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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===
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===
|}
|}
</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===
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===
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===
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===
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===
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===
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===
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===
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)===
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)===
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)===
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===
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''===
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===
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)===
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)===
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)===
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===
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===
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===
</font><br>
</font><br>
===Figure 37-a. Tacit Extension of J (Areal)===
===Figure 37-a. Tacit Extension of ''J'' (Areal)===
<pre>
<pre>
</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'' (Areal)'''</font></center></p>
===Figure 37-b. Tacit Extension of ''J'' (Bundle)===
<pre>
<pre>
</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'' (Bundle)'''</font></center></p>
===Figure 37-c. Tacit Extension of ''J'' (Compact)===
<pre>
<pre>
</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'' (Compact)'''</font></center></p>
===Figure 37-d. Tacit Extension of ''J'' (Digraph)===
<pre>
<pre>
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'' (Digraph)'''</font></center></p>
===Table 38. Computation of EJ (Method 1)===
===Table 38. Computation of EJ (Method 1)===
</font><br>
</font><br>
===Figure 40-a. Enlargement of J (Areal)===
===Figure 40-a. Enlargement of ''J'' (Areal)===
<pre>
<pre>
</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'' (Areal)'''</font></center></p>
===Figure 40-b. Enlargement of ''J'' (Bundle)===
<pre>
<pre>
</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'' (Bundle)'''</font></center></p>
===Figure 40-c. Enlargement of ''J'' (Compact)===
<pre>
<pre>
</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'' (Compact)'''</font></center></p>
===Figure 40-d. Enlargement of ''J'' (Digraph)===
<pre>
<pre>
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'' (Digraph)'''</font></center></p>
===Table 41. Computation of DJ (Method 1)===
===Table 41. Computation of DJ (Method 1)===
</font><br>
</font><br>
===Figure 44-a. Difference Map of J (Areal)===
===Figure 44-a. Difference Map of ''J'' (Areal)===
<pre>
<pre>
</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'' (Areal)'''</font></center></p>
===Figure 44-b. Difference Map of ''J'' (Bundle)===
<pre>
<pre>
</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'' (Bundle)'''</font></center></p>
===Figure 44-c. Difference Map of ''J'' (Compact)===
<pre>
<pre>
</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'' (Compact)'''</font></center></p>
===Figure 44-d. Difference Map of ''J'' (Digraph)===
<pre>
<pre>
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'' (Digraph)'''</font></center></p>
===Table 45. Computation of dJ===
===Table 45. Computation of dJ===
</font><br>
</font><br>
===Figure 46-a. Differential of J (Areal)===
===Figure 46-a. Differential of ''J'' (Areal)===
<pre>
<pre>
</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'' (Areal)'''</font></center></p>
===Figure 46-b. Differential of ''J'' (Bundle)===
<pre>
<pre>
</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'' (Bundle)'''</font></center></p>
===Figure 46-c. Differential of ''J'' (Compact)===
<pre>
<pre>
</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'' (Compact)'''</font></center></p>
===Figure 46-d. Differential of ''J'' (Digraph)===
<pre>
<pre>
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'' (Digraph)'''</font></center></p>
===Table 47. Computation of rJ===
===Table 47. Computation of rJ===
</font><br>
</font><br>
===Figure 48-a. Remainder of J (Areal)===
===Figure 48-a. Remainder of ''J'' (Areal)===
<pre>
<pre>
</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'' (Areal)'''</font></center></p>
===Figure 48-b. Remainder of ''J'' (Bundle)===
<pre>
<pre>
</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'' (Bundle)'''</font></center></p>
===Figure 48-c. Remainder of ''J'' (Compact)===
<pre>
<pre>
</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'' (Compact)'''</font></center></p>
===Figure 48-d. Remainder of ''J'' (Digraph)===
<pre>
<pre>
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'' (Digraph)'''</font></center></p>
===Table 49. Computation Summary for J===
===Table 49. Computation Summary for J===
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)===
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===
</pre>
</pre>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 2. Fundamental Notations for Propositional Calculus'''
|+ '''Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators'''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
! Symbol
! Item
! Notation
! Notation
! Description
! Description
! Type
! Type
|-
|-
| <font face="lucida calligraphy">A<font>
| ''U''<sup> •</sup>
| {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}
| = [''u'', ''v'']
| Alphabet
| Source Universe
| [''n''] = '''n'''
| ['''B'''<sup>2</sup>]
|-
| ''X''<sup> •</sup>
| = [''x'']
| Target Universe
| ['''B'''<sup>1</sup>]
|-
| E''U''<sup> •</sup>
| = [''u'', ''v'', d''u'', d''v'']
| Extended Source Universe
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>]
|-
| E''X''<sup> •</sup>
| = [''x'', d''x'']
| Extended Target Universe
| ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>]
|-
| ''J''
| ''J'' : ''U'' → '''B'''
| Proposition
| ('''B'''<sup>2</sup> → '''B''') ∈ ['''B'''<sup>2</sup>]
|-
| ''J''
| ''J'' : ''U''<sup> •</sup> → ''X''<sup> •</sup>
| Transformation, or Mapping
| ['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
|-
|-
| ''A''<sub>''i''</sub>
| valign="top" |
| {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Dimension ''i''
| W
| '''B'''
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| W :
|-
|-
| ''A''
| ''U''<sup> •</sup> → E''U''<sup> •</sup> ,
∏<sub>''i''</sub> ''A''<sub>''i''</sub>
|-
|-
| ''A''*
| ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
| (''A'' → '''B''')
|-
|-
| ''A''<sup>•</sup>
| →
|-
| (E''U''<sup> •</sup> → E''X''<sup> •</sup>) ,
|-
| for each W in the set:
|-
| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}
|
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}
| Operator
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
['''B'''<sup>''n''</sup>]
|
|}</font><br>
|-
| ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] ,
|-
| ['''B'''<sup>1</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>] ,
<pre>
|-
| (['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>])
|-
| | Operator | Proposition | Map |
| →
|-
| | | | |
| (['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>])
|-
|
|-
| | | | |
|
|}
| | | | |
|-
| Trope | !h! : | !h!J : | !h!J : |
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| <math>\epsilon</math>
| | | | |
|-
| <math>\eta</math>
|-
| E
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
|-
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| D
|-
| d
| | | | |
|}
| Difference | D : | DJ : | DJ : |
| valign="top" |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| colspan="2" |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| | | | |
| Tacit Extension Operator || <math>\epsilon</math>
|-
| Trope Extension Operator || <math>\eta</math>
|-
| Enlargement Operator || E
|-
| Difference Operator || D
|-
| Differential Operator || d
|}
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''W'''</font>
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''W'''</font> :
|-
| ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> ,
|-
| ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
|-
| →
|-
| (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) ,
|-
| for each <font face=georgia>'''W'''</font> in the set:
|-
| {<font face=georgia>'''e'''</font>, <font face=georgia>'''E'''</font>, <font face=georgia>'''D'''</font>, <font face=georgia>'''T'''</font>}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Operator
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
|
|-
| ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] ,
|-
| ['''B'''<sup>1</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>] ,
|-
| (['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>])
|-
| →
|-
| (['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>])
|-
|
|-
|
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font>
|-
| <font face=georgia>'''E'''</font>
|-
| <font face=georgia>'''D'''</font>
|-
| <font face=georgia>'''T'''</font>
|}
| valign="top" |
| colspan="2" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| Radius Operator || <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>›
|-
| Secant Operator || <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›
|-
| Chord Operator || <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D›
|-
| Tangent Functor || <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d›
|}
|}<br>
===Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes===
<pre>
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
| | Operator | Proposition | Map |
o--------------o----------------------o--------------------o----------------------o
o--------------o----------------------o--------------------o----------------------o
| | | | |
| | | | |
| Differential | d : | dJ : | dJ : |
| Tacit | !e! : | !e!J : | !e!J : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] |
| | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Trope | !h! : | !h!J : | !h!J : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
| | | | |
o--------------o----------------------o--------------------o----------------------o
o--------------o----------------------o--------------------o----------------------o
| | | | |
| | | | |
| Remainder | r : | rJ : | rJ : |
| Enlargement | E : | EJ : | EJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
o--------------o----------------------o--------------------o----------------------o
o--------------o----------------------o--------------------o----------------------o
| | | | |
| | | | |
| Radius | $e$ = <!e!, !h!> : | | $e$J : |
| Difference | D : | DJ : | DJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Differential | d : | dJ : | dJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Remainder | r : | rJ : | rJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Radius | $e$ = <!e!, !h!> : | | $e$J : |
| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
</pre>
</pre>
===Figure 56-a1. Radius Map of the Conjunction J = uv===
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes'''
|- style="background:paleturquoise"
!
