
 +
|-
 +
| ''F''
 +
|-
 +
| ‹''f'',&nbsp;''g''&nbsp;›
 +
|-
 +
| &uarr;
 +
|}
 +
|-
 +
| &nbsp;
 +
| ((''u'')(''v''))
 +
| ((''u'',&nbsp;''v''))
 +
| ''u''&nbsp;''v''
 +
| (''u'',&nbsp;''v'')
 +
| (''u'')(''v'')
 +
| (&nbsp;)
 +
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''&nbsp;]
 +
|}
 +
<br>
 +
 
 +
===Table 65.  Induced Transformation on Propositions===
 +
 
 +
<pre>
 +
Table 65.  Induced Transformation on Propositions
 +
o------------o---------------------------------o------------o
 +
|    X%    |  <---  F  =  <f , g>  <---  |    U%    |
 +
o------------o----------o-----------o----------o------------o
 +
|            |      u = |  1 1 0 0  | = u      |            |
 +
|            |      v = |  1 0 1 0  | = v      |            |
 +
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
 +
|            |      x = |  1 1 1 0  | = f<u,v> |            |
 +
|            |      y = |  1 0 0 1  | = g<u,v> |            |
 +
o------------o----------o-----------o----------o------------o
 
|            |          |          |          |            |
 
|            |          |          |          |            |
−
|    f_10   |      y  |  1 0 1 0  | ((u, v)) |    f_9     |
+
|    f_0    |    ()   |  0 0 0 0  |   ()   |    f_0     |
 
|            |          |          |          |            |
 
|            |          |          |          |            |
−
|    f_11    |  (x (y)) |  1 0 1 1  | ((u, v)) |    f_9     |
+
|    f_1    |  (x)(y) |  0 0 0 1  |   ()   |    f_0     |
 
|            |          |          |          |            |
 
|            |          |          |          |            |
−
|    f_12    |  x      |  1 1 0 0  | ((u)(v)) |    f_14    |
+
|    f_2    |  (x) y  |  0 0 1 0  |  (u)(v)  |    f_1    |
 +
|            |          |          |          |            |
 +
|    f_3    |  (x)    |  0 0 1 1  |  (u)(v)  |    f_1    |
 +
|            |          |          |          |            |
 +
|    f_4    |  x (y)  |  0 1 0 0  |  (u, v)  |    f_6    |
 +
|            |          |          |          |            |
 +
|    f_5    |    (y)  |  0 1 0 1  |  (u, v)  |    f_6    |
 +
|            |          |          |          |            |
 +
|    f_6    |  (x, y)  |  0 1 1 0  |  (u  v)  |    f_7    |
 +
|            |          |          |          |            |
 +
|    f_7    |  (x  y)  |  0 1 1 1  |  (u  v)  |    f_7    |
 +
|            |          |          |          |            |
 +
o------------o----------o-----------o----------o------------o
 +
|            |          |          |          |            |
 +
|    f_8    |  x  y  |  1 0 0 0  |  u  v  |    f_8    |
 +
|            |          |          |          |            |
 +
|    f_9    | ((x, y)) |  1 0 0 1  |  u  v  |    f_8    |
 +
|            |          |          |          |            |
 +
|    f_10    |      y  |  1 0 1 0  | ((u, v)) |    f_9    |
 +
|            |          |          |          |            |
 +
|    f_11    |  (x (y)) |  1 0 1 1  | ((u, v)) |    f_9    |
 +
|            |          |          |          |            |
 +
|    f_12    |  x      |  1 1 0 0  | ((u)(v)) |    f_14    |
 
|            |          |          |          |            |
 
|            |          |          |          |            |
 
|    f_13    | ((x) y)  |  1 1 0 1  | ((u)(v)) |    f_14    |
 
|    f_13    | ((x) y)  |  1 1 0 1  | ((u)(v)) |    f_14    |
Line 9,135: Line 9,455:  
|            |          |          |          |            |
 
|            |          |          |          |            |
 
o------------o----------o-----------o----------o------------o
 
o------------o----------o-----------o----------o------------o
−
</pre>
  −
  −
===Formula Display 14===
  −
  −
<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|  EG_i  =  G_i <u + du, v + dv>                |
  −
|                                                |
  −
o-------------------------------------------------o
   
</pre>
 
</pre>
    
<br><font face="courier new">
 
<br><font face="courier new">
−
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ Table 65.  Induced Transformation on Propositions
 +
|- style="background:paleturquoise"
 +
| ''X''<sup>&nbsp;&bull;</sup>
 +
| colspan="3" |
 +
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"
 +
| &larr;
 +
| ''F''&nbsp;=&nbsp;‹''f''&nbsp;,&nbsp;''g''›
 +
| &larr;
 +
|}
 +
| ''U''<sup>&nbsp;&bull;</sup>
 +
|- style="background:paleturquoise"
 +
| rowspan="2" | ''f''<sub>''i''</sub>‹''x'',&nbsp;''y''›
 +
|
 +
{| align="right" style="background:paleturquoise; text-align:right"
 +
| ''u'' =
 +
|-
 +
| ''v'' =
 +
|}
 
