Changes

65,618 bytes added , 15:00, 25 August 2007

add image stub

Line 1,141: Line 1,141:

Figure 12. The Anchor

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]

+

<center>'''Figure 12. The Anchor'''</center>

===Figure 13. The Tiller===

Line 1,174: Line 1,178:

Figure 13. The Tiller

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]

+

<center>'''Figure 13. The Tiller'''</center>

===Table 14. Differential Propositions===

Line 1,667: Line 1,675:

|}

+

===Figure 16. A Couple of Fourth Gear Orbits===

+

+

[[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]

+

<center>'''Figure 16. A Couple of Fourth Gear Orbits'''</center>

===Figure 16-a. A Couple of Fourth Gear Orbits: 1===

Line 2,064: Line 2,078:

Figure 18-a. Extension from 1 to 2 Dimensions: Areal

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]

+

<center>'''Figure 18-a. Extension from 1 to 2 Dimensions: Areal'''</center>

===Figure 18-b. Extension from 1 to 2 Dimensions: Bundle===

Line 2,093: Line 2,111:

Figure 18-b. Extension from 1 to 2 Dimensions: Bundle

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]

+

<center>'''Figure 18-b. Extension from 1 to 2 Dimensions: Bundle'''</center>

===Figure 18-c. Extension from 1 to 2 Dimensions: Compact===

Line 2,124: Line 2,146:

Figure 18-c. Extension from 1 to 2 Dimensions: Compact

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]

+

<center>'''Figure 18-c. Extension from 1 to 2 Dimensions: Compact'''</center>

===Figure 18-d. Extension from 1 to 2 Dimensions: Digraph===

Line 2,143: Line 2,169:

Figure 18-d. Extension from 1 to 2 Dimensions: Digraph

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]

+

<center>'''Figure 18-d. Extension from 1 to 2 Dimensions: Digraph'''</center>

===Figure 19-a. Extension from 2 to 4 Dimensions: Areal===

Line 2,186: Line 2,216:

Figure 19-a. Extension from 2 to 4 Dimensions: Areal

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]

+

<center>'''Figure 19-a. Extension from 2 to 4 Dimensions: Areal'''</center>

===Figure 19-b. Extension from 2 to 4 Dimensions: Bundle===

Line 2,247: Line 2,281:

Figure 19-b. Extension from 2 to 4 Dimensions: Bundle

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]

+

<center>'''Figure 19-b. Extension from 2 to 4 Dimensions: Bundle'''</center>

===Figure 19-c. Extension from 2 to 4 Dimensions: Compact===

Line 2,287: Line 2,325:

Figure 19-c. Extension from 2 to 4 Dimensions: Compact

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]

+

<center>'''Figure 19-c. Extension from 2 to 4 Dimensions: Compact'''</center>

===Figure 19-d. Extension from 2 to 4 Dimensions: Digraph===

Line 2,330: Line 2,372:

Figure 19-d. Extension from 2 to 4 Dimensions: Digraph

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]

+

<center>'''Figure 19-d. Extension from 2 to 4 Dimensions: Digraph'''</center>

===Figure 20-i. Thematization of Conjunction (Stage 1)===

Line 2,360: Line 2,406:

Figure 20-i. Thematization of Conjunction (Stage 1)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]

+

<center>'''Figure 20-i. Thematization of Conjunction (Stage 1)'''</center>

===Figure 20-ii. Thematization of Conjunction (Stage 2)===

Line 2,407: Line 2,457:

Figure 20-ii. Thematization of Conjunction (Stage 2)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]

+

<center>'''Figure 20-ii. Thematization of Conjunction (Stage 2)'''</center>

===Figure 20-iii. Thematization of Conjunction (Stage 3)===

Line 2,450: Line 2,504:

Figure 20-iii. Thematization of Conjunction (Stage 3)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]

+

<center>'''Figure 20-iii. Thematization of Conjunction (Stage 3)'''</center>

===Figure 21. Thematization of Disjunction and Equality===

Line 2,516: Line 2,574:

Figure 21. Thematization of Disjunction and Equality

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]

+

<center>'''Figure 21. Thematization of Disjunction and Equality'''</center>

===Table 22. Disjunction ''f'' and Equality ''g''===

Line 3,673: Line 3,735:

Figure 30. Generic Frame of a Logical Transformation

</pre>

+

'''Note.''' The following image was corrupted in transit between software platforms.

+

+

[[Image:Diff Log Dyn Sys -- Figure 30 -- Generic Frame of a Logical Transformation.gif|center]]

+

<center>'''Figure 30. Generic Frame of a Logical Transformation'''</center>

===Formula Display 3===

Line 3,729: Line 3,797:

Figure 31. Operator Diagram (1)

</pre>

+

'''Note.''' The following image was corrupted in transit between software platforms.

+

+

[[Image:Diff Log Dyn Sys -- Figure 31 -- Operator Diagram (1).gif|center]]

+

<center>'''Figure 31. Operator Diagram (1)'''</center>

===Figure 32. Operator Diagram (2)===

Line 3,754: Line 3,828:

Figure 32. Operator Diagram (2)

</pre>

+

'''Note.''' The following image was corrupted in transit between software platforms.

+

+

[[Image:Diff Log Dyn Sys -- Figure 32 -- Operator Diagram (2).gif|center]]

+

<center>'''Figure 32. Operator Diagram (2)'''</center>

===Figure 33-i. Analytic Diagram (1)===

Line 3,774: Line 3,854:

Figure 33-i. Analytic Diagram (1)

</pre>

+

'''Note.''' The following image was corrupted in transit between software platforms.

+

+

[[Image:Diff Log Dyn Sys -- Figure 33-i -- Analytic Diagram (1).gif|center]]

+

<center>'''Figure 33-i. Analytic Diagram (1)'''</center>

===Figure 33-ii. Analytic Diagram (2)===

Line 3,794: Line 3,880:

Figure 33-ii. Analytic Diagram (2)

</pre>

+

'''Note.''' The following image was corrupted in transit between software platforms.

+

+

[[Image:Diff Log Dyn Sys -- Figure 33-ii -- Analytic Diagram (2).gif|center]]

+

<center>'''Figure 33-ii. Analytic Diagram (2)'''</center>

===Formula Display 4===

Line 4,012: Line 4,104:

Figure 34. Tangent Functor Diagram

</pre>

+

'''Note.''' The following image was corrupted in transit between software platforms.

+

+

[[Image:Diff Log Dyn Sys -- Figure 34 -- Tangent Functor Diagram.gif|center]]

+

<center>'''Figure 34. Tangent Functor Diagram'''</center>

===Figure 35. Conjunction as Transformation===

Line 4,067: Line 4,165:

Figure 35. Conjunction as Transformation

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]

+

<center>'''Figure 35. Conjunction as Transformation'''</center>

===Table 36. Computation of !e!J===

Line 4,140: Line 4,242:

−

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

+

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

<pre>

Line 4,183: Line 4,285:

</pre>

−

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

+

+

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

+

<center>'''Figure 37-a. Tacit Extension of ''J''  (Areal)'''</center>

+

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

<pre>

Line 4,252: Line 4,358:

</pre>

−

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

+

+

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

+

<center>'''Figure 37-b. Tacit Extension of ''J''  (Bundle)'''</center>

+

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

<pre>

Line 4,292: Line 4,402:

</pre>

−

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

+

+

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

+

<center>'''Figure 37-c. Tacit Extension of ''J''  (Compact)'''</center>

+

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

<pre>

Line 4,333: Line 4,447:

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

</pre>

+

+

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

+

<center>'''Figure 37-d. Tacit Extension of ''J''  (Digraph)'''</center>

===Table 38. Computation of EJ (Method 1)===

Line 4,504: Line 4,622:

−

===Figure 40-a. Enlargement of J (Areal)===

+

===Figure 40-a. Enlargement of ''J''  (Areal)===

<pre>

Line 4,547: Line 4,665:

</pre>

−

===Figure 40-b. Enlargement of J (Bundle)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-a. Enlargement of ''J''  (Areal)'''</center>

+

===Figure 40-b. Enlargement of ''J''  (Bundle)===

<pre>

Line 4,616: Line 4,738:

</pre>

−

===Figure 40-c. Enlargement of J (Compact)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-b. Enlargement of ''J''  (Bundle)'''</center>

+

===Figure 40-c. Enlargement of ''J''  (Compact)===

<pre>

Line 4,656: Line 4,782:

</pre>

−

===Figure 40-d. Enlargement of J (Digraph)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-c. Enlargement of ''J''  (Compact)'''</center>

+

===Figure 40-d. Enlargement of ''J''  (Digraph)===

<pre>

Line 4,697: Line 4,827:

Figure 40-d. Enlargement of J (Digraph)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-d. Enlargement of ''J''  (Digraph)'''</center>

===Table 41. Computation of DJ (Method 1)===

Line 4,964: Line 5,098:

−

===Figure 44-a. Difference Map of J (Areal)===

+

===Figure 44-a. Difference Map of ''J''  (Areal)===

<pre>

Line 5,007: Line 5,141:

</pre>

−

===Figure 44-b. Difference Map of J (Bundle)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]

+

<center>'''Figure 44-a. Difference Map of ''J''  (Areal)'''</center>

+

===Figure 44-b. Difference Map of ''J''  (Bundle)===

<pre>

Line 5,076: Line 5,214:

</pre>

−

===Figure 44-c. Difference Map of J (Compact)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]

+

<center>'''Figure 44-b. Difference Map of ''J''  (Bundle)'''</center>

+

===Figure 44-c. Difference Map of ''J''  (Compact)===

<pre>

Line 5,117: Line 5,259:

</pre>

−

===Figure 44-d. Difference Map of J (Digraph)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]

