MyWikiBiz

Changes

User:Jon Awbrey/ATLAS (view source)

Revision as of 15:00, 25 August 2007

24,835 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 8,902: 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 8,953: 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 9,030: 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 9,205: 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="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+
{| align="right" style="background:paleturquoise; text-align:right"
−
| width="8%"  | E''G''<sub>''i''</sub>
+
| ''u'' =
−
| width="4%"  | =
+
|-
−
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
+
| ''v'' =
 
|}
 
|}
 +
|
 +
{| align="center" style="background:paleturquoise; text-align:center"
 +
| 1 1 0 0
 +
|-
 +
| 1 0 1 0
 
|}
 
|}
−
</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="left" style="background:paleturquoise; text-align:left"
−
| width="8%"  | D''G''<sub>''i''</sub>
+
| = ''u''
−
| 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;
+
| = ''v''
−
| width="4%"  | =
  −
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
  −
| width="4%"  | +
  −
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''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 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%"
+
{| align="center" style="background:paleturquoise; text-align:center"
−
| width="8%"  | E''f''
+
| 1 1 1 0
−
| width="4%"  | =
  −
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
   
|-
 
|-
−
| width="8%"  | E''g''
+
| 1 0 0 1
−
| width="4%"  | =
  −
| width="88%" | ((''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 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''
+
| ''f''<sub>0</sub>
−
| width="4%"  | =
+
|-
−
| width="20%" | ((''u'')(''v''))
+
| ''f''<sub>1</sub>
−
| width="4%"  | +
  −
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
   
|-
 
|-
−
| width="8%"  | D''g''
+
| ''f''<sub>2</sub>
−
| width="4%"  | =
+
|-
−
| width="20%" | ((''u'', ''v''))
+
| ''f''<sub>3</sub>
−
| width="4%"  | +
+
|-
−
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
+
| ''f''<sub>4</sub>
−
|}
+
|-
 +
| ''f''<sub>5</sub>
 +
|-
 +
| ''f''<sub>6</sub>
 +
|-
 +
| ''f''<sub>7</sub>
 
|}
 
|}
−
</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''
+
| ()
−
| = || ''uv''        || <math>\cdot</math> || 1
+
|-
−
| + || ''u''(''v'')   || <math>\cdot</math> || 1
+
| &nbsp;(''x'')(''y'')&nbsp;
−
| + || (''u'')''v''   || <math>\cdot</math> || 1
+
|-
−
| + || (''u'')(''v'') || <math>\cdot</math> || 0
+
| &nbsp;(''x'')&nbsp;''y''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;(''x'')&nbsp;&nbsp;&nbsp;&nbsp;
 
|-
 
|-
−
| E''f''
+
| &nbsp;&nbsp;''x''&nbsp;(''y'')&nbsp;
−
| = || ''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''
+
| &nbsp;&nbsp;&nbsp;&nbsp;(''y'')&nbsp;
−
| = || ''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''
+
| &nbsp;(''x'',&nbsp;''y'')&nbsp;
−
| = || ''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''
+
| &nbsp;(''x''&nbsp;&nbsp;''y'')&nbsp;
−
| = || ''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''
+
{| cellpadding="2" style="background:lightcyan"
−
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
+
| 0 0 0 0
−
|}
+
|-
−
|}
+
| 0 0 0 1
−
</font><br>
+
|-
−
 
+
| 0 0 1 0
−
===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''
+
| 0 0 1 1
−
| = || ''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''
+
| 0 1 0 0
−
| = || ''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''
+
| 0 1 0 1
−
| = || ''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''
+
| 0 1 1 0
−
| = || ''uv''         || <math>\cdot</math> || 0
+
|-
−
| + || ''u''(''v'')   || <math>\cdot</math> || 0
+
| 0 1 1 1
−
| + || (''u'')''v''   || <math>\cdot</math> || 0
+
|}
−
| + || (''u'')(''v'') || <math>\cdot</math> || 0
+
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ()
 +
|-
 +
| ()
 +
|-
 +
| &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;
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ''f''<sub>0</sub>
 +
|-
 +
| ''f''<sub>0</sub>
 +
|-
 +
| ''f''<sub>1</sub>
 +
|-
 +
| ''f''<sub>1</sub>
 +
|-
 +
| ''f''<sub>6</sub>
 +
|-
 +
| ''f''<sub>6</sub>
 +
|-
 +
| ''f''<sub>7</sub>
 +
|-
 +
| ''f''<sub>7</sub>
 
|}
 
|}
 +
|-
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>9</sub>
 +
|-
 +
| ''f''<sub>10</sub>
 +
|-
 +
| ''f''<sub>11</sub>
 +
|-
 +
| ''f''<sub>12</sub>
 +
|-
 +
| ''f''<sub>13</sub>
 +
|-
 +
| ''f''<sub>14</sub>
 +
|-
 +
| ''f''<sub>15</sub>
 
|}
 
|}
−
</font><br>
+
|
−
 
+
{| cellpadding="2" style="background:lightcyan"
−
===Table 67.  Computation of an Analytic Series in Terms of Coordinates===
+
| &nbsp;&nbsp;''x''&nbsp;&nbsp;''y''&nbsp;&nbsp;
−
 