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+
{| align="center" style="background:paleturquoise; text-align:center"
−
| width="8%"  | E''G''<sub>''i''</sub>
+
| 1 1 0 0
−
| width="4%"  | =
+
|-
−
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
+
| 1 0 1 0
 
|}
 
|}
 +
|
 +
{| align="left" style="background:paleturquoise; text-align:left"
 +
| = ''u''
 +
|-
 +
| = ''v''
 +
|}
 +
| rowspan="2" | ''f''<sub>''j''</sub>‹''u'',&nbsp;''v''›
 +
|- style="background:paleturquoise"
 +
|
 +
{| align="right" style="background:paleturquoise; text-align:right"
 +
| ''x'' =
 +
|-
 +
| ''y'' =
 
|}
 
|}
−
</font><br>
  −
  −
===Formula Display 15===
  −
  −
<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|  DG_i  =  G_i <u, v>  +  EG_i <u, v, du, dv>  |
  −
|                                                |
  −
|        =  G_i <u, v>  +  G_i <u + du, v + dv>  |
  −
|                                                |
  −
o-------------------------------------------------o
  −
</pre>
  −
  −
<br><font face="courier new">
  −
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
   
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+
{| align="center" style="background:paleturquoise; text-align:center"
−
| width="8%"  | D''G''<sub>''i''</sub>
+
| 1 1 1 0
−
| width="4%"  | =
  −
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
  −
| width="4%"  | +
  −
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
   
|-
 
|-
−
| width="8%"  | &nbsp;
+
| 1 0 0 1
−
| width="4%"  | =
  −
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
  −
| width="4%"  | +
  −
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
   
|}
 
|}
 +
|
 +
{| align="left" style="background:paleturquoise; text-align:left"
 +
| = ''f''‹''u'',&nbsp;''v''›
 +
|-
 +
| = ''g''‹''u'',&nbsp;''v''›
 
|}
 
|}
−
</font><br>
+
|-
−
 
  −
===Formula Display 16===
  −
 
  −
<pre>
  −
o-------------------------------------------------o
  −
|                                                 |
  −
|  Ef  =  ((u + du)(v + dv))                    |
  −
|                                                |
  −
|  Eg  =  ((u + du, v + dv))                    |
  −
|                                                |
  −
o-------------------------------------------------o
  −
</pre>
  −
 
  −
<br><font face="courier new">
  −
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
   
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+
{| cellpadding="2" style="background:lightcyan"
−
| width="8%"  | E''f''
+
| ''f''<sub>0</sub>
−
| width="4%"  | =
+
|-
−
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
+
| ''f''<sub>1</sub>
 +
|-
 +
| ''f''<sub>2</sub>
 +
|-
 +
| ''f''<sub>3</sub>
 +
|-
 +
| ''f''<sub>4</sub>
 +
|-
 +
| ''f''<sub>5</sub>
 +
|-
 +
| ''f''<sub>6</sub>
 
|-
 
|-
−
| width="8%"  | E''g''
+
| ''f''<sub>7</sub>
−
| width="4%"  | =
  −
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
   
|}
 
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ()
 +
|-
 +
| &nbsp;(''x'')(''y'')&nbsp;
 +
|-
 +
| &nbsp;(''x'')&nbsp;''y''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;(''x'')&nbsp;&nbsp;&nbsp;&nbsp;
 +
|-
 +
| &nbsp;&nbsp;''x''&nbsp;(''y'')&nbsp;
 +
|-
 +
| &nbsp;&nbsp;&nbsp;&nbsp;(''y'')&nbsp;
 +
|-
 +
| &nbsp;(''x'',&nbsp;''y'')&nbsp;
 +
|-
 +
| &nbsp;(''x''&nbsp;&nbsp;''y'')&nbsp;
 
|}
 
|}
−
</font><br>
  −
  −
===Formula Display 17===
  −
  −
<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|  Df  =  ((u)(v))  +  ((u + du)(v + dv))        |
  −
|                                                |
  −
|  Dg  =  ((u, v))  +  ((u + du, v + dv))        |
  −
|                                                |
  −
o-------------------------------------------------o
  −
</pre>
  −
  −
<br><font face="courier new">
  −
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
   
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+
{| cellpadding="2" style="background:lightcyan"
−
| width="8%"  | D''f''
+
| 0 0 0 0
−
| width="4%"  | =
+
|-
−
| width="20%" | ((''u'')(''v''))
+
| 0 0 0 1
−
| width="4%"  | +
+
|-
−
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
+
| 0 0 1 0
 +
|-
 +
| 0 0 1 1
 +
|-
 +
| 0 1 0 0
 +
|-
 +
| 0 1 0 1
 +
|-
 +
| 0 1 1 0
 
|-
 
|-
−
| width="8%" | D''g''
+
| 0 1 1 1
−
| width="4%"  | =
+
|}
−
| width="20%" | ((''u'', ''v''))
+
|
−
| width="4%"  | +
+
{| cellpadding="2" style="background:lightcyan"
−
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
+
| ()
 +
|-
 +
| ()
 +
|-
 +
| &nbsp;(''u'')(''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u'')(''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u'',&nbsp;''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u'',&nbsp;''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;
 