+

<center>'''Figure 44-c. Difference Map of ''J''  (Compact)'''</center>

+

===Figure 44-d. Difference Map of ''J''  (Digraph)===

<pre>

Line 5,155: Line 5,301:

Figure 44-d. Difference Map of J (Digraph)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]

+

<center>'''Figure 44-d. Difference Map of ''J''  (Digraph)'''</center>

===Table 45. Computation of dJ===

Line 5,193: Line 5,343:

−

===Figure 46-a. Differential of J (Areal)===

+

===Figure 46-a. Differential of ''J''  (Areal)===

<pre>

Line 5,236: Line 5,386:

</pre>

−

===Figure 46-b. Differential of J (Bundle)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]

+

<center>'''Figure 46-a. Differential of ''J''  (Areal)'''</center>

+

===Figure 46-b. Differential of ''J''  (Bundle)===

<pre>

Line 5,305: Line 5,459:

</pre>

−

===Figure 46-c. Differential of J (Compact)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]

+

<center>'''Figure 46-b. Differential of ''J''  (Bundle)'''</center>

+

===Figure 46-c. Differential of ''J''  (Compact)===

<pre>

Line 5,342: Line 5,500:

</pre>

−

===Figure 46-d. Differential of J (Digraph)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]

+

<center>'''Figure 46-c. Differential of ''J''  (Compact)'''</center>

+

===Figure 46-d. Differential of ''J''  (Digraph)===

<pre>

Line 5,378: Line 5,540:

Figure 46-d. Differential of J (Digraph)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]

+

<center>'''Figure 46-d. Differential of ''J''  (Digraph)'''</center>

===Table 47. Computation of rJ===

Line 5,439: Line 5,605:

−

===Figure 48-a. Remainder of J (Areal)===

+

===Figure 48-a. Remainder of ''J''  (Areal)===

<pre>

Line 5,482: Line 5,648:

</pre>

−

===Figure 48-b. Remainder of J (Bundle)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]

+

<center>'''Figure 48-a. Remainder of ''J''  (Areal)'''</center>

+

===Figure 48-b. Remainder of ''J''  (Bundle)===

<pre>

Line 5,551: Line 5,721:

</pre>

−

===Figure 48-c. Remainder of J (Compact)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]

+

<center>'''Figure 48-b. Remainder of ''J''  (Bundle)'''</center>

+

===Figure 48-c. Remainder of ''J''  (Compact)===

<pre>

Line 5,591: Line 5,765:

</pre>

−

===Figure 48-d. Remainder of J (Digraph)===

+

+

[[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]

+

<center>'''Figure 48-c. Remainder of ''J''  (Compact)'''</center>

+

===Figure 48-d. Remainder of ''J''  (Digraph)===

<pre>

Line 5,627: Line 5,805:

Figure 48-d. Remainder of J (Digraph)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]

+

<center>'''Figure 48-d. Remainder of ''J''  (Digraph)'''</center>

===Table 49. Computation Summary for J===

Line 6,228: Line 6,410:

Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]

+

<center>'''Figure 52. Decomposition of E''J'''''</center>

===Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===

Line 6,279: Line 6,465:

Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]

+

<center>'''Figure 53. Decomposition of D''J'''''</center>

===Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators===

Line 6,348: Line 6,538:

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

</pre>

−

~~'''W'''~~

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"

Line 6,389: Line 6,577:

| ['''B'''2] → ['''B'''1]

|-

−

| valign="top" | ~~W~~

+

| valign="top" |

−

| ~~valign~~="~~top~~" ~~| W&nbsp~~;:~~~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~''U''&nbsp~~;~~• → E''U'' • ,~~

+

| W

−

~~''X'' • → E''X'' • ,~~

+

|}

−

~~(''U'' • → ''X'' •) → ~~

+

| valign="top" |

−

~~E''U'' • → E''X'' •) ,~~

−

~~For each~~ W ~~in the set: ~~

−

~~{<math>\epsilon</math>, <math>\eta</math>, E, D, d~~}~~~~

−

| valign="top" ~~| Operator~~

−

~~| valign="top" |  ~~

−

~~['''B'''2] → ['''B'''2 × '''D'''2] ,~~

−

~~['''B'''1] → ['''B'''1 × '''D'''1] ,~~

−

~~(['''B'''2] → ['''B'''1]) → ~~

−

~~(['''B'''2 × '''D'''2] → ['''B'''1 × '''D'''1])~~

−

~~   ~~

−

|-

−

|

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~<math>\epsilon</math>~~

+

| W :

|-

−

| <~~math~~>~~\eta~~</~~math~~>

+

| ''U'' • → E''U'' • ,

|-

−

| E

+

| ''X'' • → E''X'' • ,

|-

−

| D

+

| (''U'' • → ''X'' •)

|-

−

~~| d~~

+

| →

−

|}

−

~~| valign="top"~~ | &~~nbsp~~;

−

~~| 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~~>

+

| (E''U'' • → E''X'' •) ,

|-

−

| ~~Enlargement Operator || E~~

+

| for each W in the set:

|-

−

| ~~Difference Operator ||~~ D

+

| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}

−

|-

−

~~| Differential Operator ||~~ 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'''2] → ['''B'''2 × '''D'''2] ,

+

|-

+

| ['''B'''1] → ['''B'''1 × '''D'''1] ,

+

|-

+

| (['''B'''2] → ['''B'''1])

+

|-

+

| →

+

|-

+

| (['''B'''2 × '''D'''2] → ['''B'''1 × '''D'''1])

|-

|  

+

|-

|  

−

|  

+

|}

+

|-

+

|

+

{| 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%"

+

| '''W'''

+

|}

+

| valign="top" |

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''W''' :

+

|-

+

| ''U'' • → '''T'''''U'' • = E''U'' • ,

+

|-

+

| ''X'' • → '''T'''''X'' • = E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •)

+

|-

+

| →

+

|-

+

| ('''T'''''U'' • → '''T'''''X'' •) ,

+

|-

+

| for each '''W''' in the set:

+

|-

+

| {'''e''', '''E''', '''D''', '''T'''}

+

|}

+

| 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"

|  

|-

−

| ~~A~~

+

| ['''B'''2] → ['''B'''2 × '''D'''2] ,

−

~~| {~~''a''<~~sub~~>1</~~sub~~>, &~~hellip~~;, ''a''<~~sub~~>~~''n''~~</~~sub~~>}

−

~~| Alphabet~~

−

~~| [~~''n'~~'] = '''n~~'''

|-

−

| ''A''<~~sub~~>~~''i''~~</~~sub~~>

+

| ['''B'''1] → ['''B'''1 × '''D'''1] ,

−

~~| {(~~''a''<~~sub~~>~~''i''~~</~~sub~~>), ''a''<~~sub~~>~~''i''~~</~~sub~~>}

−

~~| Dimension ''i''~~

−

~~| '''B'''~~

|-

−

| ''~~A''~~

+

| (['''B'''2] → ['''B'''1])

−

|

−

~~〈A〉 ~~

−

〈''a''<~~sub~~>1</~~sub>, …, ''a''''n''〉 ~~

−

~~{‹''a''1</sub~~>, &~~hellip~~;~~, ''a''''n''›} ~~

−

~~''A''1 × … × ''A''''n'' ~~

−

~~∏''i'' ''A''''i''~~

−

|

−

~~Set of cells, ~~

−

~~coordinate tuples, ~~

−

~~points, or vectors ~~

−

~~in the universe ~~

−

~~of discourse~~

−

| '''B'''~~''n''~~

|-

−

| ''A''*

+

| →

−

~~| (hom : ''A''~~ → ~~'''B''')~~

−

~~| Linear functions~~

−

~~| ('''B'''''n'')* = '''B'''''n''~~

|-

−

| ''A''^

+

| (['''B'''2 × '''D'''2] → ['''B'''1 × '''D'''1])

−

~~| (''A~~'' &~~rarr~~; '''B''')

+

|-

−

~~| Boolean functions~~

+

|  

−

| '''B'''~~''n''~~ &~~rarr~~; '''B'''

+

|-

+

|  

+

|}

|-

−

~~| ''A''•~~

|

−

[A]

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~(''A~~'', ''A''^)

+

| '''e'''

−

~~(''A'' +→~~ '''B''')

+

|-

−

(''A''~~, (''A~~'' &~~rarr~~; '''B'''))

+

| '''E'''

−

~~[''a''~~<~~sub~~>1</~~sub~~>, &~~hellip~~;~~, ''a''~~<~~sub~~>~~''n''~~</~~sub~~>]

+

|-

−

|

+

| '''D'''

−

~~Universe of discourse~~

+

|-

−

~~based on the features ~~

+

| '''T'''

−

{''a''~~1~~</~~sub~~>, &~~hellip~~;~~, ''a''~~<~~sub~~>~~''n''~~</~~sub~~>}

+

|}

−

|

+

| valign="top" |  

−

~~('''B'~~''~~~~''n''</~~sup~~>~~, ('''B'''~~<~~sup~~>~~''n''~~</~~sup~~> &~~rarr~~; ~~'''B'''))~~

+

| colspan="2" |

−

('''B'~~''''n~~''</~~sup~~> +&~~rarr~~; ~~'''B''')~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"

−

~~['''B'''''n''~~</~~sup~~>]

+

| Radius Operator || '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>›

+

|-

+

| Secant Operator || '''E''' = ‹<math>\epsilon</math>, E›

+

|-

+

| Chord Operator || '''D''' = ‹<math>\epsilon</math>, D›

+

|-

+

| Tangent Functor || '''T''' = ‹<math>\epsilon</math>, d›

+

|}

|}

−

~~<pre>~~

−

~~| $W$ | $W$ : | Operator~~

−

~~| | U% -> $T$U% = EU%, | | [B^2] -> [B^2 x D^2],~~

−

~~| | X% -> $T$X% = EX%, | | [B^1] -> [B^1 x D^1],~~

−

~~| | (U%->X%)->($T$U%->$T$X%)| | ([B^2] -> [B^1])~~

−

~~| | for each $W$ among: | | ->~~

−

~~| | $e$, $E$, $D$, $T$ | | ([B^2 x D^2]->[B^1 x D^1])~~

−

~~-------------o~~

−

~~| $e$ || Radius Operator $e$ = <!e!, !h!>~~

−

~~| $E$ || Secant Operator $E$ = <!e!, E >~~

−

~~| $D$ || Chord Operator $D$ = <!e!, D >~~

−

~~| $T$ || Tangent Functor $T$ = <!e!, d >~~

−

~~-------------o~~

−

~~</pre>~~

===Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes===

Line 6,580: Line 6,785:

</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'''

−

~~<pre>~~

+

|- style="background:paleturquoise"

−

o

+

!  