! Operator
! Proposition
! Map
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tacit
|-
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math>''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''B'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math>''J'' :
|-
|\ / \ / \ / \ /| |\ / \ / \ / \ /|
| [''u'', ''v'', d''u'', d''v''] → [''x'']
| \ / \ / \ / \ / | | \ / \ / \ / \ / |
|-
| o o o o | | o o o o |
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
| |\ / \ / \ /| | | |\ / \ / \ /| |
|}
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Trope
|-
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
===Figure 56-a2. Secant Map of the Conjunction J = uv===
|}
|-
<pre>
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Enlargement
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| Difference
| o o o o | | o o\\\\\o o |
|-
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Differential
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Remainder
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Radius
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| 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'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Secant
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| 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'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Chord
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| 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'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tangent
|-
| Functor
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font>''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}<br>
===Figure 56-a1. Radius Map of the Conjunction J = uv===
===Figure 56-a3. Chord Map of the Conjunction J = uv===
<pre>
<pre>
o
o
//\
/X\
////\
/XXX\
oXXXXXo
/X\////X\
/X\XXX/X\
/XXX\//XXX\
/XXX\X/XXX\
oXXXXXoXXXXXo
oXXXXXoXXXXXo
/\\XXX/X\XXX/\\
/ \XXX/X\XXX/ \
/\\\\X/XXX\X/\\\\
/ \X/XXX\X/ \
o\\\\\oXXXXXo\\\\\o
o oXXXXXo o
/ \\\\/ \XXX/ \\\\/ \
/ \ / \XXX/ \ / \
/ \\/ \X/ \\/ \
/ \ / \X/ \ / \
o o o o o
o o o o o
=|\ / \ /\\ / \ /|=
=|\ / \ / \ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | \ / \ / \ / \ / | =
= | o o\\\\\o o | =
= | o o o o | =
= | |\ / \\\\/ \ /| | =
= | |\ / \ / \ /| | =
= |u | \ / \\/ \ / | v| =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
o o--+--o o o--+--o o
//\ | \ / \ / | / \
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | / \
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o o
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / /\\ /\\
//\/////\ \ / /\\\\\/\\
////\/////\ \ / /\\\\ /\\\\
////\/////\ \ / /\\\\\/\\\\
o/////o/////o o o\\\\\o\\\\\o
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/\\\\\/\\
/ \/////\//// \ = = / \\\\/\\\\\/ \
/ \/////\// \ = = /\\\\\/\\\\\/\\\\
/ \/////\// \ = = / \\/\\\\\/ \
o o/////o o = = o\\\\\o\\\\\o\\\\\o
o o/////o o = = o o\\\\\o o
/ \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \
/ \ / \//// \ / \ = = / \ / \\\\/ \ / \
/ \ / \// \ / \ = = / \\/ \\/ \\/ \
/ \ / \// \ / \ = = / \ / \\/ \ / \
o o o o o o o o o o
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
|\ / \ / \ / \ /| |\ / \ / \ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / \ / \ / | | \ / \ / \ / \ / |
| o o o o | | o o\\\\\o o |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
. | \ / \ / | /X\ | \ / \ / | .
. //\XXXXXXXXX/\\ .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ DJ
!e!J //////\XXXXX/\\\\\\ !h!J
. ////////\XXX/\\\\\\\\ .
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
o-----o o-----o
o-----o o-----o
\ /
\ /
\ / dx = (u, du)(v, dv) - uv
\ /
x = uv \ /
x = uv \ / dx = uv
\ / dx = u dv + v du + du dv
\ /
\ /
\ /
o
o
Figure 56-a3. Chord Map of the Conjunction J = uv
Figure 56-a1. Radius Map of the Conjunction J = uv
</pre>
</pre>
===Figure 56-a4. Tangent Map of the Conjunction J = uv===
<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===
<pre>
<pre>
o
o
//\
/X\
/XXX\
oXXXXXo
//\XXX//\
////\X////\
o/////o/////o
/\\XXX//\XXX/\\
/\\/////\////\\
/\\\\X////\X/\\\\
/\\\\/////\//\\\\
o\\\\\o/////o\\\\\o
o\\\\\o/////o\\\\\o
/ \\\\/\\////\\\\\/ \
/ \\\\/ \//// \\\\/ \
/ \\/\\\\//\\\\\/ \
/ \\/ \// \\/ \
o o\\\\\o\\\\\o o
o o o o o
=|\ / \\\\/ \\\\/ \ /|=
=|\ / \ /\\ / \ /|=
= | \ / \\/ \\/ \ / | =
= | \ / \ /\\\\ / \ / | =
= | o o o o | =
= | o o\\\\\o o | =
= | |\ / \ / \ /| | =
= | |\ / \\\\/ \ /| | =
= |u | \ / \ / \ / | v| =
= |u | \ / \\/ \ / | v| =
o o--+--o o o--+--o o
o o--+--o o o--+--o o
//\ | \ / \ / | / \
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | / \
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o o
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / /\\ /\\
//\/////\ \ / / \\\\/ \
////\/////\ \ / /\\\\ /\\\\
////\/////\ \ / / \\/ \
o/////o/////o o o\\\\\o\\\\\o
o/////o/////o o o o o
/ \/////\//// \ = = /\\\\\/ \\\\/\\
/ \/////\//// \ = = /\\ / \ /\\
/ \/////\// \ = = /\\\\\/ \\/\\\\
/ \/////\// \ = = /\\\\ / \ /\\\\
o o/////o o = = o\\\\\o o\\\\\o
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/\\ /\\\\\/ \
/ \ / \//// \ / \ = = / \\\\/ \ / \\\\/ \
/ \ / \// \ / \ = = / \\/\\\\ /\\\\\/ \
/ \ / \// \ / \ = = / \\/ \ / \\/ \
o o o o o o o\\\\\o\\\\\o o
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \\\\/ \\\\/ \ /|
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \\/ \\/ \ / |
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o o o |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
. | \ / \ / | /X\ | \ / \ / | .
. //\XXXXXXXXX/\\ .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ dJ
!e!J //////\XXXXX/\\\\\\ EJ
. ////////\XXX/\\\\\\\\ .
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. | \////// \\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
.| x \// \\/ dx |.
o-----o o-----o
o-----o o-----o
\ /
\ /
\ /
\ / dx = (u, du)(v, dv)
x = uv \ / dx = u dv + v du
x = uv \ /
\ / dx = uv + u dv + v du + du dv
\ /
\ /
o
o
Figure 56-a4. Tangent Map of the Conjunction J = uv
Figure 56-a2. Secant Map of the Conjunction J = uv
</pre>
</pre>
===Figure 56-b1. Radius Map of the Conjunction J = uv===
<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===
<pre>
<pre>
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX/X\XXX/\\
/\\\\X/XXX\X/\\\\
o\\\\\oXXXXXo\\\\\o
/ \\\\/ \XXX/ \\\\/ \
/ \\/ \X/ \\/ \
o o o o o
=|\ / \ /\\ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | o o\\\\\o o | =
= | |\ / \\\\/ \ /| | =
| |
= |u | \ / \\/ \ / | v| =
| |
o o--+--o o o--+--o o
| |
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
o-----------------------o \
//\/////\ \ / /\\ /\\
| | \
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
| | \
/ \/////\//// \ = = /\\\\\/\\\\\/\\
/ \/////\// \ = = /\\\\\/\\\\\/\\\\
o o/////o o = = o\\\\\o\\\\\o\\\\\o
/ \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \
/ \ / \// \ / \ = = / \\/ \\/ \\/ \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ DJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
| o / \ o | | o\ /```\ o | |``o``````/```\``````o``|
. | \//// \\\\/ | .