|}
 
|}
−
|}
  −
</font><br>
  −
  −
===Table 66-i.  Computation Summary for f‹u, v› = ((u)(v))===
  −
  −
<pre>
  −
Table 66-i.  Computation Summary for f<u, v> = ((u)(v))
  −
o--------------------------------------------------------------------------------o
  −
|                                                                                |
  −
| !e!f  =  uv.    1      + u(v).    1      + (u)v.    1      + (u)(v).    0      |
  −
|                                                                                |
  −
|  Ef  =  uv. (du  dv)  + u(v). (du (dv)) + (u)v.((du) dv)  + (u)(v).((du)(dv)) |
  −
|                                                                                |
  −
|  Df  =  uv.  du  dv  + u(v).  du (dv)  + (u)v. (du) dv  + (u)(v).((du)(dv)) |
  −
|                                                                                |
  −
|  df  =  uv.    0      + u(v).  du      + (u)v.      dv  + (u)(v). (du, dv)  |
  −
|                                                                                |
  −
|  rf  =  uv.  du  dv  + u(v).  du  dv  + (u)v.  du  dv  + (u)(v).  du  dv  |
  −
|                                                                                |
  −
o--------------------------------------------------------------------------------o
  −
</pre>
  −
  −
<font face="courier new">
  −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
  −
|+ Table 66-i.  Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))
   
|
 
|
−
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
{| cellpadding="2" style="background:lightcyan"
−
| <math>\epsilon</math>''f''
+
| ''f''<sub>0</sub>
−
| = || ''uv''        || <math>\cdot</math> || 1
+
|-
−
| + || ''u''(''v'')  || <math>\cdot</math> || 1
+
| ''f''<sub>0</sub>
−
| + || (''u'')''v''  || <math>\cdot</math> || 1
+
|-
−
| + || (''u'')(''v'') || <math>\cdot</math> || 0
+
| ''f''<sub>1</sub>
 +
|-
 +
| ''f''<sub>1</sub>
 
|-
 
|-
−
| E''f''
+
| ''f''<sub>6</sub>
−
| = || ''uv''        || <math>\cdot</math> || (d''u'' d''v'')
  −
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u (d''v''))
  −
| + || (''u'')''v''  || <math>\cdot</math> || ((d''u'') d''v'')
  −
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
   
|-
 
|-
−
| D''f''
+
| ''f''<sub>6</sub>
−
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
  −
| + || ''u''(''v'')  || <math>\cdot</math> || d''u'' (d''v'')
  −
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'') d''v''
  −
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
   
|-
 
|-
−
| d''f''
+
| ''f''<sub>7</sub>
−
| = || ''uv''        || <math>\cdot</math> || 0
  −
| + || ''u''(''v'')  || <math>\cdot</math> || d''u''
  −
| + || (''u'')''v''  || <math>\cdot</math> || d''v''
  −
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
   
|-
 
|-
−
| r''f''
+
| ''f''<sub>7</sub>
−
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
  −
| + || ''u''(''v'')  || <math>\cdot</math> || d''u'' d''v''
  −
| + || (''u'')''v''  || <math>\cdot</math> || d''u'' d''v''
  −
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
   
|}
 
|}
−
|}
+
|-
−
</font><br>
  −
 
  −
===Table 66-ii.  Computation Summary for g‹u, v› = ((u, v))===
  −
 
  −
<pre>
  −
Table 66-ii.  Computation Summary for g<u, v> = ((u, v))
  −
o--------------------------------------------------------------------------------o
  −
|                                                                                |
  −
| !e!g  =  uv.    1      + u(v).    0      + (u)v.    0      + (u)(v).    1      |
  −
|                                                                                |
  −
|  Eg  =  uv.((du, dv)) + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v).((du, dv)) |
  −
|                                                                                |
  −
|  Dg  =  uv. (du, dv)  + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v). (du, dv)  |
  −
|                                                                                |
  −
|  dg  =  uv. (du, dv)  + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v). (du, dv)  |
  −
|                                                                                |
  −
|  rg  =  uv.    0      + u(v).    0      + (u)v.    0      + (u)(v).    0      |
  −
|                                                                                |
  −
o--------------------------------------------------------------------------------o
  −
</pre>
  −
 
  −
<font face="courier new">
  −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
  −
|+ Table 66-ii.  Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))
   
|
 
|
−
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
{| cellpadding="2" style="background:lightcyan"
−
| <math>\epsilon</math>''g''
+
| ''f''<sub>8</sub>
−
| = || ''uv''         || <math>\cdot</math> || 1
+
|-
−
| + || ''u''(''v'')  || <math>\cdot</math> || 0
+
| ''f''<sub>9</sub>
−
| + || (''u'')''v''   || <math>\cdot</math> || 0
+
|-
−
| + || (''u'')(''v'') || <math>\cdot</math> || 1
+
| ''f''<sub>10</sub>
 +
|-
 +
| ''f''<sub>11</sub>
 
|-
 
|-
−
| E''g''
+
| ''f''<sub>12</sub>
−
| = || ''uv''        || <math>\cdot</math> || ((d''u'', d''v''))
  −
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
   
|-
 
|-
−
| D''g''
+
| ''f''<sub>13</sub>
−
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
   
|-
 
|-
−
| d''g''
+
| ''f''<sub>14</sub>
−
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
   
|-
 
|-
−
| r''g''
+
| ''f''<sub>15</sub>
−
| = || ''uv''        || <math>\cdot</math> || 0
  −
| + || ''u''(''v'')  || <math>\cdot</math> || 0
  −
| + || (''u'')''v''  || <math>\cdot</math> || 0
  −
| + || (''u'')(''v'') || <math>\cdot</math> || 0
   