−

~~/X\~~

+

! Operator

−

~~/XXX\~~

+

! Proposition

−

~~oXXXXXo~~

+

! Map

−

~~/X\XXX/X\~~

+

|-

−

~~/XXX\X/XXX\~~

+

|

−

~~oXXXXXoXXXXXo~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~/ \XXX/X\XXX/ \~~

+

| Tacit

−

~~/ \X/XXX~~\X/ \

+

|-

−

~~o oXXXXXo o~~

+

| Extension

−

/ \ / ~~\XXX~~/ ~~\ / \~~

+

|}

−

~~/ \ / \~~X/ ~~\ / \~~

+

|

−

~~o o o o o~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

=|\ / \ / \ / \ /|=

+

| <math>\epsilon</math> :

−

= | ~~\ / \ / \ / \ /~~ | =

+

|-

−

= ~~| o o o o |~~ =

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~= |~~ |\ ~~/ \ / \~~ /| ~~| =~~

+

|-

−

= |u ~~| \ / \ / \ / |~~ v| =

+

| (''U'' • → ''X'' •) → (E''U'' • → ''X'' •)

−

~~o o--+--o o o--+-~~-~~o o~~

+

|}

−

//\ | ~~\ / \ /~~ | ~~/\\~~

+

|

−

~~////\~~ | du \ / ~~\ / dv~~ | ~~/\\\\~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

o///~~//o o~~---~~--o o-----o o\\\\\o~~

+

| <math>\epsilon</math>''J'' :

−

//\/////~~\ \ / /\\\\\/\\~~

+

|-

−

////~~\/////\ \ /~~ /\~~\\\\~~/~~\\\\~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''

−

~~o/////o/~~//~~//o o o\\\\\o\\\\\o~~

+

|-

−

~~/ \/////\//// \~~ = = ~~/ \\\~~\/~~\\\\\/ \~~

+

| '''B'''2 × '''D'''2 → '''B'''

−

/ \//~~///\// \~~ = = ~~/ \\/\\\\\/ \~~

+

|}

−

~~o o/////o o~~ = = ~~o o\\\\\o o~~

+

|

−

~~/ \~~ / \//// ~~\ / \ = = / \~~ / ~~\\\\~~/ \ / \

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~/ \ / \// \ / \~~ = = ~~/ \ / \\/ \ / \~~

+

| <math>\epsilon</math>''J'' :

−

~~o o o o o o o o o o~~

+

|-

−

|~~\ / \ / \ / \ /~~| ~~|\ / \ / \ / \ /~~|

+

| [''u'', ''v'', d''u'', d''v''] → [''x'']

−

| \ / ~~\ / \ / \~~ / | | ~~\ / \ / \ / \ /~~ |

+

|-

−

| ~~o o o o~~ | | ~~o o o o~~ |

+

| ['''B'''2 × '''D'''2] → ['''B'''1]

−

| |\ / \ / \ /| | | |~~\ / \ / \ /~~| |

+

|}

−

|u | ~~\ / \ / \ /~~ | v| |u | \ / \ / \ / | v|

+

|-

−

~~o--+--o o o--+--o o o--+--o o o--+-~~-o

+

|

−

. | \ / \ / | /X\ | ~~\ / \ /~~ | .

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

.| ~~du \ / \ / dv~~ | ~~/XXX\~~ | ~~du \ / \ / dv~~ |.

+

| Trope

−

o-~~----o o-----o~~ /~~XXXXX\ o-----o o--~~---o

+

|-

−

~~. \~~ / /~~XXXXXXX\ \~~ / .

+

| Extension

−

~~. \ / /XXXXXXXXX\ \ / .~~

+

|}

−

~~. o oXXXXXXXXXXXo o .~~

+

|

−

~~. //\XXXXXXXXX/\\ .~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~. ////\XXXXXXX/\\\\ .~~

+

| <math>\eta</math> :

−

~~!e!J ////~~//~~\XXXXX~~/~~\\\\\\ !h!J~~

+

|-

−

~~. /~~////~~///\XXX/\\\\\\\\ .~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~. //////////\X/\\\\\\\\\\ .~~

+

|-

−

~~. o///////////o\\\\\\\\\\\o .~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

. |~~\////////// \\\\\\\\\\/~~| .

+

|}

−

. | \//~~////// \\\\\\\\/~~ | .

+

|

−

. | ~~\////// \\\\\\/~~ | .

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

. | ~~\//// \\\\/~~ | .

+

| <math>\eta</math>''J'' :

−

.| ~~x \~~// \\/ dx |.

+

|-

−

~~o-----o o----~~-o

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~\ /~~

+

|-

−

~~\ /~~

+

| '''B'''2 × '''D'''2 → '''D'''

−

x = ~~uv \ / dx~~ = uv

+

|}

−

~~\ /~~

+

|

−

~~\ /~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

o

+

| <math>\eta</math>''J'' :

−

+

|-

−

~~Figure 56-a1. Radius Map of the Conjunction J = uv~~

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'']

−

~~</pre>~~

+

|-

−

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

−

===~~Figure 56-a2. Secant Map of the Conjunction J~~ = uv===

+

|}

−

+

|-

−

~~<pre~~>

+

|

−

o

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

/X\

+

| Enlargement

−

/~~XXX\~~

+

|-

−

~~oXXXXXo~~

+

| Operator

−

//~~\XXX~~//\

+

|}

−

~~////\~~X/~~///\~~

+

|

−

~~o/////o/////o~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~/\\/////\////\\~~

+

| E :

−

~~/\\\\/////\//\\\\~~

+

|-

−

~~o\\\\\o~~//~~///o\\\\\o~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~/ \\\\/ \//// \\\\/ \~~

+

|-

−

~~/ \\/ \// \\/ \~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~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~~ | =

+

| E''J'' :

−

= | |\ / ~~\\\\~~/ \ /| | =

+

|-

−

= |u | ~~\ / \\/ \ /~~ | ~~v| =~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~o o--+--o o o--+-~~-~~o o~~

+

|-

−

~~//\~~ | ~~\ / \ /~~ | ~~/\\~~

+

| '''B'''2 × '''D'''2 → '''D'''

−

~~////\~~ | ~~du \ / \ / dv~~ | /\\\\

+

|}

−

~~o/////o o-----o o-----o o\\\\\o~~

+

|

−

//\/////~~\ \~~ / ~~/ \\\\/ \~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~////\/////\ \ / / \\/ \~~

+

| E''J'' :

−

~~o/////o/////o o o o o~~

+

|-

−

~~/ \/////\//// \~~ = = ~~/\\ / \ /\\~~

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'']

−

~~/ \/////\// \~~ = = ~~/\\\\ / \ /\\\\~~

+

|-

−

~~o o/////o o~~ = = ~~o\\\\\o o\\\\\o~~

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

−

~~/ \ / \//// \ / \~~ = = / ~~\\\\/ \ / \\\\/ \~~

+

|}

−

~~/ \ / \// \ / \ = = / \\/ \ / \\/ \~~

+

|-

−

~~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%"

−

| ~~\ / \ / \ / \ /~~ | | ~~\ / \ /\\\\ / \ /~~ |

+

| Difference

−

| ~~o o o o~~ | ~~| o o\\\\\o o~~ |

+

|-

−

| |\ / \ ~~/ \~~ /| | ~~| |\~~ / ~~\\\\~~/ \ /~~| |~~

+

| Operator

−

~~|u | \ / \ / \~~ / | v| ~~|u | \~~ / \\/ \ / | v|

+

|}

−

o--+-~~-o o o--+--o o o--+--o o o--+--o~~

+

|

−

. | ~~\ / \ /~~ | ~~/X\~~ | ~~\ / \ / | .~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

.| ~~du \~~ / ~~\ / dv~~ | ~~/XXX\~~ | ~~du \ / \ / dv |.~~

+

| D :

−

o-~~----o o-----o~~ /~~XXXXX\ o-----o o--~~---o

+

|-

−

~~. \ / /XXXXXXX\ \ / .~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~. \ / /XXXXXXXXX\ \ / .~~

+

|-

−

~~. o oXXXXXXXXXXXo o .~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~. /~~/\~~XXXXXXXXX~~/~~\\ .~~

+

|}

−

. ////~~\XXXXXXX/\\\\ .~~

+

|

−

~~!e!J~~ ////~~//\XXXXX/\\\\\\ EJ~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~. ////////\XXX/\\\\\\\\ .~~

+

| D''J'' :

−

~~. //////////\X~~/~~\\\\\\\\\\ .~~

+

|-

−

~~. o//~~//~~///////o\\\\\\\\\\\o .~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

. |~~\////////// \\\\\\\\\\/~~| .

+

|-

−

. | ~~\//////// \\\\\\\\/~~ | .

+

| '''B'''2 × '''D'''2 → '''D'''

−

. | ~~\/////~~/ ~~\\\\\~~\/ | .

+

|}

−

. | \//// ~~\\\\~~/ | .

+

|

−

.| ~~x \~~// \\/ dx |.