| | o o | | | | \ o`````o | | |``|`````o`````o`````|``|
.| x \// \\/ dx |.
| | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``|
o-----o o-----o
\ /
| o \ / o | | o \```/ o | |``o``````\```/``````o``|
\ / dx = (u, du)(v, dv) - uv
| \ \ / / | | \ \`/ / | |```\``````\`/``````/```|
x = uv \ /
\ / dx = u dv + v du + du dv
\ /
o
Figure 56-a3. Chord Map of the Conjunction J = uv
</pre>
o-----------------------o o-----------------------o o-----------------------o
<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===
<pre>
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX//\XXX/\\
/\\\\X////\X/\\\\
o\\\\\o/////o\\\\\o
/ \\\\/\\////\\\\\/ \
/ \\/\\\\//\\\\\/ \
| o-----o | | \`````````````````````/ | | o-----o |
o o\\\\\o\\\\\o o
=|\ / \\\\/ \\\\/ \ /|=
= | \ / \\/ \\/ \ / | =
= | o o o o | =
= | |\ / \ / \ /| | =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
Figure 56-b1. Radius Map of the Conjunction J = uv
//\/////\ \ / /\\ /\\
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/ \\\\/\\
/ \/////\// \ = = /\\\\\/ \\/\\\\
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/\\ /\\\\\/ \
/ \ / \// \ / \ = = / \\/\\\\ /\\\\\/ \
o o o o o o o\\\\\o\\\\\o o
|\ / \ / \ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / \ / \ / | | \ / \\/ \\/ \ / |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ dJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ /
x = uv \ / dx = u dv + v du
\ /
\ /
o
Figure 56-a4. Tangent Map of the Conjunction J = uv
</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===
<pre>
<pre>
o-----------------------o
o-----------------------o
| / \ / \ |
| / \ / \ |
| / o \ |
| / o \ |
| / du /`\ dv \ |
| / du / \ dv \ |
| o /```\ o |
| o / \ o |
| | o`````o | |
| | o o | |
| | |`````| | |
| | | | | |
| | o`````o | |
| | o o | |
| o \```/ o |
| o \ / o |
| \ \`/ / |
| \ \ / / |
| \ o / |
| \ o / |
| \ / \ / |
| \ / \ / |
| | \
| | \
| o--o o--o | \
| o--o o--o | \
| /````\ / \ | \
| / \ / \ | \
| /``````o \ | \
| / o \ | \
| /``du``/ \ dv \ | \
| / du / \ dv \ | \
| o``````/ \ o | \
| o / \ o | \
| |`````o o | @ \
| | o o | @ \
| |`````| | | |\ \
| | | | | |\ \
| |`````o o | | \ \
| | o o | | \ \
| o``````\ / o | \ \
| o \ / o | \ \
| \``````\ / / | \ \
| \ \ / / | \ \
| \``````o / | \ \
| \ o / | \ \
| \````/ \ / | \ \
| \ / \ / | \ \
| o--o o--o | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \| \ | |```````````````````````|
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````|
| / \ / \ | |\ / \ /\ \ | |`````/````\`/````\`````|
| / o``````\ | | \ / o @ \ | |````/ o \````|
| / o \ | | \ / o @ \ | |````/``````o``````\````|
| / du / \``dv``\ | | \/ du /`\ dv \ | |```/ du / \ dv \```|
| / du / \ dv \ | | \/ du /`\ dv \ | |```/``du``/`\``dv``\```|
| o / \``````o | | o\ /```\ o | |``o / \ o``|
| o / \ o | | o\ /```\ o | |``o``````/```\``````o``|
| | o o`````| | | | \ o`````o | | |``| o o |``|
| | o o | | | | \ o`````o | | |``|`````o`````o`````|``|
| | | |`````| | | | @ |``@--|-----|------@``| | | |``|
| | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``|
| | o o`````| | | | o`````o | | |``| o o |``|
| | o o | | | | o`````o | | |``|`````o`````o`````|``|
| o \ /``````o | | o \```/ o | |``o \ / o``|
| o \ / o | | o \```/ o | |``o``````\```/``````o``|
| \ \ /``````/ | | \ \`/ / | |```\ \ / /```|
| \ \ / / | | \ \`/ / | |```\``````\`/``````/```|
| \ o``````/ | | \ o / | |````\ o /````|
| \ o / | | \ o / | |````\``````o``````/````|
| \ / \````/ | | \ / \ / | |`````\ /`\ /`````|
| \ / \ / | | \ / \ / | |`````\````/`\````/`````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | |```````````````````````|
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ / \ / \ /
\ EJ / \ J / \ EJ /
\ !h!J / \ J / \ !h!J /
\ / \ / \ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / o----------\---------/----------o \ /
o-------------------------------o
o-------------------------------o
Figure 56-b2. Secant Map of the Conjunction J = uv
Figure 56-b1. Radius Map of the Conjunction J = uv
</pre>
</pre>
===Figure 56-b3. Chord Map of the Conjunction J = uv===
<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===
<pre>
<pre>
\ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | | |
| |\ | \ \ | |```````````````````````|
| | \ | \ @ | | |
| | \ | \ @ | |```````````````````````|
| | \| \ | | |
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | | o--o o--o |
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````|
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / o``````\ | | \ / o @ \ | |````/ o \````|
| / du / \``dv``\ | | \/ du /`\ dv \ | | /``du``/`\``dv``\ |
| / du / \``dv``\ | | \/ du /`\ dv \ | |```/ du / \ dv \```|
| o / \``````o | | o\ /```\ o | | o``````/```\``````o |
| o / \``````o | | o\ /```\ o | |``o / \ o``|
| | o o`````| | | | \ o`````o | | | |`````o`````o`````| |
| | o o`````| | | | \ o`````o | | |``| o o |``|
| | | |`````| | | | @ |``@--|-----|------@ |`````|`````|`````| |
| | | |`````| | | | @ |``@--|-----|------@``| | | |``|
| | o o`````| | | | o`````o | | | |`````o`````o`````| |
| | o o`````| | | | o`````o | | |``| o o |``|
| o \ /``````o | | o \```/ o | | o``````\```/``````o |
| o \ /``````o | | o \```/ o | |``o \ / o``|
| \ \ /``````/ | | \ \`/ / | | \``````\`/``````/ |
| \ \ /``````/ | | \ \`/ / | |```\ \ / /```|
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ o``````/ | | \ o / | |````\ o /````|
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| \ / \````/ | | \ / \ / | |`````\ /`\ /`````|
| o--o o--o | | o--o o--o | | o--o o--o |
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | | |
| | | | |```````````````````````|
| | | | | |
| | | | |```````````````````````|
| | | | | |
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ / \ / \ /
\ DJ / \ J / \ DJ /
\ EJ / \ J / \ EJ /
\ / \ / \ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / o----------\---------/----------o \ /
o-------------------------------o
o-------------------------------o
Figure 56-b3. Chord Map of the Conjunction J = uv
Figure 56-b2. Secant Map of the Conjunction J = uv
</pre>
</pre>
===Figure 56-b4. Tangent Map of the Conjunction J = uv===
<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===
<pre>
<pre>
| / \ / \ |
| / \ / \ |
| / o \ |
| / o \ |
| / du / \ dv \ |
| / du /`\ dv \ |
| o / \ o |
| o /```\ o |
| | o o | |
| | o`````o | |
| | | | | |
| | |`````| | |
| | o o | |
| | o`````o | |
| o \ / o |
| o \```/ o |
| \ \ / / |
| \ \`/ / |
| \ o / |
| \ o / |
| \ / \ / |
| \ / \ / |
| /````\ / \ | \
| /````\ / \ | \
| /``````o \ | \
| /``````o \ | \
| /``du``/`\ dv \ | \
| /``du``/ \ dv \ | \
| o``````/```\ o | \
| o``````/ \ o | \
| |`````o`````o | @ \
| |`````o o | @ \
| |`````|`````| | |\ \
| |`````| | | |\ \
| |`````o`````o | | \ \
| |`````o o | | \ \
| o``````\```/ o | \ \
| o``````\ / o | \ \
| \``````\`/ / | \ \
| \``````\ / / | \ \
| \``````o / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| \````/ \ / | \ \
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / du /`\``dv``\ | | \/ du /`\ dv \ | | /``du``/ \``dv``\ |
| / du / \``dv``\ | | \/ du /`\ dv \ | | /``du``/`\``dv``\ |
| o /```\``````o | | o\ /```\ o | | o``````/ \``````o |
| o / \``````o | | o\ /```\ o | | o``````/```\``````o |
| | o`````o`````| | | | \ o`````o | | | |`````o o`````| |
| | o o`````| | | | \ o`````o | | | |`````o`````o`````| |
| | |`````|`````| | | | @ |``@--|-----|------@ |`````| |`````| |
| | | |`````| | | | @ |``@--|-----|------@ |`````|`````|`````| |
| | o`````o`````| | | | o`````o | | | |`````o o`````| |
| | o o`````| | | | o`````o | | | |`````o`````o`````| |
| o \```/``````o | | o \```/ o | | o``````\ /``````o |
| o \ /``````o | | o \```/ o | | o``````\```/``````o |
| \ \`/``````/ | | \ \`/ / | | \``````\ /``````/ |
| \ \ /``````/ | | \ \`/ / | | \``````\`/``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ / \ / \ /
\ dJ / \ J / \ dJ /
\ DJ / \ J / \ DJ /
\ / \ / \ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / o----------\---------/----------o \ /
o-------------------------------o
o-------------------------------o
Figure 56-b4. Tangent Map of the Conjunction J = uv
Figure 56-b3. Chord Map of the Conjunction J = uv
</pre>
</pre>
===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===
<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===
<pre>
<pre>
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
/ \////////// \ / \XXX/X\XXX/ \
| / du / \ dv \ |
/ \//////// \ / \X/XXX\X/ \
| o / \ o |
| | o o | |
| | | | | |
| | o o | |
o o o o o o o o
| o \ / o |
|\ / \ /| |\ / \ / \ / \ /|
| \ \ / / |
| \ / \ / | | \ / \ / \ / \ / |
| \ o / |
| \ / \ / | | o o o o |
| \ / \ / |
| \ / \ / | | |\ / \ / \ /| |
| o--o o--o |
| u \ / \ / v | |u | \ / \ / \ / | v|
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/`\ dv \ | \
| o``````/```\ o | \
| |`````o`````o | @ \
| |`````|`````| | |\ \
| |`````o`````o | | \ \
| o``````\```/ o | \ \
| \``````\`/ / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
////////\ /XXXXXXX\
o-----------------------o \ \
\ \
////////////o oXXXXXXXXXXXo
o-----------------------@ o--------\----------\---o o-----------------------o
///////////// \ //\XXXXXXXXX/\\
| |\ | \ \ | | |
///////////// \ ////\XXXXXXX/\\\\