|}
 
|}
−
|}
+
|
−
</font><br>
+
{| cellpadding="2" style="background:lightcyan"
−
 
+
| &nbsp;&nbsp;''x''&nbsp;&nbsp;''y''&nbsp;&nbsp;
−
===Table 67.  Computation of an Analytic Series in Terms of Coordinates===
+
|-
 +
| ((''x'',&nbsp;''y''))
 +
|-
 +
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''y''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;(''x''&nbsp;(''y''))
 +
|-
 +
| &nbsp;&nbsp;''x''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 +
|-
 +
| ((''x'')&nbsp;''y'')&nbsp;
 +
|-
 +
| ((''x'')(''y''))
 +
|-
 +
| (())
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| 1 0 0 0
 +
|-
 +
| 1 0 0 1
 +
|-
 +
| 1 0 1 0
 +
|-
 +
| 1 0 1 1
 +
|-
 +
| 1 1 0 0
 +
|-
 +
| 1 1 0 1
 +
|-
 +
| 1 1 1 0
 +
|-
 +
| 1 1 1 1
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
 +
|-
 +
| ((''u'',&nbsp;''v''))
 +
|-
 +
| ((''u'',&nbsp;''v''))
 +
|-
 +
| ((''u'')(''v''))
 +
|-
 +
| ((''u'')(''v''))
 +
|-
 +
| (())
 +
|-
 +
| (())
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>9</sub>
 +
|-
 +
| ''f''<sub>9</sub>
 +
|-
 +
| ''f''<sub>14</sub>
 +
|-
 +
| ''f''<sub>14</sub>
 +
|-
 +
| ''f''<sub>15</sub>
 +
|-
 +
| ''f''<sub>15</sub>
 +
|}
 +
|}
 +
</font><br>
 +
 
 +
===Formula Display 14===
    
<pre>
 
<pre>
−
Table 67.  Computation of an Analytic Series in Terms of Coordinates
+
o-------------------------------------------------o
−
o--------o-------o-------o--------o-------o-------o-------o-------o
+
|                                                 |
−
| u  v  | du dv | u' v' |  f  g  | Ef Eg | Df Dg | df dg | rf rg |
+
|   EG_i = G_i <u + du, v + dv>                |
−
o--------o-------o-------o--------o-------o-------o-------o-------o
+
|                                                 |
−
|        |      |      |        |      |      |      |      |
+
o-------------------------------------------------o
−
|  0  0  | 0  0  | 0  0  |  0  1  | 0  1  | 0  0  | 0  0  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 0  1  | 0  1  |        | 1  0  | 1  1  | 1  1  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 1  0  | 1  0  |        | 1  0  | 1  1  | 1  1  | 0  0 |
  −
|       |      |      |        |      |      |      |      |
  −
|        | 1  1  | 1  1  |        | 1  1  | 1  0  | 0  0  | 1  0  |
  −
|        |      |      |        |      |      |      |      |
  −
o--------o-------o-------o--------o-------o-------o-------o-------o
  −
|        |      |      |        |      |      |      |      |
  −
|  0  1  | 0  0  | 0  1  |  1  0  | 1  0  | 0  0  | 0  0  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 0  1  | 0  0  |        | 0  1  | 1  1  | 1  1  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 1  0  | 1  1  |        | 1  1  | 0  1  | 0  1  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 1  1  | 1  0  |        | 1  0  | 0  0  | 1  0  | 1  0  |
  −
|        |      |      |        |      |      |      |      |
  −
o--------o-------o-------o--------o-------o-------o-------o-------o
  −
|        |      |      |        |      |      |      |      |
  −
|  1  0  | 0  0  | 1  0  |  1  0  | 1  0  | 0  0  | 0  0  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 0  1  | 1  1  |        | 1  1  | 0  1  | 0  1  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 1  0  | 0  0  |        | 0  1  | 1  1  | 1  1  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 1  1  | 0  1  |        | 1  0  | 0  0  | 1  0  | 1  0  |
  −
|        |      |      |        |      |      |      |      |
  −
o--------o-------o-------o--------o-------o-------o-------o-------o
  −
|        |      |      |        |      |      |      |      |
  −
|  1  1  | 0  0  | 1  1  |  1  1  | 1  1  | 0  0  | 0  0  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 0  1  | 1  0  |        | 1  0  | 0  1  | 0  1  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 1  0  | 0  1  |        | 1  0  | 0  1  | 0  1  | 0  0  |
  −
|        |      |      |        |      |      |      |      |
  −
|        | 1  1  | 0  0  |        | 0  1  | 1  0  | 0  0  | 1  0  |
  −
|        |      |      |        |      |      |      |      |
  −
o--------o-------o-------o--------o-------o-------o-------o-------o
   
</pre>
 
</pre>
   −
===Table 68. Computation of an Analytic Series in Symbolic Terms===
+
<br><font face="courier new">
 +
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
 +
|
 +
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
 +
| width="8%"  | E''G''<sub>''i''</sub>
 +
| width="4%" | =
 +
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
 +
|}
 +
|}
 +
</font><br>
 +
 