+
|-
−
<pre>
+
| ((''x'',&nbsp;''y''))
−
Table 67.  Computation of an Analytic Series in Terms of Coordinates
+
|-
−
o--------o-------o-------o--------o-------o-------o-------o-------o
+
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''y''&nbsp;&nbsp;
−
| 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
+
| &nbsp;(''x''&nbsp;(''y''))
−
|       |      |      |        |      |      |      |      |
+
|-
−
| 0  0  | 0  0  | 0  0  |  0  1  | 0  1  | 0  0  | 0  0  | 0  0  |
+
| &nbsp;&nbsp;''x''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
−
|       |      |      |        |      |      |      |      |
+
|-
−
|       | 0  1  | 0  1  |        | 1  0  | 1  1  | 1  1  | 0  0  |
+
| ((''x'')&nbsp;''y'')&nbsp;
−
|       |      |      |        |      |      |      |      |
+
|-
−
|       | 1  0  | 1  0  |        | 1  0  | 1  1  | 1  1  | 0  0  |
+
| ((''x'')(''y''))
−
|       |      |      |        |      |      |      |      |
+
|-
−
|       | 1  1  | 1  1  |        | 1  1  | 1  0  | 0  0  | 1  0  |
+
| (())
−
|        |      |      |        |      |      |      |      |
+
|}
−
o--------o-------o-------o--------o-------o-------o-------o-------o
+
|
−
|       |      |      |        |      |      |      |      |
+
{| cellpadding="2" style="background:lightcyan"
−
| 0  1  | 0  0  | 0  1  |  1  0  | 1  0  | 0  0  | 0  0  | 0  0  |
+
| 1 0 0 0
−
|       |      |      |        |      |      |      |      |
+
|-
−
|       | 0  1  | 0  0  |        | 0  1  | 1  1  | 1  1  | 0  0  |
+
| 1 0 0 1
−
|        |      |      |        |      |      |      |      |
+
|-
−
|       | 1  0  | 1  1  |        | 1  1  | 0  1  | 0  1  | 0  0  |
+
| 1 0 1 0
−
|        |      |      |        |      |      |      |      |
+
|-
−
|        | 1  1  | 1  0  |        | 1 0  | 0 0 | 1  0 | 1  0  |
+
| 1 0 1 1
−
|       |      |      |        |      |      |      |      |
+
|-
−
o--------o-------o-------o--------o-------o-------o-------o-------o
+
| 1 1 0 0
−
|        |      |      |        |      |      |      |      |
+
|-
−
| 1 0 | 0  0  | 1  0 |  1 0  | 1  0  | 0  0  | 0  0  | 0  0  |
+
| 1 1 0 1
−
|       |      |      |        |      |      |      |      |
+
|-
−
|       | 0  1  | 1  1  |        | 1  1  | 0  1 | 0 1 | 0  0 |
+
| 1 1 1 0
−
|       |      |      |        |      |      |      |      |
+
|-
−
|        | 1 0 | 0  0  |        | 0  1  | 1  1  | 1 1 | 0  0  |
+
| 1 1 1 1
−
|       |      |      |        |      |      |      |      |
+
|}
−
|        | 1  1  | 0  1  |        | 1 0  | 0  0  | 1 0 | 1  0 |
+
|
−
|       |      |      |        |      |      |      |      |
+
{| cellpadding="2" style="background:lightcyan"
−
o--------o-------o-------o--------o-------o-------o-------o-------o
+
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
−
|       |      |      |        |      |      |      |      |
  −
|  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>
  −
 