| | \ | \ @ | | |
///////////// \ //////\XXXXX/\\\\\\
| | \| \ | | |
| o--o o--o | \ o--o \o--o | | o--o o--o |
///////////// \ //////////\X/\\\\\\\\\\
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
o//////////// o o///////////o\\\\\\\\\\\o
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
|\////////// / |\////////// \\\\\\\\\\/|
| / du /`\``dv``\ | | \/ du /`\ dv \ | | /``du``/ \``dv``\ |
| \//////// / | \//////// \\\\\\\\/ |
| o /```\``````o | | o\ /```\ o | | o``````/ \``````o |
| \////// / | \////// \\\\\\/ |
| | o`````o`````| | | | \ o`````o | | | |`````o o`````| |
| \//// / | \//// \\\\/ |
| | |`````|`````| | | | @ |``@--|-----|------@ |`````| |`````| |
| x \// / | x \// \\/ dx |
| | o`````o`````| | | | o`````o | | | |`````o o`````| |
o-----o / o-----o o-----o
| o \```/``````o | | o \```/ o | | o``````\ /``````o |
| \ \`/``````/ | | \ \`/ / | | \``````\ /``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| | | | | |
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
o-----------------------o o-----------------------o o-----------------------o
</pre>
\ / \ / \ /
\ dJ / \ J / \ dJ /
===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===
\ / \ / \ /
\ / o----------\---------/----------o \ /
<pre>
\ / | \ / | \ /
o o
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b4. Tangent Map of the Conjunction J = uv
</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===
<pre>
o o
//\ /X\
//\ /X\
////\ /XXX\
////\ /XXX\
//////\ oXXXXXo
//////\ oXXXXXo
////////\ //\XXX//\
////////\ /X\XXX/X\
//////////\ ////\X////\
//////////\ /XXX\X/XXX\
o///////////o o/////o/////o
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\/////\////\\
/ \////////// \ / \XXX/X\XXX/ \
/ \//////// \ /\\\\/////\//\\\\
/ \//////// \ / \X/XXX\X/ \
/ \////// \ o\\\\\o/////o\\\\\o
/ \////// \ o oXXXXXo o
/ \//// \ / \\\\/ \//// \\\\/ \
/ \//// \ / \ / \XXX/ \ / \
/ \// \ / \\/ \// \\/ \
/ \// \ / \ / \X/ \ / \
o o o o o o o o
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
|\ / \ /| |\ / \ / \ / \ /|
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \\\\/ \ /| |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \\/ \ / | v|
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | \ / \ / |
\ / \ /
\ / \ /
o o
o o
U% $E$ $E$U%
U% $e$ $E$U%
o------------------>o
o------------------>o
| |
| |
| |
| |
| |
| |
J | | $E$J
J | | $e$J
| |
| |
| |
| |
v v
v v
o------------------>o
o------------------>o
X% $E$ $E$X%
X% $e$ $E$X%
o o
o o
//\ /X\
//\ /X\
o o
o o
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
</pre>
</pre>
===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===
<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===
<pre>
<pre>
o o
o o
//\ //\
//\ /X\
////\ ////\
////\ /XXX\
//////\ o/////o
//////\ oXXXXXo
////////\ /X\////X\
////////\ //\XXX//\
//////////\ /XXX\//XXX\
//////////\ ////\X////\
o///////////o oXXXXXoXXXXXo
o///////////o o/////o/////o
/ \////////// \ /\\XXX/X\XXX/\\
/ \////////// \ /\\/////\////\\
/ \//////// \ /\\\\X/XXX\X/\\\\
/ \//////// \ /\\\\/////\//\\\\
/ \////// \ o\\\\\oXXXXXo\\\\\o
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/ \XXX/ \\\\/ \
/ \//// \ / \\\\/ \//// \\\\/ \
/ \// \ / \\/ \X/ \\/ \
/ \// \ / \\/ \// \\/ \
o o o o o o o o
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
|\ / \ /| |\ / \ /\\ / \ /|
\ / \ /
\ / \ /
o o
o o
U% $D$ $E$U%
U% $E$ $E$U%
o------------------>o
o------------------>o
| |
| |
| |
| |
| |
| |
J | | $D$J
J | | $E$J
| |
| |
| |
| |
v v
v v
o------------------>o
o------------------>o
X% $D$ $E$X%
X% $E$ $E$X%
o o
o o
//\ /X\
//\ /X\
o o
o o
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
</pre>
</pre>
===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===
<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===
<pre>
<pre>
//////////\ /XXX\//XXX\
//////////\ /XXX\//XXX\
o///////////o oXXXXXoXXXXXo
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\XXX//\XXX/\\
/ \////////// \ /\\XXX/X\XXX/\\
/ \//////// \ /\\\\X////\X/\\\\
/ \//////// \ /\\\\X/XXX\X/\\\\
/ \////// \ o\\\\\o/////o\\\\\o
/ \////// \ o\\\\\oXXXXXo\\\\\o
/ \//// \ / \\\\/\\////\\\\\/ \
/ \//// \ / \\\\/ \XXX/ \\\\/ \
/ \// \ / \\/\\\\//\\\\\/ \
/ \// \ / \\/ \X/ \\/ \
o o o o o\\\\\o\\\\\o o
o o o o o o o o
|\ / \ /| |\ / \\\\/ \\\\/ \ /|
|\ / \ /| |\ / \ /\\ / \ /|
| \ / \ / | | \ / \\/ \\/ \ / |
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | o o o o |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | |\ / \ / \ /| |
| \ / \ / | | |\ / \\\\/ \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
| u \ / \ / v | |u | \ / \\/ \ / | v|
o-----o o-----o o--+--o o o--+--o
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | \ / \ / |
\ / \ /
\ / \ /
o o
o o
U% $T$ $E$U%
U% $D$ $E$U%
o------------------>o
o------------------>o
| |
| |
| |
| |
| |
| |
J | | $T$J
J | | $D$J
| |
| |
| |
| |
v v
v v
o------------------>o
o------------------>o
X% $T$ $E$X%
X% $D$ $E$X%
o o
o o
//\ /X\
//\ /X\
o o
o o
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
</pre>
</pre>
===Formula Display 11===
<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===
<pre>
<pre>
o-----------------------------------------------------------o
o o
| |
//\ //\
| F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] |
////\ ////\
| |
//////\ o/////o
| where f = F_1 : [u, v] -> [x] |
////////\ /X\////X\
| |
//////////\ /XXX\//XXX\
| and g = F_2 : [u, v] -> [y] |
o///////////o oXXXXXoXXXXXo
| |
/ \////////// \ /\\XXX//\XXX/\\
o-----------------------------------------------------------o
/ \//////// \ /\\\\X////\X/\\\\
</pre>
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/\\////\\\\\/ \
/ \// \ / \\/\\\\//\\\\\/ \
o o o o o\\\\\o\\\\\o o
|\ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / | | \ / \\/ \\/ \ / |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $T$J
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
</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===
<pre>
o-----------------------------------------------------------o
| |
| F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] |
| |
| where f = F_1 : [u, v] -> [x] |
| |
| and g = F_2 : [u, v] -> [y] |
| |
o-----------------------------------------------------------o
</pre>
<br><font face="courier new">
<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="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| ''F''
| =
| =
| ‹''f'', ''g''›
| align="center" | ‹''f'', ''g''›
| =
| =
| ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
| align="center" | ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
| :
| :
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>
| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki>
| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki>
|-
|-
| align="left" colspan="2" | where
| colspan="2" | where
| ''f''
| align="center" | ''f''
| =
| =
| ''F''<sub>1</sub>
| align="center" | ''F''<sub>1</sub>
| :
| :
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>
| <nowiki>[</nowiki>''x''<nowiki>]</nowiki>
| <nowiki>[</nowiki>''x''<nowiki>]</nowiki>
|-
|-
| align="left" colspan="2" | and
| colspan="2" | and
| ''g''
| align="center" | ''g''
| =
| align="center" | ''F''<sub>2</sub>
| :
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>
| →
| <nowiki>[</nowiki>''y''<nowiki>]</nowiki>
|}
|}
</font><br>
<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%"
| align="left" | ''F''
| =
| ‹''f'', ''g''›
| =
| ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
| :
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>
| →
| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki>
|-
| align="left" colspan="2" | where
| ''f''
| =
| ''F''<sub>1</sub>
| :
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>
| →
| <nowiki>[</nowiki>''x''<nowiki>]</nowiki>
|-
| align="left" colspan="2" | and
| ''g''
| =
| =
| ''F''<sub>2</sub>
| ''F''<sub>2</sub>
</pre>
</pre>
===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''
|- style="background:paleturquoise"
! Item
! Notation
! Description
| | or | or | or |
! Type
| | Operand | Component | Mapping |
|-
| valign="top" | ''U''<sup> •</sup>
| | | | |
| valign="top" | <font face="courier new">= </font>[''u'', ''v'']
| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| valign="top" | Source Universe
| | | | |
| valign="top" | ['''B'''<sup>''n''</sup>]
| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
|-
| | | | |
| valign="top" | ''X''<sup> •</sup>
| valign="top" |
| | | | |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tacit | !e! : | !e!F_i : | !e!F : |
| <font face="courier new">= </font>[''x'', ''y'']
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
|-
| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| <font face="courier new">= </font>[''f'', ''g'']
| | | | |
|}
| valign="top" | Target Universe
| | | | |
| valign="top" | ['''B'''<sup>''k''</sup>]
| Trope | !h! : | !h!F_i : | !h!F : |
|-
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| valign="top" | E''U''<sup> •</sup>
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| valign="top" | <font face="courier new">= </font>[''u'', ''v'', d''u'', d''v'']
| | | | |
| valign="top" | Extended Source Universe