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~o-----o o---~~--o

+

| D''J'' :

−

~~\ /~~

+

|-

−

~~\ / dx = (~~u, ~~du)(~~v, ~~dv)~~

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'']

−

~~x = uv \ /~~

+

|-

−

\ / ~~dx = uv + u dv + v du + du dv~~

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

−

~~\ /~~

+

|}

−

o

+

|-

−

+

|

−

~~Figure 56-a2. Secant Map of the Conjunction J = uv~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

</~~pre~~>

+

| Differential

−

+

|-

−

===~~Figure 56-a3. Chord Map of the Conjunction J = uv=~~==

+

| Operator

−

+

|}

−

<~~pre~~>

+

|

−

o

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

//\

+

| d :

−

~~////\~~

+

|-

−

~~o/////o~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

/X\/~~///X\~~

+

|-

−

~~/XXX\//XXX\~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~oXXXXXoXXXXXo~~

+

|}

−

~~/\\XXX/X\XXX/\\~~

+

|

−

~~/\\\\X/XXX\X/\\\\~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~o\\\\\oXXXXXo\\\\\o~~

+

| d''J'' :

−

~~/ \\\\/ \XXX/ \\\\/ \~~

+

|-

−

~~/ \\/ \X/ \\/ \~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~o o o o o~~

+

|-

−

~~=|\ / \ /\\ / \ /|=~~

+

| '''B'''2 × '''D'''2 → '''D'''

−

~~= | \ / \ /\\\\ / \ / | =~~

+

|}

−

~~= | o o\\\\\o o | =~~

+

|

−

~~= | |\ / \\\\/ \ /| | =~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~= |u | \ / \\/ \ / | v| =~~

+

| d''J'' :

−

~~o o--+--o o o--+--o o~~

+

|-

−

~~//\ | \ / \ / | / \~~

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'']

−

~~////\ | du \ / \ / dv | / \~~

+

|-

−

~~o/////o o-----o o-----o o o~~

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

−

~~//\/////\ \ / /\\ /\\~~

+

|}

−

~~////\/////\ \ / /\\\\ /\\\\~~

+

|-

−

~~o/////o/////o o o\\\\\o\\\\\o~~

+

|

−

~~/ \/////\//// \ = = /\\\\\/\\\\\/\\~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~/ \/////\// \ = = /\\\\\/\\\\\/\\\\~~

+

| Remainder

−

~~o o/////o o = = o\\\\\o\\\\\o\\\\\o~~

+

|-

−

~~/ \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \~~

+

| Operator

−

~~/ \ / \// \ / \ = = / \\/ \\/ \\/ \~~

+

|}

−

~~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%"

−

~~| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |~~

+

| r :

−

~~| o o o o | | o o\\\\\o o |~~

+

|-

−

~~| |\ / \ / \ /| | | |\ / \\\\/ \ /| |~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|~~

+

|-

−

~~o--+--o o o--+--o o o--+--o o o--+--o~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~. | \ / \ / | /X\ | \ / \ / | .~~

+

|}

−

~~.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.~~

+

|

−

~~o-----o o-----o /XXXXX\ o-----o o-----o~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~. \ / /XXXXXXX\ \ / .~~

+

| r''J'' :

−

~~. \ / /XXXXXXXXX\ \ / .~~

+

|-

−

~~. o oXXXXXXXXXXXo o .~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~. //\XXXXXXXXX/\\ .~~

+

|-

−

~~. ////\XXXXXXX/\\\\ .~~

+

| '''B'''2 × '''D'''2 → '''D'''

−

~~!e!J //////\XXXXX/\\\\\\ DJ~~

+

|}

−

~~. ////////\XXX/\\\\\\\\ .~~

+

|

−

~~. //////////\X/\\\\\\\\\\ .~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~. o///////////o\\\\\\\\\\\o .~~

+

| r''J'' :

−

~~. |\////////// \\\\\\\\\\/| .~~

+

|-

−

~~. | \//////// \\\\\\\\/ | .~~

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'']

−

~~. | \////// \\\\\\/ | .~~

+

|-

−

~~. | \//// \\\\/ | .~~

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

−

~~.| x \// \\/ dx |.~~

+

|}

−

~~o-----o o-----o~~

+

|-

−

~~\ /~~

+

|

−

~~\ / dx = (u, du)(v, dv) - uv~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~x = uv \ /~~

+

| Radius

−

~~\ / dx = u dv + v du + du dv~~

+

|-

−

~~\ /~~

+

| Operator

−

o

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>› :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

|  

+

|-

+

|  

+

|-

+

|  

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''e'''''J'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']

+

|-

+

| ['''B'''2 × '''D'''2] → ['''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%"

+

| '''E''' = ‹<math>\epsilon</math>, E› :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

|  

+

|-

+

|  

+

|-

+

|  

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''E'''''J'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']

+

|-

+

| ['''B'''2 × '''D'''2] → ['''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%"

+

| '''D''' = ‹<math>\epsilon</math>, D› :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

|  

+

|-

+

|  

+

|-

+

|  

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''D'''''J'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']

+

|-

+

| ['''B'''2 × '''D'''2] → ['''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%"

+

| '''T''' = ‹<math>\epsilon</math>, d› :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

+

|}

+

|

+

{| 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'''2 × '''D'''2 → '''D'''

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''T'''''J'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']

+

|-

+

| ['''B'''2 × '''D'''2] → ['''B''' × '''D''']

+

|}

+

|}

−

~~Figure 56-a3. Chord Map of the Conjunction J = uv~~

+

===Figure 56-a1. Radius Map of the Conjunction J = uv===

−

~~</pre>~~

−

===Figure 56-a4. ~~Tangent~~ Map of the Conjunction J = uv===

<pre>

o

−

//\

+

/X\

−

/~~///~~\

+

/XXX\

−

~~o/////o~~

+

oXXXXXo

−

/X\~~///~~/X\

+

/X\XXX/X\

−

/XXX\//XXX\

+

/XXX\X/XXX\

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 | =

= | |\ / \ / \ /| | =

= |u | \ / \ / \ / | v| =

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 |

| |\ / \ / \ /| | | |\ / \ / \ /| |

Line 6,832: Line 7,152:

. //\XXXXXXXXX/\\ .

. ////\XXXXXXX/\\\\ .

−

!e!J //////\XXXXX/\\\\\\ dJ

+

!e!J //////\XXXXX/\\\\\\ !h!J

. ////////\XXX/\\\\\\\\ .

. //////////\X/\\\\\\\\\\ .

Line 6,844: Line 7,164:

\ /

−

x = uv \ / dx = ~~u dv + v du~~

+

x = uv \ / dx = uv

\ /

o

−

Figure 56-a4. ~~Tangent~~ Map of the Conjunction J = uv

+

Figure 56-a1. Radius Map of the Conjunction J = uv

</pre>

−

===Figure 56-b1. ~~Radius~~ Map of the Conjunction J = uv===

+

+

[[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]

+

<center>'''Figure 56-a1. Radius Map of the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 56-a2. Secant Map of the Conjunction J = uv===

<pre>

−

~~o-----------------------~~o

+

o

−

~~| |~~

+

/X\

−

~~| |~~

+

/XXX\

−

~~| |~~

+

oXXXXXo

−

~~| o--o o--o |~~

+

//\XXX//\

−

| / \ / \ |

+

////\X////\

−

| / 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 |

+

= | |\ / \\\\/ \ /| | =

−

| |

+

= |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 o o o o

−

| | \

+

/ \/////\//// \ = = /\\ / \ /\\

−

~~| o--o o--o~~ | \

+

/ \/////\// \ = = /\\\\ / \ /\\\\

−

| / \ / \ | \

+

o o/////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 |~~ \ \

+

| |\ / \ / \ /| | | |\ / \\\\/ \ /| |

−

| \ \ / / | \ \

+

|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|

−

~~| \~~ o / ~~| \ \~~

+

o--+--o o o--+--o o o--+--o o o--+--o

−

~~| \~~ / \ / ~~| \ \~~

+

. | \ / \ / | /X\ | \ / \ / | .

−

| o--o o--o | \ \

+

.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.

−

~~| |~~ \ \

+

o-----o o-----o /XXXXX\ o-----o o-----o

−

~~| |~~ \ \

+

. \ / /XXXXXXX\ \ / .

−

~~| |~~ \ \

+

. \ / /XXXXXXXXX\ \ / .

−

~~o-----------------------o~~ \ \

+

. o oXXXXXXXXXXXo o .

−

\ \

+

. //\XXXXXXXXX/\\ .

−

o~~-----------------------@~~ o~~--------~~\~~----------~~\~~---~~o o~~-----------------------~~o

+

. ////\XXXXXXX/\\\\ .

−

~~| |~~\ | \ \ | ~~|```````````````````````|~~

+

!e!J //////\XXXXX/\\\\\\ EJ

−

~~| |~~ \ | \ ~~@ |~~ ~~|```````````````````````|~~

+

. ////////\XXX/\\\\\\\\ .

−

~~| |~~ \| \ | ~~|```````````````````````|~~

+

. //////////\X/\\\\\\\\\\ .

−

+

. o///////////o\\\\\\\\\\\o .

−

| / \ / \ | |\ / \ /\ \ | ~~|`````~~/~~````~~\`/~~````~~\~~`````~~|

+

. |\////////// \\\\\\\\\\/| .