 +
===Formula Display 15===
    
<pre>
 
<pre>
−
Table 68.  Computation of an Analytic Series in Symbolic Terms
+
o-------------------------------------------------o
−
o-----o-----o------------o----------o----------o----------o----------o----------o
+
|                                                 |
−
| u v | f g |     Df    |   Dg    |   df    |   dg    |   rf    |    rf    |
+
|   DG_i  =  G_i <u, v>  +  EG_i <u, v, du, dv>  |
−
o-----o-----o------------o----------o----------o----------o----------o----------o
+
|                                                 |
−
|    |    |            |          |          |          |          |          |
+
|         =  G_i <u, v>  +  G_i <u + du, v + dv>  |
−
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) |  du  dv |   ()    |
+
|                                                |
−
|     |     |           |         |          |          |          |         |
+
o-------------------------------------------------o
−
| 0 1 | 1 0 |  (du) dv  | (du, dv) |   dv    | (du, dv) | du  dv |   ()    |
+
</pre>
−
|     |    |            |          |          |          |          |          |
+
 
−
| 1 0 | 1 0 |   du (dv) | (du, dv) |   du    | (du, dv) | du  dv  |    ()    |
+
<br><font face="courier new">
−
|    |    |            |          |          |          |          |          |
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
−
| 1 1 | 1 1 |  du  dv  | (du, dv) |    ()    | (du, dv) |  du  dv  |    ()    |
+
|
−
|    |    |            |          |          |          |          |          |
+
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
−
o-----o-----o------------o----------o----------o----------o----------o----------o
+
| width="8%" | D''G''<sub>''i''</sub>
−
</pre>
+
| width="4%"  | =
−
 
+
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
−
===Formula Display 18===
+
| width="4%"  | +
−
 
+
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
−
<pre>
+
|-
−
o-------------------------------------------------------------------------o
+
| width="8%" | &nbsp;
−
|                                                                         |
+
| width="4%"  | =
−
|  Df  = uv. du  dv  + u(v). du (dv) + (u)v.(du) dv  + (u)(v).((du)(dv)) |
+
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
−
|                                                                        |
+
| width="4%" | +
−
|  Dg  = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv)  |
+
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
−
|                                                                         |
+
|}
−
o-------------------------------------------------------------------------o
+
|}
−
</pre>
+
</font><br>
−
 
+
 
−
<br><font face="courier new">
+
===Formula Display 16===
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+
 
−
|
+
<pre>
−
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
o-------------------------------------------------o
−
| &nbsp;
+
|                                                |
−
|-
+
|  Ef  = ((u + du)(v + dv))                    |
−
| D''f''
+
|                                                |
−
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
+
|  Eg  = ((u + du, v + dv))                    |
−
| + || ''u''(''v'')   || <math>\cdot</math> || d''u'' (d''v'')
+
|                                                |
−
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'') d''v''
+
o-------------------------------------------------o
−
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
+
</pre>
−
|-
+
 
−
| &nbsp;
+
<br><font face="courier new">
−
|-
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
−
| D''g''
+
|
−
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
+
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
−
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
+
| width="8%"  | E''f''
−
| + || (''u'')''v''   || <math>\cdot</math> || (d''u'', d''v'')
+
| width="4%"  | =
−
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
−
|-
+
|-
−
| &nbsp;
+
| width="8%"  | E''g''
−
|}
+
| width="4%" | =
−
|}
+
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
−
</font><br>
+
|}
−
 
+
|}
−
===Figure 69.  Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›===
+
</font><br>
−
 
+
 
−
<pre>
+
===Formula Display 17===
−
o-----------------------------------o o-----------------------------------o
+
 
−
| U                                | |`U`````````````````````````````````|
+
<pre>
−
|                                  | |```````````````````````````````````|
+
o-------------------------------------------------o
−
|                ^                | |```````````````````````````````````|
+
|                                                 |
−
|                |                | |```````````````````````````````````|
+
|   Df  =  ((u)(v))  + ((u + du)(v + dv))       |
−
|      o-------o | o-------o      | |```````o-------o```o-------o```````|
+
|                                                 |
−
| ^    /`````````\|/`````````\    ^ | | ^ ```/      ^  \`/  ^      \``` ^ |
+
|   Dg  =  ((u, v))  +  ((u + du, v + dv))       |
−
|  \  /```````````|```````````\  /  | |``\``/        \  o  /        \``/``|
+
|                                                 |
−
|  \/`````u`````/|\`````v`````\/  | |```\/    u    \/`\/    v    \/```|
+
o-------------------------------------------------o
−
|  /\``````````/`|`\``````````/\  | |```/\          /\`/\          /\```|
+
</pre>
−
|  o``\````````o``@``o````````/``o  | |``o  \        o``@``o        /  o``|
+
 
 +
<br><font face="courier new">
 +
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
 +
|
 +
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
 +
| width="8%"  | D''f''
 +
| width="4%"  | =
 +
| width="20%" | ((''u'')(''v''))
 +
| width="4%"  | +
 +
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
 +
|-
 +
| width="8%"  | D''g''
 +
| width="4%"  | =
 +
| width="20%" | ((''u'', ''v''))
 +
| width="4%"  | +
 +
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
 +
|}
 +
|}
 +
</font><br>
 +
 