| valign="top" | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
|-
| valign="top" | E''X''<sup> •</sup>
| valign="top" |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face="courier new">= </font>[''x'', ''y'', d''x'', d''y'']
|-
| | | | |
| <font face="courier new">= </font>[''f'', ''g'', d''f'', d''g'']
| Difference | D : | DF_i : | DF : |
|}
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| valign="top" | Extended Target Universe
| valign="top" | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|-
| ''F''
| ''F'' = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup>
| Transformation, or Mapping
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
| ''f''
|-
| ''g''
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ''f'', ''g'' : ''U'' → '''B'''
|-
| ''f'' : ''U'' → [''x''] ⊆ ''X''<sup> •</sup>
|-
| ''g'' : ''U'' → [''y''] ⊆ ''X''<sup> •</sup>
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Proposition
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
| '''B'''<sup>''n''</sup> → '''B'''
|-
| ∈ ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> → '''B''')
| | | | [B^k x D^k] |
|-
| = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
|}
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| W
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| W :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> ,
|-
| ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
|-
| →
|-
| (E''U''<sup> •</sup> → E''X''<sup> •</sup>) ,
|-
| for each W in the set:
|-
| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Operator
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
|
|-
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
|-
| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
|-
| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
|-
| →
|-
| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
|-
|
|-
|
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math>
|-
| <math>\eta</math>
|-
| E
|-
| D
|-
| d
|}
| valign="top" |
| colspan="2" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| Tacit Extension Operator || <math>\epsilon</math>
|-
| Trope Extension Operator || <math>\eta</math>
|-
| Enlargement Operator || E
|-
| Difference Operator || D
|-
| Differential Operator || d
|}
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''W'''</font>
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''W'''</font> :
|-
| ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> ,
|-
| ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
|-
| →
|-
| (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) ,
|-
| for each <font face=georgia>'''W'''</font> in the set:
|-
| {<font face=georgia>'''e'''</font>, <font face=georgia>'''E'''</font>, <font face=georgia>'''D'''</font>, <font face=georgia>'''T'''</font>}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Operator
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
|
|-
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
|-
| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
|-
| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
|-
| →
|-
| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
|-
|
|-
|
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font>
|-
| <font face=georgia>'''E'''</font>
|-
| <font face=georgia>'''D'''</font>
|-
| <font face=georgia>'''T'''</font>
|}
| valign="top" |
| colspan="2" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| Radius Operator || <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>›
|-
| Secant Operator || <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›
|-
| Chord Operator || <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D›
|-
| Tangent Functor || <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d›
|}
|}<br>
===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
<pre>
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
| | Operator | Proposition | Transformation |
| | or | or | or |
| | Operand | Component | Mapping |
o--------------o----------------------o--------------------o----------------------o
o--------------o----------------------o--------------------o----------------------o
| | | | |
| | | | |
| Chord | $D$ = <!e!, D> : | | $D$F : |
| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| Operator | | | |
| | | | |
| | | | |
| | | | [B^n x D^n] -> |
| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
| | | | |
| | | | |
o--------------o----------------------o--------------------o----------------------o
o--------------o----------------------o--------------------o----------------------o
| | | | |
| | | | |
| Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |
| Tacit | !e! : | !e!F_i : | !e!F : |
| Functor | | | |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
| | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Trope | !h! : | !h!F_i : | !h!F : |
| Extension | 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
| | | | |
| Enlargement | E : | EF_i : | EF : |
| 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
| | | | |
| Difference | D : | DF_i : | DF : |
| 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
| | | | |
| Differential | d : | dF_i : | dF : |
| 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
| | | | |
| 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], |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | |
| | | B^n x D^n -> D | [B^n x D^n] -> |
| | | | [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] |
| | | | [B^k x D^k] |
| | | | |
| | | | |
</pre>
</pre>
{| 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'''
|- style="background:paleturquoise"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
| =
|
|
| align="center" | '''Operator<br>or<br>Operand'''
| align="center" | '''Proposition<br>or<br>Component'''
| align="center" | '''Transformation<br>or<br>Mapping'''
|-
| Operand
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ''F'' = ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
|-
| ''F'' = ‹''f'', ''g''› : ''U'' → ''X''
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ''F''<sub>''i''</sub> : 〈''u'', ''v''〉 → '''B'''
|-
| ''F''<sub>''i''</sub> : '''B'''<sup>''n''</sup> → '''B'''
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
| ''F'' : [''u'', ''v''] → [''x'', ''y'']
|-
|-
| ''F'' : '''B'''<sup>''n''</sup> → '''B'''<sup>''k''</sup>
| ''y''
|}
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tacit
|-
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math>''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B'''
|}
|}
|
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ‹''x'', ''y''›
| <math>\epsilon</math>''F'' :
| =
|-
| ''F''‹''u'', ''v''›
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
|}
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Trope
|-
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <math>\eta</math>''F''<sub>''i''</sub> :
|-
| =
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
| ''F''‹''u'', ''v''›
|-
| =
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
| ‹((''u'')(''v'')), ((''u'', ''v''))›
|}
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|}
|-
|
===Table 60. Propositional Transformation===
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Enlargement
|-
| Operator
|}
| u | v | f | g |
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | | | |
| E :
| 0 | 0 | 0 | 1 |
|-
| | | | |
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| 0 | 1 | 1 | 0 |
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
| 1 | 0 | 1 | 0 |
|}
| | | | |
|
| 1 | 1 | 1 | 1 |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | | | |
| E''F''<sub>''i''</sub> :
|-
| | | ((u)(v)) | ((u, v)) |
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
</pre>
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
===Figure 61. Propositional Transformation===
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
<pre>
| E''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Difference
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
|}
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
|-
| ////////o///////\ | |\\\\/ o \\\\\|
|
| //////////\///////\ | |\\\/ /\\ \\\\|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| o///////o///o///////o | |\\o o\\\o o\\|
| Differential
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
|-
| o///////o///o///////o | |\\o o\\\o o\\|
| Operator
| \///////\////////// | |\\\\ \\/ /\\\|
|}
| \///////o//////// | |\\\\\ o /\\\\|
|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Remainder
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Radius
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| 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>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Secant
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| 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>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Chord
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| 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>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tangent
|-
| Functor
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|}<br>
===Figure 63. Transformation of Positions===