−

| / 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`````|``|~~

+

\ /

−

~~| o~~ \ / o | | o \~~```~~/ ~~o |~~ ~~|``o``````~~\~~```~~/~~``````o``~~|

+

\ / dx = (u, du)(v, dv)

−

| \ \ / / ~~| |~~ \ \`/ ~~/ | |```\``````\`/``````/```~~|

+

x = uv \ /

−

| \ o ~~/ | | \~~ o / ~~| |````~~\~~``````~~o~~``````/````|~~

+

\ / dx = uv + u dv + v du + du dv

−

| \ / \ / ~~| |~~ \ / \ / ~~| |`````~~\~~````/`~~\~~````~~/~~`````|~~

+

\ /

−

+

o

−

~~| | | | |```````````````````````|~~

+

−

~~| | | | |```````````````````````|~~

+

Figure 56-a2. Secant Map of the Conjunction J = uv

−

~~| | | | |```````````````````````|~~

+

</pre>

−

o~~-----------------------o o-----------------------~~o ~~o-----------------------o~~

−

\ / \ / \ /

−

\ ~~!h!J~~ / ~~\ J~~ / \ ~~!h!J~~ /

−

\ / \ / \ /

−

~~\ /~~ 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-b1. ~~Radius~~ Map of the Conjunction J = uv

+

−

</~~pre~~>

+

[[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]

+

<center>'''Figure 56-a2. Secant Map of the Conjunction ''J'' = ''uv'''''</center>

−

===Figure 56-b2. ~~Secant~~ Map of the Conjunction J = uv===

+

===Figure 56-a3. Chord Map of the Conjunction J = uv===

<pre>

−

~~o-----------------------~~o

+

o

−

~~| |~~

+

//\

−

~~| |~~

+

////\

−

~~| |~~

+

o/////o

−

| o~~--o o--~~o |

+

/X\////X\

−

| / \ / \ |

+

/XXX\//XXX\

−

| / o \ |

+

oXXXXXoXXXXXo

−

| / du /`\ ~~dv \ |~~

+

/\\XXX/X\XXX/\\

−

~~| o~~ /~~```~~\ ~~o |~~

+

/\\\\X/XXX\X/\\\\

−

~~| |~~ o~~`````o | |~~

+

o\\\\\oXXXXXo\\\\\o

−

~~| | |`````| | |~~

+

/ \\\\/ \XXX/ \\\\/ \

−

~~| | o`````~~o ~~| |~~

+

/ \\/ \X/ \\/ \

−

~~| 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

−

| |

+

//\ | \ / \ / | / \

−

~~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 \ | \~~

+

/ \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \

−

| /~~``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 |~~ \ \

+

| |\ / \ / \ /| | | |\ / \\\\/ \ /| |

−

| \~~``````~~\ ~~/ / |~~ \ \

+

|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|

−

~~| \``````~~o / ~~| \ \~~

+

o--+--o o o--+--o o o--+--o o o--+--o

−

~~| \````~~/ \ / ~~| \ \~~

+

. | \ / \ / | /X\ | \ / \ / | .

−

| o--o o--o | \ \

+

.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.

−

~~| |~~ \ \

+

o-----o o-----o /XXXXX\ o-----o o-----o

−

~~| |~~ \ \

+

. \ / /XXXXXXX\ \ / .

−

~~| |~~ \ \

+

. \ / /XXXXXXXXX\ \ / .

−

~~o-----------------------o~~ \ \

+

. o oXXXXXXXXXXXo o .

−

\ \

+

. //\XXXXXXXXX/\\ .

−

~~o-----------------------@~~ ~~o--------~~\~~----------~~\~~---o~~ ~~o-----------------------o~~

+

. ////\XXXXXXX/\\\\ .

−

~~| |~~\ | \ \ ~~| |```````````````````````|~~

+

!e!J //////\XXXXX/\\\\\\ DJ

−

~~| |~~ \ | \ ~~@ | |```````````````````````|~~

+

. ////////\XXX/\\\\\\\\ .

−

~~| | \| \ | |```````````````````````|~~

+

. //////////\X/\\\\\\\\\\ .

−

+

. o///////////o\\\\\\\\\\\o .

−

| / \ /~~````~~\ ~~| |\~~ / \ /\ \ | ~~|`````~~/ \`/ \~~`````|~~

+

. |\////////// \\\\\\\\\\/| .

−

| / ~~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 |``|~~

+

\ /

−

~~| o~~ \ /~~``````o~~ | ~~| o~~ \~~```/~~ o | ~~|``o~~ \ / ~~o``|~~

+

\ / dx = (u, du)(v, dv) - uv

−

| \ \ /~~``````/~~ | ~~| \ \`/~~ / ~~| |```~~\ ~~\ / /```|~~

+

x = uv \ /

−

| \ ~~o``````~~/ ~~| |~~ \ ~~o / |~~ ~~|````\ o~~ /~~````|~~

+

\ / 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~~

+

−

\ / \ / \ /

+

−

~~\ EJ~~ / ~~\ J~~ / ~~\ EJ~~ /

+

[[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]

−

\ / \ / \ /

+

<center>'''Figure 56-a3. Chord Map of the Conjunction ''J'' = ''uv'''''</center>

−

\ / ~~o----------~~\~~---------~~/~~----------o~~ \ /

+

−

\ ~~/ |~~ \ / | \ /

+

===Figure 56-a4. Tangent Map of the Conjunction J = uv===

−

\ / | \ / | \ /

+

−

\ / | o-----o-----o | \ /

+

<pre>

−

\ / ~~| /`````````````~~\ ~~| \~~ /

+

o

−

\ / ~~| /```````````````\ |~~ \ /

+

//\

−

o-~~-----\---~~/------o | /~~`````````````````\ |~~ ~~o------\---~~/~~------o~~

+

////\

−

~~| \~~ / ~~| |~~ /~~```````````````````\ |~~ | \ / |

+

o/////o

−

| o--o~~--o | |~~ /~~`````````````````````~~\ ~~| | o--o--o |~~

+

/X\////X\

−

| /~~```````~~\ ~~| | o```````````````````````o | |~~ /~~```````~~\ |

+

/XXX\//XXX\

−

| /~~`````````~~\ ~~| | |```````````````````````| | |~~ /~~`````````~~\ |

+

oXXXXXoXXXXXo

−

~~| o```````````~~o ~~| | |```````````````````````| | |~~ o~~```````````o |~~

+

/\\XXX//\XXX/\\

−

~~| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |~~

+

/\\\\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 o | =

−

| \~~`````````````~~/ |

+

= | |\ / \ / \ /| | =

−

| o-----------o |

+

= |u | \ / \ / \ / | v| =

−

~~| |~~

+

o o--+--o o o--+--o o

−

~~| |~~

+

//\ | \ / \ / | / \

−

o-~~------------------------------o~~

+

////\ | du \ / \ / dv | / \

−

+

o/////o o-----o o-----o o o

−

~~Figure 56-b2. Secant Map of the Conjunction J = uv~~

+

//\/////\ \ / /\\ /\\

−

</~~pre~~>

+

////\/////\ \ / /\\\\ /\\\\

−

+

o/////o/////o o o\\\\\o\\\\\o

−

===Figure 56-b3. ~~Chord~~ 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 |

+

| |\ / \ / \ /| | | |\ / \ / \ /| |

+

|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>

+

+

[[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]

+

<center>'''Figure 56-a4. Tangent Map of the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 56-b1. Radius Map of the Conjunction J = uv===

+

<pre>

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

Line 7,062: Line 7,402:

| / \ / \ |

| / o \ |

−

| / du /`\ dv \ |

+

| / du / \ dv \ |

−

| o /~~```~~\ o |

+

| o / \ o |

−

| | o~~`````~~o | |

+

| | o o | |

−

| | |~~`````~~| | |

+

| | | | | |

−

| | o~~`````~~o | |

+

| | o o | |

−

| o \~~```~~/ o |

+

| o \ / o |

−

| \ \`/ / |

+

| \ \ / / |

| \ o / |

| \ / \ / |

Line 7,082: Line 7,422:

| | \

| 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 | \ \

| | \ \

Line 7,100: Line 7,440:

\ \

o-----------------------@ o--------\----------\---o o-----------------------o

−

| |\ | \ \ | | |

+

| |\ | \ \ | |```````````````````````|

−

| | \ | \ @ | | |

+

| | \ | \ @ | |```````````````````````|

−

| | \| \ | | |

+

| | \| \ | |```````````````````````|

−

+

−

| / \ /~~````~~\ | |\ / \ /\ \ | | /````\ /````\ |

+

| / \ / \ | |\ / \ /\ \ | |`````/````\`/````\`````|

−

| / o~~``````~~\ | | \ / o @ \ | | /``````o``````\ |

+

| / o \ | | \ / o @ \ | |````/``````o``````\````|

−

+

−

| o / \~~``````~~o | | o\ /```\ o | | o``````/```\``````o |

+

| o / \ o | | o\ /```\ o | |``o``````/```\``````o``|

−

| | o o~~`````~~| | | | \ o`````o | | | |`````o`````o`````| |

+

| | o o | | | | \ o`````o | | |``|`````o`````o`````|``|

−

| | | |~~`````~~| | | | @ |``@--|-----|------@ |`````|`````|`````| |

+

| | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``|

−

| | o o~~`````~~| | | | o`````o | | | |`````o`````o`````| |

+

| | o o | | | | o`````o | | |``|`````o`````o`````|``|

−

| 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 /

+

\ !h!J / \ J / \ !h!J /

\ / \ / \ /

\ / o----------\---------/----------o \ /

Line 7,149: Line 7,489:

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

−

Figure 56-b3. ~~Chord~~ Map of the Conjunction J = uv

+

Figure 56-b1. Radius Map of the Conjunction J = uv

</pre>

−

===Figure 56-b4. ~~Tangent~~ Map of the Conjunction J = uv===

+

+

[[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]

+

<center>'''Figure 56-b1. Radius Map of the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 56-b2. Secant Map of the Conjunction J = uv===