 +
===Table 66-i.  Computation Summary for f‹u, v› = ((u)(v))===
 +
 
 +
<pre>
 +
Table 66-i.  Computation Summary for f<u, v> = ((u)(v))
 +
o--------------------------------------------------------------------------------o
 +
|                                                                                |
 +
| !e!f  =  uv.    1      + u(v).    1      + (u)v.    1      + (u)(v).    0      |
 +
|                                                                                |
 +
|  Ef  =  uv. (du  dv)  + u(v). (du (dv)) + (u)v.((du) dv)  + (u)(v).((du)(dv)) |
 +
|                                                                                |
 +
|  Df  =  uv.  du  dv  + u(v).  du (dv)  + (u)v. (du) dv  + (u)(v).((du)(dv)) |
 +
|                                                                                |
 +
|  df  =  uv.    0      + u(v).  du      + (u)v.      dv  + (u)(v). (du, dv)  |
 +
|                                                                                |
 +
|  rf  =  uv.  du  dv  + u(v).  du  dv  + (u)v.  du  dv  + (u)(v).  du  dv  |
 +
|                                                                                |
 +
o--------------------------------------------------------------------------------o
 +
</pre>
 +
 
 +
<font face="courier new">
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ Table 66-i.  Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))
 +
|
 +
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| <math>\epsilon</math>''f''
 +
| = || ''uv''        || <math>\cdot</math> || 1
 +
| + || ''u''(''v'')  || <math>\cdot</math> || 1
 +
| + || (''u'')''v''  || <math>\cdot</math> || 1
 +
| + || (''u'')(''v'') || <math>\cdot</math> || 0
 +
|-
 +
| E''f''
 +
| = || ''uv''        || <math>\cdot</math> || (d''u'' d''v'')
 +
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u (d''v''))
 +
| + || (''u'')''v''  || <math>\cdot</math> || ((d''u'') d''v'')
 +
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
 +
|-
 +
| D''f''
 +
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
 +
| + || ''u''(''v'')  || <math>\cdot</math> || d''u'' (d''v'')
 +
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'') d''v''
 +
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
 +
|-
 +
| d''f''
 +
| = || ''uv''        || <math>\cdot</math> || 0
 +
| + || ''u''(''v'')  || <math>\cdot</math> || d''u''
 +
| + || (''u'')''v''  || <math>\cdot</math> || d''v''
 +
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
 +
|-
 +
| r''f''
 +
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
 +
| + || ''u''(''v'')  || <math>\cdot</math> || d''u'' d''v''
 +
| + || (''u'')''v''  || <math>\cdot</math> || d''u'' d''v''
 +
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
 +
|}
 +
|}
 +
</font><br>
 +
 
 +
===Table 66-ii.  Computation Summary for g‹u, v› = ((u, v))===
 +
 
 +
<pre>
 +
Table 66-ii.  Computation Summary for g<u, v> = ((u, v))
 +
o--------------------------------------------------------------------------------o
 +
|                                                                                |
 +
| !e!g  =  uv.    1      + u(v).    0      + (u)v.    0      + (u)(v).    1      |
 +
|                                                                                |
 +
|  Eg  =  uv.((du, dv)) + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v).((du, dv)) |
 +
|                                                                                |
 +
|  Dg  =  uv. (du, dv)  + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v). (du, dv)  |
 +
|                                                                                |
 +
|  dg  =  uv. (du, dv)  + u(v). (du, dv)  + (u)v. (du, dv)  + (u)(v). (du, dv)  |
 +
|                                                                                |
 +
|  rg  =  uv.    0      + u(v).    0      + (u)v.    0      + (u)(v).    0      |
 +
|                                                                                |
 +
o--------------------------------------------------------------------------------o
 +
</pre>
 +
 
 +
<font face="courier new">
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ Table 66-ii.  Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))
 +
|
 +
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| <math>\epsilon</math>''g''
 +
| = || ''uv''        || <math>\cdot</math> || 1
 +
| + || ''u''(''v'')  || <math>\cdot</math> || 0
 +
| + || (''u'')''v''  || <math>\cdot</math> || 0
 +
| + || (''u'')(''v'') || <math>\cdot</math> || 1
 +
|-
 +
| E''g''
 +
| = || ''uv''        || <math>\cdot</math> || ((d''u'', d''v''))
 +
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
 +
|-
 +
| D''g''
 +
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
 +
|-
 +
| d''g''
 +
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
 +
|-
 +
| r''g''
 +
| = || ''uv''        || <math>\cdot</math> || 0
 +
| + || ''u''(''v'')  || <math>\cdot</math> || 0
 +
| + || (''u'')''v''  || <math>\cdot</math> || 0
 +
| + || (''u'')(''v'') || <math>\cdot</math> || 0
 +
|}
 +
|}
 +
</font><br>
 +
 