===Formula Display 12===
<pre>
<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%"
|
| ''x''
| =
| ''f''‹''u'', ''v''›
| =
| ((''u'')(''v''))
|
|-
|
| ''y''
| =
| ''g''‹''u'', ''v''›
| =
| U | |\ | | |`U```````````````````````|
| ((''u'', ''v''))
| o---o o---o | | \ | | |``````o---o```o---o``````|
|
|}
| /'''''''o'''''''\ | | \ | | |````/ o \````|
|}
| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
</font><br>
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
</pre>
===Table 64. Transformation of Positions===
===Formula Display 13===
<pre>
<pre>
o-----------------------------------------------------------o
o-----o----------o----------o-------o-------o--------o--------o-------------o
| |
| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |
| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
| |
o-----------------------------------------------------------o
o-----o----------o----------o-------o-------o--------o--------o-------------o
</pre>
</pre>
===Table 65. Induced Transformation on Propositions===
<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%"
| ‹''x'', ''y''›
| =
| ''F''‹''u'', ''v''›
| =
| ‹((''u'')(''v'')), ((''u'', ''v''))›
|}
|}
</font><br>
<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%"
|
| ‹''x'', ''y''›
| =
| ''F''‹''u'', ''v''›
| =
| ‹((''u'')(''v'')), ((''u'', ''v''))›
|
|}
|}
</font><br>
===Table 60. Propositional Transformation===
<pre>
<pre>
Table 65. Induced Transformation on Propositions
Table 60. Propositional Transformation
o------------o---------------------------------o------------o
o-------------o-------------o-------------o-------------o
| u | v | f | g |
o-------------o-------------o-------------o-------------o
| | | | |
| 0 | 0 | 0 | 1 |
| | | | |
| 0 | 1 | 1 | 0 |
| | | | |
o------------o----------o-----------o----------o------------o
| 1 | 0 | 1 | 0 |
| | | | | |
| | | | |
| f_0 | () | 0 0 0 0 | () | f_0 |
| 1 | 1 | 1 | 1 |
| | | | | |
| | | | |
| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |
o-------------o-------------o-------------o-------------o
| | | | | |
| | | ((u)(v)) | ((u, v)) |
| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |
o-------------o-------------o-------------o-------------o
| | | | | |
| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |
| | | | | |
o------------o----------o-----------o----------o------------o
| | | | | |
o------------o----------o-----------o----------o------------o
</pre>
</pre>
===Formula Display 14===
<font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 60. Propositional Transformation'''
|- style="background:paleturquoise"
| |
| width="25%" | ''u''
| EG_i = G_i <u + du, v + dv> |
| width="25%" | ''v''
| |
| width="25%" | ''f''
| width="25%" | ''g''
|-
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| 0
|
|-
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| 0
| width="8%" | E''G''<sub>''i''</sub>
|-
| width="4%" | =
| 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%" |
| width="25%" |
| width="25%" | ((''u'')(''v''))
| width="25%" | ((''u'', ''v''))
|}
|}
</font><br>
</font><br>
===Formula Display 15===
===Figure 61. Propositional Transformation===
<pre>
<pre>
o-------------------------------------------------o
o-----------------------------------------------------o
| |
| U |
| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |
| |
| |
| o-----------o o-----------o |
| = G_i <u, v> + G_i <u + du, v + dv> |
| / \ / \ |
| |
| / o \ |
o-------------------------------------------------o
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
|
| | | | | |
| o o o o |
| width="8%" | D''G''<sub>''i''</sub>
| \ \ / / |
| width="4%" | =
| \ \ / / |
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
| \ o / |
| width="4%" | +
| \ / \ / |
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
| o-----------o o-----------o |
|-
| |
| width="8%" |
| |
| width="4%" | =
o-----------------------------------------------------o
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
/ \ / \
| width="4%" | +
/ \ / \
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
/ \ / \
|}
/ \ / \
|}
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
o-------------------------------------------------o
/ \ / \
| |
/ \ / \
| Ef = ((u + du)(v + dv)) |
o-------------------------o o-------------------------o
| |
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| Eg = ((u + du, v + dv)) |
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| |
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
o-------------------------------------------------o
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ | | /
\ | | /
\ | | /
\ f | | g /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
o-------\----|---------------------------|----/-------o
| X \ | | / |
| \| |/ |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 61. Propositional Transformation
</pre>
</pre>
<br><font face="courier new">
<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>
| width="8%" | E''f''
</font><br>
===Formula Display 17===
===Figure 62. Propositional Transformation (Short Form)===
<pre>
<pre>
o-------------------------------------------------o
o-------------------------o o-------------------------o
| |
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| Df = ((u)(v)) + ((u + du)(v + dv)) |
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| |
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| Dg = ((u, v)) + ((u + du, v + dv)) |
| ////////o///////\ | |\\\\/ o \\\\\|
| |
| //////////\///////\ | |\\\/ /\\ \\\\|
o-------------------------------------------------o
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| width="8%" | D''f''
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
| width="4%" | =
o-------------------------o o-------------------------o
| width="20%" | ((''u'')(''v''))
\ / \ /
| width="4%" | +
\ / \ /
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
\ / \ /
|-
\ f / \ g /
| width="8%" | D''g''
\ / \ /
| width="4%" | =
\ / \ /
| width="20%" | ((''u'', ''v''))
\ / \ /
| width="4%" | +
\ / \ /
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
\ / \ /
o---------\-----/---------------------\-----/---------o
| X \ / \ / |
</font><br>
| \ / \ / |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 62. Propositional Transformation (Short Form)
</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>
===Table 66-i. Computation Summary for f‹u, v› = ((u)(v))===
===Figure 63. Transformation of Positions===
<pre>
<pre>
o-----------------------------------------------------o
o--------------------------------------------------------------------------------o
|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
|` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
|` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
|` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
| |
|` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
| Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) |
|` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
| |
|` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
| |
|` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
| rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv |
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
| |
|` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
|` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
|` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
|` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
|` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
|` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
|+ Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))
|` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
|
|` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
o-----------\----|---------|---------|----------------o
" " \ | | | " "
| = || ''uv'' || <math>\cdot</math> || 1
" " \ | | | " "
| + || ''u''(''v'') || <math>\cdot</math> || 1
" " \ | | | " "
| + || (''u'')''v'' || <math>\cdot</math> || 1
" " \| | | " "
o-------------------------o \ | | o-------------------------o
| U | |\ | | |`U```````````````````````|
| o---o o---o | | \ | | |``````o---o```o---o``````|
| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')
| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
| /'''''''o'''''''\ | | \ | | |````/ o \````|
| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')
| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|
| \'''''''o'''''''/ | | \ | |````\ o /````|
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
| \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
| o---o o---o | | | \ | |``````o---o```o---o``````|
| | | | \ * |`````````````````````````|
o-------------------------o | | \ / o-------------------------o
\ | | | \ / | /
\ ((u)(v)) | | | \/ | ((u, 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''
\ | | /| | | /
o-------\----|---|-------/-|---------|---|----/-------o
| X \ | | / | | | / |
</font><br>
| \| | / | | |/ |
| o---|----/--o | o-------|---o |
| /' ' | ' / ' '\|/` ` ` ` | ` `\ |
| / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |
| /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |
| / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |
| @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |
| \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |
| \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 63. Transformation of Positions
</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 66-ii. Computation Summary for g‹u, v› = ((u, v))===
===Table 64. Transformation of Positions===
<pre>
<pre>
Table 66-ii. Computation Summary for g<u, v> = ((u, v))
Table 64. Transformation of Positions
o--------------------------------------------------------------------------------o
o-----o----------o----------o-------o-------o--------o--------o-------------o
| |
| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |
| !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| |
| | | | | | | | ^ |
| Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) |
| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |
| |
| | | | | | | | |
| Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |
| | | | | | | | = |
| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |
| | | | | | | | |
| rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 |
| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |
| | | | | | | | | |
o--------------------------------------------------------------------------------o
o-----o----------o----------o-------o-------o--------o--------o-------------o
| | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] |
o-----o----------o----------o-------o-------o--------o--------o-------------o
</pre>
</pre>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
{| 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''))
|+ '''Table 64. Transformation of Positions'''
|- style="background:paleturquoise"
| ''u'' ''v''
| ''x''
| ''y''
| ''x'' ''y''
| ''x'' (''y'')
| (''x'') ''y''
| (''x'')(''y'')
| ''X''<sup> •</sup> = [''x'', ''y'' ]
|-
|-
| E''g''
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| 0 0
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
|-
|-
| D''g''
| 0 1
|-
|-
| d''g''
| 1 0
|-
|-
| r''g''
| 1 1
| = || ''uv'' || <math>\cdot</math> || 0
|}
| + || ''u''(''v'') || <math>\cdot</math> || 0
| width="12%" |
| + || (''u'')''v'' || <math>\cdot</math> || 0
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| + || (''u'')(''v'') || <math>\cdot</math> || 0
| 0
|-
| 1
|-
| 1
|-
| 1
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 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%"
===Table 67. Computation of an Analytic Series in Terms of Coordinates===
| 0
|-
| 1
|-
| 1
| u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |
|-
| 0
| | | | | | | | |
|}
| 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 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%"
| | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
| 1
| | | | | | | | |
|-
| | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
| 0
| | | | | | | | |
|-
| | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 |
| 0
| | | | | | | | |
|-
| 0
|}
| 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 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%"
| | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| 0
| | | | | | | | |
|-
| | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| 0
| | | | | | | | |
|-
| | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 |
| 0
|-
| 0
| | | | | | | | |
|}
| 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 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%"
| | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| ↑
| | | | | | | | |
|-
| | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| ''F''
| | | | | | | | |
|-
| | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 |
| ‹''f'', ''g'' ›
| | | | | | | | |
|-
| ↑
| | | | | | | | |
|}
| 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 |
| ((''u'')(''v''))
| | | | | | | | |
| ((''u'', ''v''))
| | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 |
| ''u'' ''v''
| | | | | | | | |
| (''u'', ''v'')
| | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 |
| (''u'')(''v'')
| | | | | | | | |
| ( )
| ''U''<sup> •</sup> = [''u'', ''v'' ]
</pre>
|}
<br>
===Table 68. Computation of an Analytic Series in Symbolic Terms===
===Table 65. Induced Transformation on Propositions===
<pre>
<pre>
Table 68. Computation of an Analytic Series in Symbolic Terms
Table 65. Induced Transformation on Propositions
o-----o-----o------------o----------o----------o----------o----------o----------o
o------------o---------------------------------o------------o
| u v | f g | Df | Dg | df | dg | rf | rf |
| X% | <--- F = <f , g> <--- | U% |
o-----o-----o------------o----------o----------o----------o----------o----------o
o------------o----------o-----------o----------o------------o
| | u = | 1 1 0 0 | = u | |
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |
| | v = | 1 0 1 0 | = v | |
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
| 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |
| | x = | 1 1 1 0 | = f<u,v> | |
| | y = | 1 0 0 1 | = g<u,v> | |
| 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |
o------------o----------o-----------o----------o------------o
| | | | | |
| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |
| f_0 | () | 0 0 0 0 | () | f_0 |
| | | | | |
o-----o-----o------------o----------o----------o----------o----------o----------o
| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |
| | | | | |
| 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 |
| |
| | | | | |
| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |
| |
| | | | | |
| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |
| |
| | | | | |
o-------------------------------------------------------------------------o
| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |
</pre>
| | | | | |
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_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |
| | | | | |
| f_15 | (()) | 1 1 1 1 | (()) | f_15 |
| | | | | |
o------------o----------o-----------o----------o------------o
</pre>
<br><font face="courier new">
<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="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> •</sup>
| colspan="3" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"
| ←
| ''F'' = ‹''f'' , ''g''›
| ←
|}
| ''U''<sup> •</sup>
|- style="background:paleturquoise"
| rowspan="2" | ''f''<sub>''i''</sub>‹''x'', ''y''›
|
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
{| align="right" style="background:paleturquoise; text-align:right"
|
| ''u'' =
|-
|-
| D''f''
| ''v'' =
| = || ''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''
{| align="center" style="background:paleturquoise; text-align:center"
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
| 1 1 0 0
|-
|-
|
| 1 0 1 0
|}
|
{| align="left" style="background:paleturquoise; text-align:left"
| = ''u''
|-
|-
| = ''v''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
|}
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| rowspan="2" | ''f''<sub>''j''</sub>‹''u'', ''v''›
|- style="background:paleturquoise"
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|
{| align="right" style="background:paleturquoise; text-align:right"
| ''x'' =
|-
|-
|
| ''y'' =
|}
|}
|
{| align="center" style="background:paleturquoise; text-align:center"
| 1 1 1 0
|-
| 1 0 0 1
|}
|}
|
{| align="left" style="background:paleturquoise; text-align:left"
===Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›===
| = ''f''‹''u'', ''v''›
|-
| = ''g''‹''u'', ''v''›
|}
|-
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>0</sub>
|-
| ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |
| ''f''<sub>1</sub>
|-
| ''f''<sub>2</sub>
| /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```|
|-
| ''f''<sub>3</sub>
| |```\```````|`````|```````/```| | |``| \ |`````| / |``|
|-
| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| ''f''<sub>4</sub>
|-
| ''f''<sub>5</sub>
|-
| \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| ''f''<sub>6</sub>
| \`````\`````|`````/`````/ | |`````\ \ o / /`````|
|-
| ''f''<sub>7</sub>
| o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````|
|}
| \ | / | |``````````````\`````/``````````````|
|
| \ | / | |```````````````\```/```````````````|
{| cellpadding="2" style="background:lightcyan"
| \|/ | |````````````````\`/````````````````|
| ()
| @ | |`````````````````@`````````````````|
|-
| (''x'')(''y'')
|-
| (''x'') ''y''
|-
| (''x'')
|-
| ''x'' (''y'')
| X \ / \ / |
|-
| \ / \ / |
| (''y'')
| \ / \ / |
|-
| o----------------o o----------------o |
| (''x'', ''y'')
| / \ / \ |
|-
| / o \ |
| (''x'' ''y'')
| / / \ \ |
|}
| / / \ \ |
|
{| cellpadding="2" style="background:lightcyan"
| / / \ \ |
| 0 0 0 0
| / / \ \ |
|-
| o o o o |
| 0 0 0 1
| | | | | |
| | | | | |
| | f | | g | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| |
| |
| |
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
| |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
|
|-
|-
| d''f''
| 0 0 1 0
| = || ''uv'' || <math>\cdot</math> || 0
|-
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| 0 0 1 1
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
|-
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| 0 1 0 0