<pre>

Line 7,162: Line 7,506:

| / \ / \ |

| / o \ |

−

| / du / \ dv \ |

+

| / du /`\ dv \ |

−

| o / \ o |

+

| o /```\ o |

−

| | o o | |

+

| | o`````o | |

−

| | | | | |

+

| | |`````| | |

−

| | o o | |

+

| | o`````o | |

−

| o \ / o |

+

| o \```/ o |

−

| \ \ / / |

+

| \ \`/ / |

| \ o / |

| \ / \ / |

Line 7,184: Line 7,528:

| /````\ / \ | \

| /``````o \ | \

−

| /``du``/`\ dv \ | \

+

| /``du``/ \ dv \ | \

−

| o``````/~~```~~\ o | \

+

| o``````/ \ o | \

−

| |`````o~~`````~~o | @ \

+

| |`````o o | @ \

−

| |`````|~~`````~~| | |\ \

+

| |`````| | | |\ \

−

| |`````o~~`````~~o | | \ \

+

| |`````o o | | \ \

−

| o``````\~~```~~/ o | \ \

+

| o``````\ / o | \ \

−

| \``````\`/ / | \ \

+

| \``````\ / / | \ \

| \``````o / | \ \

| \````/ \ / | \ \

Line 7,200: Line 7,544:

\ \

o-----------------------@ o--------\----------\---o o-----------------------o

−

| |\ | \ \ | | |

+

| |\ | \ \ | |```````````````````````|

−

| | \ | \ @ | | |

+

| | \ | \ @ | |```````````````````````|

−

| | \| \ | | |

+

| | \| \ | |```````````````````````|

−

+

−

| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |

+

| / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````|

−

| / o``````\ | | \ / o @ \ | | ~~/``~~````o````~~``\~~ |

+

| / o``````\ | | \ / o @ \ | |````/ o \````|

−

+

−

| o /~~```~~\``````o | | o\ /```\ o | | ~~o````~~``/ \``~~````o~~ |

+

| o / \``````o | | o\ /```\ o | |``o / \ o``|

−

| | o~~`````~~o`````| | | | \ o`````o | | | |``~~```~~o o``~~```|~~ |

+

| | o o`````| | | | \ o`````o | | |``| o o |``|

−

| | |~~`````~~|`````| | | | @ |``@--|-----|------@ ~~|```~~``| |``~~```|~~ |

+

| | | |`````| | | | @ |``@--|-----|------@``| | | |``|

−

| | o~~`````~~o`````| | | | o`````o | | | |``~~```~~o o~~```~~``| |

+

| | o o`````| | | | o`````o | | |``| o o |``|

−

| 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 \ /

Line 7,249: Line 7,593:

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

−

Figure 56-b4. ~~Tangent~~ Map of the Conjunction J = uv

+

Figure 56-b2. Secant Map of the Conjunction J = uv

</pre>

−

===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===

+

+

[[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]

+

<center>'''Figure 56-b2. Secant Map of the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 56-b3. Chord Map of the Conjunction J = uv===

+

<pre>

+

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 \ | \

+

| /``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

+

\ / \ / \ /

+

\ DJ / \ J / \ DJ /

+

\ / \ / \ /

+

\ / 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-b3. Chord Map of the Conjunction J = uv

+

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]

+

<center>'''Figure 56-b3. Chord Map of the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 56-b4. Tangent Map of the Conjunction J = uv===

+

<pre>

+

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 \ | \

+

| /``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

+

\ / \ / \ /

+

\ dJ / \ J / \ dJ /

+

\ / \ / \ /

+

\ / 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>

+

+

[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J.gif|center]]

+

<center>'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===

<pre>

Line 7,258: Line 7,814:

//\ /X\

////\ /XXX\

−

//////\ oXXXXXo

+

//////\ oXXXXXo

−

////////\ /X\XXX/X\

+

////////\ /X\XXX/X\

−

//////////\ /XXX\X~~/XXX\~~

+

//////////\ /XXX\X/XXX\

−

~~o///////////o oXXXXXoXXXXXo~~

−

~~/ \////////// \ / \XXX/X\XXX/ \~~

−

~~/ \//////// \ / \X/XXX\X/ \~~

−

~~/ \////// \ o oXXXXXo o~~

−

~~/ \//// \ / \ / \XXX/ \ / \~~

−

~~/ \// \ / \ / \X/ \ / \~~

−

~~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% $e$ $E$U%~~

−

~~o------------------>o~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~J | | $e$J~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~v v~~

−

~~o------------------>o~~

−

~~X% $e$ $E$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 57-1. Radius Operator Diagram for the Conjunction J = uv~~

−

~~</pre>~~

−

~~===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===~~

−

~~<pre>~~

−

~~o o~~

−

~~//\ /X\~~

−

~~////\ /XXX\~~

−

~~//////\ oXXXXXo~~

−

~~////////\ //\XXX//\~~

−

~~//////////\ ////\X////\~~

−

~~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~~

−

~~\ / | \ / \ / |~~

−

~~\ / | du \ / \ / dv |~~

−

~~\ / o-----o o-----o~~

−

~~\ / \ /~~

−

~~\ / \ /~~

−

~~o o~~

−

~~U% $E$ $E$U%~~

−

~~o------------------>o~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~J | | $E$J~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~v v~~

−

~~o------------------>o~~

−

~~X% $E$ $E$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 57-2. Secant Operator Diagram for the Conjunction J = uv~~

−

~~</pre>~~

−

~~===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===~~

−

~~<pre>~~

−

~~o o~~

−

~~//\ //\~~

−

~~////\ ////\~~

−

~~//////\ o/////o~~

−

~~////////\ /X\////X\~~

−

~~//////////\ /XXX\//XXX\~~

−

~~o///////////o oXXXXXoXXXXXo~~

−

~~/ \////////// \ /\\XXX/X\XXX/\\~~

−

~~/ \//////// \ /\\\\X/XXX\X/\\\\~~

−

~~/ \////// \ o\\\\\oXXXXXo\\\\\o~~

−

~~/ \//// \ / \\\\/ \XXX/ \\\\/ \~~

−

~~/ \// \ / \\/ \X/ \\/ \~~

−

~~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% $D$ $E$U%~~

−

~~o------------------>o~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~J | | $D$J~~

−

~~| |~~

−

~~| |~~

−

~~| |~~

−

~~v v~~

−

~~o------------------>o~~

−

~~X% $D$ $E$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 57-3. Chord Operator Diagram for the Conjunction J = uv~~

−

~~</pre>~~

−

~~===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===~~

−

~~<pre>~~

−

~~o o~~

−

~~//\ //\~~

−

~~////\ ////\~~

−

~~//////\ o/////o~~

−

~~////////\ /X\////X\~~

−

~~//////////\ /XXX\/~~/XXX\

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 |

| \ / \ / | | |\ / \ / \ /| |

Line 7,490: Line 7,836:

\ / \ /

o o

−

U% $T$ $E$U%

+

U% $e$ $E$U%

o------------------>o

| |

Line 7,496: Line 7,842:

| |

−

J | | $T$J

+

J | | $e$J

| |

Line 7,502: Line 7,848:

v v

o------------------>o

−

X% $T$ $E$X%

+

X% $e$ $E$X%

o o

//\ /X\

Line 7,529: Line 7,875:

o o

−

Figure 57-4. ~~Tangent Functor~~ Diagram for the Conjunction J = uv

+

Figure 57-1. Radius Operator Diagram for the Conjunction J = uv

</pre>

−

===~~Formula Display 11~~===

+

+

[[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]

+

<center>'''Figure 57-1. Radius Operator Diagram for the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===

<pre>

−

o~~-----------------------------------------------------------~~o

+

o o

−

| |

+

//\ /X\

−

| F = ~~<f, g> = <F_1, F_2> : [u, v] -> [x, y]~~ |

+

////\ /XXX\

−

| |

+

//////\ oXXXXXo

−

| ~~where f = F_1~~ : ~~[u, v]~~ -> ~~[x]~~ |

+

////////\ //\XXX//\

−

| |

+

//////////\ ////\X////\

−

| ~~and g = F_2 :~~ [u~~, v]~~ -> ~~[y]~~ |

+

o///////////o o/////o/////o

−

| |

+

/ \////////// \ /\\/////\////\\

−

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

+

/ \//////// \ /\\\\/////\//\\\\

−

~~</pre>~~

+

/ \////// \ o\\\\\o/////o\\\\\o

−

+

/ \//// \ / \\\\/ \//// \\\\/ \

−

~~ ~~

+

/ \// \ / \\/ \// \\/ \

−

~~{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font~~-~~weight:bold; text~~-~~align:left; width:96%"~~

+

o o o o o o o o

−

|

+

|\ / \ /| |\ / \ /\\ / \ /|

−

~~{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font~~-~~weight:bold; text~~-~~align:left; width:100~~%"

+

| \ / \ / | | \ / \ /\\\\ / \ / |

−

~~| ''F''~~

+

| \ / \ / | | o o\\\\\o o |

−

~~| =~~

+

| \ / \ / | | |\ / \\\\/ \ /| |

−

~~| align="center" | ‹''f'', ''g''›~~

+

| u \ / \ / v | |u | \ / \\/ \ / | v|

−

~~| =~~

+

o-----o o-----o o--+--o o o--+--o

−

~~| align="center" | ‹''F''1<~~/~~sub>, ''F''2<~~/~~sub>›~~

+

\ / | \ / \ / |

−

~~| :~~

+

\ / | du \ / \ / dv |

−

~~| <nowiki>[<~~/~~nowiki>''u'', ''v''<nowiki>]<~~/~~nowiki>~~

+

\ / o-----o o-----o

−

~~| →~~

+

\ / \ /

−

~~| <nowiki>[<~~/~~nowiki>''x'', ''y''<nowiki>]<~~/~~nowiki>~~

+

\ / \ /

−

|-

+

o o

−

~~| colspan="2" | where~~

+

U% $E$ $E$U%

−

~~| align="center" | ''f''~~

+

o------------------>o

−

~~| =~~

+

| |

−

| ~~align="center"~~ | ~~''F''1<~~/~~sub>~~

+

| |

−

| :