 +
===Table 67.  Computation of an Analytic Series in Terms of Coordinates===
 +
 
 +
<pre>
 +
Table 67.  Computation of an Analytic Series in Terms of Coordinates
 +
o--------o-------o-------o--------o-------o-------o-------o-------o
 +
|  u  v  | du dv | u' v' |  f  g  | Ef Eg | Df Dg | df dg | rf rg |
 +
o--------o-------o-------o--------o-------o-------o-------o-------o
 +
|        |      |      |        |      |      |      |      |
 +
|  0  0  | 0  0  | 0  0  |  0  1  | 0  1  | 0  0  | 0  0  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 0  1  | 0  1  |        | 1  0  | 1  1  | 1  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  0  | 1  0  |        | 1  0  | 1  1  | 1  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  1  | 1  1  |        | 1  1  | 1  0  | 0  0  | 1  0  |
 +
|        |      |      |        |      |      |      |      |
 +
o--------o-------o-------o--------o-------o-------o-------o-------o
 +
|        |      |      |        |      |      |      |      |
 +
|  0  1  | 0  0  | 0  1  |  1  0  | 1  0  | 0  0  | 0  0  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 0  1  | 0  0  |        | 0  1  | 1  1  | 1  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  0  | 1  1  |        | 1  1  | 0  1  | 0  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  1  | 1  0  |        | 1  0  | 0  0  | 1  0  | 1  0  |
 +
|        |      |      |        |      |      |      |      |
 +
o--------o-------o-------o--------o-------o-------o-------o-------o
 +
|        |      |      |        |      |      |      |      |
 +
|  1  0  | 0  0  | 1  0  |  1  0  | 1  0  | 0  0  | 0  0  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 0  1  | 1  1  |        | 1  1  | 0  1  | 0  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  0  | 0  0  |        | 0  1  | 1  1  | 1  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  1  | 0  1  |        | 1  0  | 0  0  | 1  0  | 1  0  |
 +
|        |      |      |        |      |      |      |      |
 +
o--------o-------o-------o--------o-------o-------o-------o-------o
 +
|        |      |      |        |      |      |      |      |
 +
|  1  1  | 0  0  | 1  1  |  1  1  | 1  1  | 0  0  | 0  0  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 0  1  | 1  0  |        | 1  0  | 0  1  | 0  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  0  | 0  1  |        | 1  0  | 0  1  | 0  1  | 0  0  |
 +
|        |      |      |        |      |      |      |      |
 +
|        | 1  1  | 0  0  |        | 0  1  | 1  0  | 0  0  | 1  0  |
 +
|        |      |      |        |      |      |      |      |
 +
o--------o-------o-------o--------o-------o-------o-------o-------o
 +
</pre>
 +
 
 +
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ Table 67.  Computation of an Analytic Series in Terms of Coordinates
 +
|
 +
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| ''u''
 +
| ''v''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| d''u''
 +
| d''v''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| ''u''<font face="courier new">’</font>
 +
| ''v''<font face="courier new">’</font>
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 1
 +
|-
 +
| 0 || 0
 +
|-
 +
| 1 || 1
 +
|-
 +
| 1 || 0
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|-
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 0 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|}
 +
|}
 +
|
 +
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| <math>\epsilon</math>''f''
 +
| <math>\epsilon</math>''g''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| E''f''
 +
| E''g''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| D''f''
 +
| D''g''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| d''f''
 +
| d''g''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| d<sup>2</sup>''f''
 +
| d<sup>2</sup>''g''
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 1 || 1
 +
|-
 +
| 1 || 1
 +
|-
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 1 || 1
 +
|-
 +
| 1 || 1
 +
|-
 +
| 0 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 1 || 0
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 1
 +
|-
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 1 || 1
 +
|-
 +
| 0 || 1
 +
|-
 +
| 0 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 1 || 1
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 1 || 0
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 0
 +
|-
 +
| 1 || 1
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 1
 +
|-
 +
| 0 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 1
 +
|-
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 1 || 0
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 1 || 1
 +
|-
 +
| 1 || 0
 +
|-
 +
| 1 || 0
 +
|-
 +
| 0 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 0 || 1
 +
|-
 +
| 1 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 1
 +
|-
 +
| 0 || 1
 +
|-
 +
| 0 || 0
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 0 || 0
 +
|-
 +
| 1 || 0
 +
|}
 +
|}
 +
|}
 +
<br>
 +
 