|-
| 0 1 0 1
|-
| 0 1 1 0
|-
| 0 1 1 1
|}
|
{| cellpadding="2" style="background:lightcyan"
| ()
|-
| ()
|-
| (''u'')(''v'')
|-
| (''u'')(''v'')
|-
| (''u'', ''v'')
|-
|-
|
| (''u'', ''v'')
|-
|-
| d''g''
| (''u'' ''v'')
|-
|-
|
| (''u'' ''v'')
|}
|}
|
{| 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>
|}
|}
|-
|
===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>8</sub>
<pre>
|-
| ''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>
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''x'' ''y''
|\ / \ /| |\ / \ / / \ / / \ /|
|-
| \ / \ / | | \ / \ / \ / \ / |
| ((''x'', ''y''))
| \ / \ / | | \ / O \ / O \ / O \ / |
|-
| \ / \ / | | o /@ o @\ o /@ o |
| ''y''
| \ / \ / | | |\ / \ / \ / \ / \ /| |
|-
| \ / \ / | | | \ / \ / \ / | |
| (''x'' (''y''))
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
|-
| ''x''
|-
| ((''x'') ''y'')
|-
| ((''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"
| ''u'' ''v''
|-
| ''u'' ''v''
|-
| ((''u'', ''v''))
|-
| ((''u'', ''v''))
|-
| ((''u'')(''v''))
|-
| ((''u'')(''v''))
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
|-
| \ / \ / | | \ / \ / \ / \ / |
| (())
| \ / \ / | | \ / O \ / O \ / O \ / |
|-
| \ / \ / | | o /@ o @ o @\ o |
| (())
| \ / \ / | | |\ / / \ / \ / \ \ /| |
|}
| \ / \ / | | | \ / \ / \ / | |
|
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
{| 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>
o-------------------------------------------------o
| |
| EG_i = G_i <u + du, v + dv> |
| |
o-------------------------------------------------o
</pre>
</pre>
===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
<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>
o-----------------------o o-----------------------o o-----------------------o
o-------------------------------------------------o
| dU | | dU | | dU |
| |
| o--o o--o | | o--o o--o | | o--o 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 | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
o-------------------------------------------------o
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\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''G''<sub>''i''</sub>
| width="4%" | =
| | | | | |
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
o-----------------------o o-----------------------o o-----------------------o
| width="4%" | +
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
|-
| width="8%" |
| width="4%" | =
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
| width="4%" | +
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
|}
|}
</font><br>
===Formula Display 16===
o-----------------------o o-----------------------o o-----------------------o
<pre>
| dU | | dU | | dU |
o-------------------------------------------------o
| o--o o--o | | o--o o--o | | o--o o--o |
| |
| Ef = ((u + du)(v + dv)) |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| Eg = ((u + du, v + dv)) |
| |
o-------------------------------------------------o
</pre>
| | | | | |
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%"
| o--o o--o | | o--o o--o | | o--o o--o |
|
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| width="8%" | E''f''
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| width="4%" | =
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
|-
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| width="8%" | E''g''
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| width="4%" | =
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
|}
|}
</font><br>
===Formula Display 17===
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
<pre>
o-------------------------------------------------o
| |
| Df = ((u)(v)) + ((u + du)(v + dv)) |
| |
| Dg = ((u, v)) + ((u + du, v + dv)) |
| |
o-------------------------------------------------o
</pre>
o-----------------------o o-----------------------o o-----------------------o
<br><font face="courier new">
| dU | | dU | | dU |
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| o--o o--o | | o--o o--o | | o--o o--o |
|
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| width="8%" | D''f''
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| width="4%" | =
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| width="20%" | ((''u'')(''v''))
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| width="4%" | +
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
|-
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| width="8%" | D''g''
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| width="4%" | =
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| 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'' ''v''
| ''f'' ''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 0
|-
| 0 1
|-
| 1 0
|-
| 1 1
|}
|
{| align="center" border="0" cellpadding="2" 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="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%"
| (d''u'', d''v'')
|-
| d''v''
|-
| d''u''
|-
| ( )
|}
|
{| 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%"
| ( )
|-
| ( )
|-
| ( )
|-
| ( )
|}
|}
<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%"
|
|-
| 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''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'')
|-
|
|}
|}
</font><br>
===Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, 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``|
| |```\```````|`````|```````/```| | |``| \ |`````| / |``|
| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| |```````````|`````|```````````| | |``| |`````| |``|
| o```````````o` ^ `o```````````o | |``o o`````o o``|
| \```````````\`|`/```````````/ | |```\ \```/ /```|
| \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| \`````\`````|`````/`````/ | |`````\ \ o / /`````|
| \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````|
| o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````|
| \ | / | |``````````````\`````/``````````````|
| \ | / | |```````````````\```/```````````````|
| \|/ | |````````````````\`/````````````````|
| @ | |`````````````````@`````````````````|
o-----------------------------------o o-----------------------------------o
\ / \ /
\ / \ /
\ ((u)(v)) / \ ((u, v)) /
\ / \ /
\ / \ /
o----------\-------------/-----------------------\-------------/----------o
| X \ / \ / |
| \ / \ / |
| \ / \ / |
| o----------------o o----------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | f | | g | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o----------------o o----------------o |
| |
| |
| |
o-------------------------------------------------------------------------o
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
</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, g› = ‹((u)(v)), ((u, v))›'''</font></center></p>
===Formula Display 19===
<pre>
o-------------------------------------------------------------------------------o
| |
| df = uv. 0 + u(v). du + (u)v. 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%"
|
|-
| 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'')
|-
|
|-
| 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'')
|-
|
|}
|}
</font><br>
===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===
<pre>
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ \ / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o /@ o
/ \ / \ / \ \ / \ / \ \ / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o /@ o o /@ o /@ o /@ o /@ o
|\ / \ /| |\ / \ / / \ / / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @\ o /@ o |
| \ / \ / | | |\ / \ / \ / \ / \ /| |
| \ / \ / | | | \ / \ / \ / | |
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
o-------o @\ o-------o o---+---o @\ o @\ o---+---o
\ / | \ / \ / \ / \ / |
\ / | \ / \ / |
\ / | du \ / O \ / dv |
\ / o-------o @\ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
F | | $T$F
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ / / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o @\ o
/ \ / \ / \ \ / \ / \ / \ / / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o @\ o o /@ o /@ o @\ o @\ o
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @ o @\ o |
| \ / \ / | | |\ / / \ / \ / \ \ /| |
| \ / \ / | | | \ / \ / \ / | |
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
o-------o @ o-------o o---+---o @ o @ o---+---o
\ / | \ / / \ \ / |
\ / | \ / \ / |
\ / | dx \ / O \ / dy |
\ / o-------o @ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
</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, v› = ‹((u)(v)), ((u, v))›'''</font></center></p>
===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
<pre>
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
= du' @ u v o-----------------------o dv' @ u v =
= du' @ u v o-----------------------o dv' @ u v =
= | dU' | =
= | dU' | =
= | o--o o--o | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| o--o o--o |
| |
| |
o-----------------------o
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o o-----------------------o o-----------------------o
= u' o-----------------------o v' =
= | U' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
</pre>
[[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>]]
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
</pre>