+

| |

−

| ~~<nowiki>[<~~/~~nowiki>''u'', ''v''<nowiki>]<~~/~~nowiki>~~

+

| |

−

| ~~→~~

+

J | | $E$J

−

| ~~<nowiki>[<~~/~~nowiki>''x''<nowiki>]<~~/~~nowiki>~~

+

| |

−

|-

+

| |

−

| ~~colspan="2" | and~~

+

| |

−

| ~~align="center"~~ | ~~''g''~~

+

v v

−

| =

+

o------------------>o

−

| ~~align="center"~~ | ~~''F''2<~~/~~sub>~~

+

X% $E$ $E$X%

−

~~| :~~

+

o o

−

~~| <nowiki>[<~~/~~nowiki>''u'', ''v''<nowiki>]<~~/~~nowiki>~~

+

//\ /X\

−

~~| →~~

+

////\ /XXX\

−

~~| <nowiki>[<~~/~~nowiki>''y''<nowiki>]<~~/~~nowiki>~~

+

//////\ /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 57-2. Secant Operator Diagram for the Conjunction J = uv

−

~~{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font~~-~~weight:bold; text-align:center; width:96%"~~

+

</pre>

−

|

+

−

~~{| align="center" border~~=~~"0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"~~

+

−

~~| align="left" | ''F''~~

+

[[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]

−

~~| =~~

+

<center>'''Figure 57-2. Secant Operator Diagram for the Conjunction ''J'' = ''uv'''''</center>

−

~~| ‹''f'', ''g''›~~

−

~~| =~~

−

~~| ‹''F''1, ''F''2~~</~~sub~~>›

−

~~| :~~

−

| <~~nowiki>[</nowiki~~>~~''u'', ''v''<nowiki>]</nowiki>~~

−

~~| →~~

−

| <~~nowiki~~>[~~</nowiki>''x'', ''y''<nowiki>]</nowiki>~~

−

|-

−

~~| align="left" colspan="~~2" | ~~where~~

−

~~| ''f''~~

−

~~| =~~

−

~~| ''F''1~~</~~sub~~>

−

~~| :~~

−

| <~~nowiki~~>[<~~/nowiki~~>~~''u'', ''v''~~<~~nowiki>]</nowiki>~~

−

~~| →~~

−

~~| <nowiki~~>~~[</nowiki>'~~'x''~~<nowiki>]</nowiki>~~

−

|-

−

~~| align="left" colspan="~~2~~" | and~~

−

| ''g''

−

| =

−

~~| ''F''2~~

−

~~| :~~

−

~~| <nowiki>[</nowiki>'~~'u'', ''v''~~<nowiki>]~~</~~nowiki~~>

−

~~| →~~

−

~~| <nowiki>[~~</~~nowiki~~>~~''y''<nowiki>]</nowiki>~~

−

|}

−

|}

−

</~~font><br~~>

−

===~~Table 58~~. ~~Cast of Characters: Expansive Subtypes of Objects and Operators~~===

+

===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===

<pre>

−

~~Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators~~

+

o o

−

o~~------~~o~~-------------------------o------------------~~o~~----------------------------~~o

+

//\ //\

−

+

////\ ////\

−

o~~------o-------------------------o------------------o----------------------------~~o

+

//////\ o/////o

−

~~| | | | |~~

+

////////\ /X\////X\

−

~~| U%~~ ~~| = [u, v] | Source Universe | [B^n] |~~

+

//////////\ /XXX\//XXX\

−

~~| | | | |~~

+

o///////////o oXXXXXoXXXXXo

−

o~~------~~o~~-------------------------~~o~~------------------~~o~~----------------------------~~o

+

/ \////////// \ /\\XXX/X\XXX/\\

−

| | | | |

+

/ \//////// \ /\\\\X/XXX\X/\\\\

−

| X% | ~~= [x, y]~~ | ~~Target Universe | [B^k]~~ |

+

/ \////// \ o\\\\\oXXXXXo\\\\\o

−

| | ~~= [f, g]~~ | ~~| |~~

+

/ \//// \ / \\\\/ \XXX/ \\\\/ \

−

~~| | | | |~~

+

/ \// \ / \\/ \X/ \\/ \

−

o~~------~~o~~-------------------------~~o~~------------------o----------------------------~~o

+

o o o o o o o o

−

| | | | |

+

|\ / \ /| |\ / \ /\\ / \ /|

−

+

| \ / \ / | | \ / \ /\\\\ / \ / |

−

~~| | | Source Universe~~ | |

+

| \ / \ / | | o o\\\\\o o |

−

| | ~~| |~~ |

+

| \ / \ / | | |\ / \\\\/ \ /| |

−

o------o---------~~----------------~~o~~------------------~~o--~~------------------------~~--o

+

| u \ / \ / v | |u | \ / \\/ \ / | v|

−

~~| | | | |~~

+

o-----o o-----o o--+--o o o--+--o

−

+

\ / | \ / \ / |

−

| ~~| = [f, g, df, dg] | Target Universe | |~~

+

\ / | du \ / \ / dv |

−

~~| | | |~~ |

+

\ / o-----o o-----o

−

o------o~~--------------------~~-----o~~------------------~~o~~----------------------------~~o

+

\ / \ /

−

~~| | |~~ ~~| |~~

+

\ / \ /

−

~~| F | F = <f, g> :~~ U% ~~-> X~~% ~~| Transformation, | [B^n] -> [B^k] |~~

+

o o

−

~~| | | or Mapping | |~~

+

U% $D$ $E$U%

−

~~| | | | |~~

+

o------------------>o

−

~~o------o-------------------------~~o------------------~~o----------------------------~~o

+

| |

−

| | | | |

+

| |

−

| ~~| f, g : U -> B | Proposition, | B^n -> B~~ |

+

| |

−

| ~~| | special case |~~ |

+

| |

−

+

J | | $D$J

−

| | | ~~or component |~~ |

+

| |

−

+

| |

−

~~| | | | |~~

+

| |

−

o~~------o-------------------------o------------------o----------~~------------------o

+

v v

−

~~| | |~~ ~~| |~~

+

o------------------>o

−

~~| W | W : | Operator | |~~

+

X% $D$ $E$X%

−

~~| | U~~% ~~-> EU%, | | [B^n] -> [B^n x~~ D~~^n], |~~

+

o o

−

~~| |~~ X% ~~-> EX%, | | [B^k] -> [B^k x D^k], |~~

+

//\ /X\

−

+

////\ /XXX\

−

~~| | for each W among: | | -> |~~

+

//////\ /XXXXX\

−

~~| | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) |~~

+

////////\ /XXXXXXX\

−

~~| |~~ ~~| | |~~

+

//////////\ /XXXXXXXXX\

−

o~~------o-------------------------o------------------o----------------------------o~~

+

////////////o oXXXXXXXXXXXo

−

| ~~| | |~~

+

///////////// \ //\XXXXXXXXX/\\

−

~~| !e! | | Tacit Extension Operator~~ ~~!e! |~~

+

///////////// \ ////\XXXXXXX/\\\\

−

~~| !h! | | Trope Extension Operator !h! |~~

+

///////////// \ //////\XXXXX/\\\\\\

−

~~| E~~ ~~| | Enlargement Operator E |~~

+

///////////// \ ////////\XXX/\\\\\\\\

−

| ~~D | | Difference Operator~~ ~~D |~~

+

///////////// \ //////////\X/\\\\\\\\\\

−

~~| d | | Differential Operator d |~~

+

o//////////// o o///////////o\\\\\\\\\\\o

−

~~| | | |~~

+

|\////////// / |\////////// \\\\\\\\\\/|

−

o~~------~~o~~-------------------------~~o~~------------------~~o~~----------------------------~~o

+

| \//////// / | \//////// \\\\\\\\/ |

−

| | | | |

+

| \////// / | \////// \\\\\\/ |

−

| ~~$W$~~ | ~~$W$ : | Operator |~~ |

+

| \//// / | \//// \\\\/ |

−

+

| x \// / | x \// \\/ dx |

−

| ~~| X% -> $T$X% = EX%, | | [B^k] -> [B^k~~ x ~~D^k], |~~

+

o-----o / o-----o o-----o

−

+

\ / \ /

−

~~| | for each $W$ among: | | -> |~~

+

\ / \ /

−

~~| | $e$, $E$, $D$, $T$ | | ([B^n~~ x ~~D^n]->[B^k x D^k]) |~~

+

\ / \ /

−

~~| | | |~~ |

+

\ / \ /

−

o------o~~--------------------~~-----o~~------------------~~o~~-----------------------~~-----o

+

\ / \ /

−

~~| |~~ ~~| |~~

+

o o

−

~~| $e$ | | Radius Operator $e$ = <!e!, !h!> |~~

+

−

~~| $E$ | | Secant Operator $E$ = <!e!, E >~~ |

+

Figure 57-3. Chord Operator Diagram for the Conjunction J = uv

−

~~| $D$ | | Chord Operator $D$ = <!e!, D >~~ |

−

~~| $T$ | | Tangent Functor $T$ = <!e!, d > |~~

−

~~| | | |~~

−

o~~------~~o-~~------------------------o-----------------------------------------------o~~

</pre>

−

===~~Table 59~~. ~~Synopsis of Terminology: Restrictive and Alternative Subtypes~~===

+

+

[[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]

+

<center>'''Figure 57-3. Chord Operator Diagram for the Conjunction ''J'' = ''uv'''''</center>

+

===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===

<pre>

−

~~Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes~~

+

o o

−

~~o--------------~~o~~----------------------o--------------------o----------------------~~o