  −
===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    |    rf    |
  −
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 64.  Transformation of Positions'''
  −
|- style="background:paleturquoise"
  −
| ''u''&nbsp;&nbsp;''v''
  −
| ''x''
  −
| ''y''
  −
| ''x''&nbsp;''y''
  −
| ''x''&nbsp;(''y'')
  −
| (''x'')&nbsp;''y''
  −
| (''x'')(''y'')
  −
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y''&nbsp;]
   
|-
 
|-
−
| width="12%" |
+
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
−
{| 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
+
| ((''u'',&nbsp;''v''))
 
|-
 
|-
−
| 1&nbsp;&nbsp;0
+
| ((''u'',&nbsp;''v''))
 
|-
 
|-
−
| 1&nbsp;&nbsp;1
+
| ((''u'')(''v''))
−
|}
  −
| width="12%" |
  −
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
  −
| 0
   
|-
 
|-
−
| 1
+
| ((''u'')(''v''))
 
|-
 
|-
−
| 1
+
| (())
 
|-
 
|-
−
| 1
+
| (())
 
|}
 
|}
−
| width="12%" |
+
|
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
{| cellpadding="2" style="background:lightcyan"
−
| 1
+
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>9</sub>
 
|-
 
|-
−
| 0
+
| ''f''<sub>9</sub>
 
|-
 
|-
−
| 0
+
| ''f''<sub>14</sub>
 
|-
 
|-
−
| 1
+
| ''f''<sub>14</sub>
−
|}
  −
| width="12%" |
  −
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
  −
| 0
  −
|-
  −
| 0
   
|-
 
|-
−
| 0
+
| ''f''<sub>15</sub>
 
|-
 
|-
−
| 1
+
| ''f''<sub>15</sub>
 
|}
 
|}
−
| 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%" |
+
</font><br>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
 
−
| 1
+
===Formula Display 14===
−
|-
+
 
−
| 0
+
<pre>
−
|-
+
o-------------------------------------------------o
−
| 0
+
|                                                |
−
|-
+
|  EG_i  =  G_i <u + du, v + dv>                |
−
| 0
+
|                                                |
 +
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%"
 +
| width="8%"  | E''G''<sub>''i''</sub>
 +
| width="4%"  | =
 +
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
 
|}
 
|}
−
| 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%" |
+
</font><br>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
 
−
| &uarr;
+
===Formula Display 15===
−
|-
+
 
−
| ''F''
+
<pre>
−
|-
+
o-------------------------------------------------o
−
| ‹''f'',&nbsp;''g''&nbsp;›
+
|                                                 |
−
|-
+
|   DG_i  =  G_i <u, v>  +  EG_i <u, v, du, dv>  |
−
| &uarr;
+
|                                                 |
−
|}
+
|         =  G_i <u, v>  +  G_i <u + du, v + dv>  |
−
|-
+
|                                                 |
−
| &nbsp;
+
o-------------------------------------------------o
−
| ((''u'')(''v''))
+
</pre>
−
| ((''u'',&nbsp;''v''))
+
 
−
| ''u''&nbsp;''v''
+
<br><font face="courier new">
−
| (''u'',&nbsp;''v'')
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
−
| (''u'')(''v'')
+
|
−
| (&nbsp;)
+
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
−
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''&nbsp;]
+
| width="8%"  | D''G''<sub>''i''</sub>
−
|}
+
| width="4%"  | =
−
<br>
+
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
−
 
+
| width="4%" | +
−
===Formula Display 18===
+
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
−
 
+
|-
−
<pre>
+
| width="8%"  | &nbsp;
−
o-------------------------------------------------------------------------o
+
| width="4%"  | =
−
|                                                                        |
+
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
−
| Df  = uv. du  dv  + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
+
| width="4%"  | +
−
|                                                                        |
+
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
−
|  Dg  =  uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv)  |
+
|}
−
|                                                                         |
+
|}
−
o-------------------------------------------------------------------------o
+
</font><br>
−
</pre>
+
 
−
 
+
===Formula Display 16===
−
<br><font face="courier new">
+
 
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+
<pre>
−
|
+
o-------------------------------------------------o
−
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
|                                                 |
−
| &nbsp;
+
|   Ef  =  ((u + du)(v + dv))                    |
−
|-
+
|                                                |
−
| D''f''
+
|   Eg  = ((u + du, v + dv))                    |
−
| = || ''uv''        || <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>
−
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
+
 
−
|-
+
<br><font face="courier new">
−
| &nbsp;
+
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
−
|-
+
|
−
| D''g''
+
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
−
| = || ''uv''         || <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''))
−
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+
|-
−
|-
+
| width="8%"  | E''g''
−
| &nbsp;
+
| width="4%"  | =
−
|}
+
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
−
|}
+
|}
−
</font><br>
+
|}
−
 
+
</font><br>
−
===Figure 69.  Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›===
+
 
−
 
+
===Formula Display 17===
−
<pre>
+
 
−
o-----------------------------------o o-----------------------------------o
+
<pre>
−
| U                                | |`U`````````````````````````````````|
+
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-------------------------------------------------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%"
 +
| 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  /        \``/``|
 
|  \  /```````````|```````````\  /  | |``\``/        \  o  /        \``/``|
Line 9,706: 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===
Line 9,744: Line 10,557:  
</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>
Line 9,829: Line 10,642:  
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))›===
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/49309...50423"