 +
===Table 68.  Computation of an Analytic Series in Symbolic Terms===
 +
 
 +
<pre>
 +
Table 68.  Computation of an Analytic Series in Symbolic Terms
 +
o-----o-----o------------o----------o----------o----------o----------o----------o
 +
| u v | f g |    Df    |    Dg    |    df    |    dg    |    rf    |    rg    |
 +
o-----o-----o------------o----------o----------o----------o----------o----------o
 +
|    |    |            |          |          |          |          |          |
 +
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) |  du  dv  |    ()    |
 +
|    |    |            |          |          |          |          |          |
 +
| 0 1 | 1 0 |  (du) dv  | (du, dv) |    dv    | (du, dv) |  du  dv  |    ()    |
 +
|    |    |            |          |          |          |          |          |
 +
| 1 0 | 1 0 |  du (dv)  | (du, dv) |    du    | (du, dv) |  du  dv  |    ()    |
 +
|    |    |            |          |          |          |          |          |
 +
| 1 1 | 1 1 |  du  dv  | (du, dv) |    ()    | (du, dv) |  du  dv  |    ()    |
 +
|    |    |            |          |          |          |          |          |
 +
o-----o-----o------------o----------o----------o----------o----------o----------o
 +
</pre>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ '''Table 68.  Computation of an Analytic Series in Symbolic Terms'''
 +
|- style="background:paleturquoise"
 +
| ''u''&nbsp;&nbsp;''v''
 +
| ''f''&nbsp;&nbsp;''g''
 +
| D''f''
 +
| D''g''
 +
| d''f''
 +
| d''g''
 +
| d<sup>2</sup>''f''
 +
| d<sup>2</sup>''g''
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0&nbsp;&nbsp;0
 +
|-
 +
| 0&nbsp;&nbsp;1
 +
|-
 +
| 1&nbsp;&nbsp;0
 +
|-
 +
| 1&nbsp;&nbsp;1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0&nbsp;&nbsp;1
 +
|-
 +
| 1&nbsp;&nbsp;0
 +
|-
 +
| 1&nbsp;&nbsp;0
 +
|-
 +
| 1&nbsp;&nbsp;1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| ((d''u'')(d''v''))
 +
|-
 +
| (d''u'')&nbsp;d''v''&nbsp;
 +
|-
 +
| &nbsp;d''u''&nbsp;(d''v'')
 +
|-
 +
| d''u''&nbsp;&nbsp;d''v''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (d''u'', d''v'')
 +
|-
 +
| d''v''
 +
|-
 +
| d''u''
 +
|-
 +
| (&nbsp;)
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| d''u'' d''v''
 +
|-
 +
| d''u'' d''v''
 +
|-
 +
| d''u'' d''v''
 +
|-
 +
| d''u'' d''v''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (&nbsp;)
 +
|-
 +
| (&nbsp;)
 +
|-
 +
| (&nbsp;)
 +
|-
 +
| (&nbsp;)
 +
|}
 +
|}
 +
<br>
 +
 
 +
===Formula Display 18===
 +
 
 +
<pre>
 +
o-------------------------------------------------------------------------o
 +
|                                                                        |
 +
|  Df  =  uv. du  dv  + u(v). du (dv) + (u)v.(du) dv  + (u)(v).((du)(dv)) |
 +
|                                                                        |
 +
|  Dg  =  uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv)  |
 +
|                                                                        |
 +
o-------------------------------------------------------------------------o
 +
</pre>
 +
 
 +
<br><font face="courier new">
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|
 +
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| &nbsp;
 +
|-
 +
| D''f''
 +
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
 +
| + || ''u''(''v'')  || <math>\cdot</math> || d''u'' (d''v'')
 +
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'') d''v''
 +
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
 +
|-
 +
| &nbsp;
 +
|-
 +
| D''g''
 +
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || ''u''(''v'')  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
 +
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
 +
|-
 +
| &nbsp;
 +
|}
 +
|}
 +
</font><br>
 +
 
 +
===Figure 69.  Difference Map of F = ‹f,&nbsp;g› = ‹((u)(v)),&nbsp;((u,&nbsp;v))›===
 +
 
 +
<pre>
 +
o-----------------------------------o o-----------------------------------o
 +
| U                                | |`U`````````````````````````````````|
 +
|                                  | |```````````````````````````````````|
 +
|                ^                | |```````````````````````````````````|
 +
|                |                | |```````````````````````````````````|
 +
|      o-------o | o-------o      | |```````o-------o```o-------o```````|
 +
| ^    /`````````\|/`````````\    ^ | | ^ ```/      ^  \`/  ^      \``` ^ |
 +
|  \  /```````````|```````````\  /  | |``\``/        \  o  /        \``/``|
 +
|  \/`````u`````/|\`````v`````\/  | |```\/    u    \/`\/    v    \/```|
 +
|  /\``````````/`|`\``````````/\  | |```/\          /\`/\          /\```|
 +
|  o``\````````o``@``o````````/``o  | |``o  \        o``@``o        /  o``|
 
|  |```\```````|`````|```````/```|  | |``|  \      |`````|      /  |``|
 
|  |```\```````|`````|```````/```|  | |``|  \      |`````|      /  |``|
 
|  |````@``````|`````|``````@````|  | |``|    @-------->`<--------@    |``|
 
|  |````@``````|`````|``````@````|  | |``|    @-------->`<--------@    |``|
Line 9,532: Line 10,515:  
Figure 69.  Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
 
Figure 69.  Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 69.  Difference Map of F = ‹f,&nbsp;g› = ‹((u)(v)),&nbsp;((u,&nbsp;v))›'''</font></center></p>
    
===Formula Display 19===
 
===Formula Display 19===
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</font><br>
 
</font><br>
   −
===Figure 70-a.  Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===
+
===Figure 70-a.  Tangent Functor Diagram for F‹u,&nbsp;v› = ‹((u)(v)),&nbsp;((u,&nbsp;v))›===
    
<pre>
 
<pre>
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Figure 70-a.  Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
 
Figure 70-a.  Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
 
</pre>
 
</pre>
 +
 +
<br>
 +
<p>[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]</p>
 +
<p><center><font size="+1">'''Figure 70-a.  Tangent Functor Diagram for F‹u,&nbsp;v› = ‹((u)(v)),&nbsp;((u,&nbsp;v))›'''</font></center></p>
    
===Figure 70-b.  Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
 
===Figure 70-b.  Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
−
  −
[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|<font size="3">'''Figure 70-b.  Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]
      
<pre>
 
<pre>
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Figure 70-b.  Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
 
Figure 70-b.  Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
 
</pre>
 
</pre>
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[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|<font size="3">'''Figure 70-b.  Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]
Jon Awbrey
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