+

//\ //\

−

+

////\ ////\

−

~~| | or | or | or |~~

+

//////\ o/////o

−

+

////////\ /X\////X\

−

o~~--------------~~o~~----------------------~~o~~--------------------o----------------------~~o

+

//////////\ /XXX\//XXX\

−

~~| | | | |~~

+

o///////////o oXXXXXoXXXXXo

−

~~| Operand~~ ~~| F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |~~

+

/ \////////// \ /\\XXX//\XXX/\\

−

~~| | | | |~~

+

/ \//////// \ /\\\\X////\X/\\\\

−

+

/ \////// \ o\\\\\o/////o\\\\\o

−

~~| | | | |~~

+

/ \//// \ / \\\\/\\////\\\\\/ \

−

o~~--------------~~o~~----------------------~~o~~--------------------o----------------------~~o

+

/ \// \ / \\/\\\\//\\\\\/ \

−

~~| | | | |~~

+

o o o o o\\\\\o\\\\\o o

−

~~| Tacit | !e! : | !e!F_i : | !e!F :~~ |

+

|\ / \ /| |\ / \\\\/ \\\\/ \ /|

−

~~| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y]~~ |

+

| \ / \ / | | \ / \\/ \\/ \ / |

−

+

| \ / \ / | | o o o o |

−

~~| | | | |~~

+

| \ / \ / | | |\ / \ / \ /| |

−

o~~--------------~~o~~----------------------~~o~~--------------------~~o~~----------------------~~o

+

| u \ / \ / v | |u | \ / \ / \ / | v|

−

| | | | |

+

o-----o o-----o 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] |~~

+

\ / | du \ / \ / dv |

−

+

\ / o-----o o-----o

−

~~| | | | |~~

+

\ / \ /

−

o~~--------------~~o~~----------------------~~o~~--------------------o----------------------~~o

+

\ / \ /

−

| | | | |

+

o o

−

~~| Enlargement~~ ~~| E : | EF_i : | EF :~~ |

+

U% $T$ $E$U%

−

~~| Operator | U%->EU%, X%->EX%, | <~~|u,v~~,du,dv~~|~~> -> D~~ | [u~~,v,du,dv]->[dx,dy]~~ |

+

o------------------>o

−

+

| |

−

~~| | | |~~ |

+

| |

−

o--------------o~~----------------------~~o~~--------------------~~o--~~------------------~~--o

+

| |

−

| ~~| | | |~~

+

| |

−

~~| Difference~~ ~~| D : | DF_i : | DF :~~ |

+

J | | $T$J

−

+

| |

−

+

| |

−

~~| | | | |~~

+

| |

−

o~~---------~~-----o~~----------------------~~o~~--------------------o-----------------~~-----o

+

v v

−

~~| | | | |~~

+

o------------------>o

−

~~| Differential | d : | dF_i : | dF : |~~

+

X% $T$ $E$X%

−

~~| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |~~

+

o o

−

+

//\ /X\

−

~~| | | | |~~

+

////\ /XXX\

−

o~~--------------o----------------------o--------------------o----------------------~~o

+

//////\ /XXXXX\

−

~~| | | | |~~

+

////////\ /XXXXXXX\

−

~~| Remainder | r :~~ ~~| rF_i : | rF : |~~

+

//////////\ /XXXXXXXXX\

−

~~| Operator |~~ U%~~->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |~~

+

////////////o oXXXXXXXXXXXo

−

+

///////////// \ //\XXXXXXXXX/\\

−

~~| | | | |~~

+

///////////// \ ////\XXXXXXX/\\\\

−

o~~--------------o----------------------o--------------------o----~~------------------o

+

///////////// \ //////\XXXXX/\\\\\\

−

| ~~| | |~~ |

+

///////////// \ ////////\XXX/\\\\\\\\

−

| ~~Radius | $e$ = <!e!, !h!> : | | $e$F :~~ |

+

///////////// \ //////////\X/\\\\\\\\\\

−

| ~~Operator | | |~~ |

+

o//////////// o o///////////o\\\\\\\\\\\o

−

+

|\////////// / |\////////// \\\\\\\\\\/|

−

+

| \//////// / | \//////// \\\\\\\\/ |

−

| ~~| | |~~ |

+

| \////// / | \////// \\\\\\/ |

−

| ~~| | | [B^n x D^n] ->~~ |

+

| \//// / | \//// \\\\/ |

−

| ~~| | | [B^k x D^k]~~ |

+

| x \// / | x \// \\/ dx |

−

~~| | | | |~~

+

o-----o / o-----o o-----o

−

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], |~~

+

o o

−

~~| | | | |~~

+

−

~~| | | | [B^n x D^n] -> |~~

+

Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv

−

~~| | | | [B^k x D^k] |~~

−

~~| | | | |~~

−

o~~--------------o----------------------o--------------------o----------------------o~~

−

~~| | | | |~~

−

~~| Chord | $D$ = <!e!, D> :~~ ~~| | $D$F : |~~

−

~~| Operator~~ ~~| | | |~~

−

−

−

| | ~~| |~~ |

−

| | ~~| | [B^n x D^n] ->~~ |

−

| ~~| |~~ | ~~[B^k~~ x ~~D^k] |~~

−

~~| | | |~~ |

−

o-----~~---------~~o~~-----------------~~-----o~~--------------------~~o~~-----------------~~-----o

−

~~| | | | |~~

−

~~| Tangent | $T$ = <!e!, d> :~~ ~~| dF_i : | $T$F : |~~

−

~~| Functor | | | |~~

−

~~| | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |~~

−

~~| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |~~

−

~~| | | | |~~

−

~~| | | B^n x D^n -> D~~ ~~| [B^n x D^n] -> |~~

−

~~| | | | [B^k x D^k] |~~

−

~~| | | | |~~

−

o~~--------------~~o-~~---------------------o--------------------o----------------------o~~

</pre>

−

===Formula Display 12===

+

+

[[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]

+

<center>'''Figure 57-4. Tangent Functor Diagram for the Conjunction ''J'' = ''uv'''''</center>

+

===Formula Display 11===

<pre>

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

| |

−

| x = f(u, v) = ((u)(v)) |

+

| F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] |

+

| |

+

| where f = F_1 : [u, v] -> [x] |

| |

−

| ~~y =~~ g~~(u, v)~~ = ((u, v)) |

+

| and g = F_2 : [u, v] -> [y] |

| |

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

Line 7,796: Line 8,119:

−

{| 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''

−

| ''x''

| =

−

| ''f~~''‹''u~~'', ''v''›

+

| align="center" | ‹''f'', ''g''›

| =

−

| ((''u'')(''v''))

+

| align="center" | ‹''F''1, ''F''2›

−

| &~~nbsp~~;

+

| :

+

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

+

| →

+

| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki>

|-

−

| ~~ ~~

+

| colspan="2" | where

−

| ''y''

+

| align="center" | ''f''

| =

−

| ''g''‹''u'', ''v''›

+

| align="center" | ''F''1

+

| :

+

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

+

| →

+

| <nowiki>[</nowiki>''x''<nowiki>]</nowiki>

+

|-

+

| colspan="2" | and

+

| align="center" | ''g''

| =

−

| ((''u'', ''v''))

+

| align="center" | ''F''2

−

| &~~nbsp~~;

+

| :

+

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

+

| →

+

| <nowiki>[</nowiki>''y''<nowiki>]</nowiki>

|}

−

~~===Formula Display 13===~~

−

~~<pre>~~

−

~~o-----------------------------------------------------------o~~

−

~~| |~~

−

~~| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |~~

−

~~| |~~

−

~~o-----------------------------------------------------------o~~

−

~~</pre>~~

Line 7,832: Line 8,157:

|

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

−

| ‹''x'', ''y''›

+

| align="left" | ''F''

+

| =

+

| ‹''f'', ''g''›

| =

−

| ''F''‹''u'', ''v''›

+

| ‹''F''1, ''F''2›

+

| :

+

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

+

| →

+

| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki>

+

|-

+

| align="left" colspan="2" | where

+

| ''f''

| =

−

| ~~‹((~~''u''~~)(''v'')), ((~~''u'', ''v''~~))›~~

+

| ''F''1

−

|}

+

| :

−

|}

+

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

−

<~~/font~~>

+

| →

−

+

| <nowiki>[</nowiki>''x''<nowiki>]</nowiki>

−

<~~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" colspan="2" | and

−

|

+

| ''g''

−

{| align="~~center" border="0" cellpadding="8~~" ~~cellspacing~~="0" ~~style="background:lightcyan; font-weight:bold; text-align:center; width:100%"~~

−

| ~~ ~~

−

| ~~‹''x'',~~ ''y''›

| =

−

| ''F''‹''u'', ''v''›

+

| ''F''2

−

| =

+

| :

−

| ~~‹((''u'')(''v'')), ((''u'',~~ ''v''~~))›~~

+

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

−

~~|  ~~

+

| →

+

| <nowiki>[</nowiki>''y''<nowiki>]</nowiki>

|}

−

===Table 60. ~~Propositional Transformation~~===

+

===Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators===

<pre>

−

Table 60. ~~Propositional Transformation~~

+

Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators

−

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

+

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

−

| u | v | f | g |

+

−

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

+

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

−

| | | | |

+

| | | | |

−

| 0 | 0 | 0 | 1 |

+

| U% | = [u, v] | Source Universe | [B^n] |

−

| | | | |

+

| | | | |

−

| 0 | 1 | 1 | 0 |

+

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

−

| | | | |

+

| | | | |

−

| 1 | 0 | 1 | 0 |

+

| X% | = [x, y] | Target Universe | [B^k] |

−

| | | | |

+

| | = [f, g] | | |

−

| 1 | 1 | 1 | 1 |

+

| | | | |

−

| | | | |

+

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

−

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

+

| | | | |

−

| | | ((u)(v)) | (~~(u, v~~)) |

+

−

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

+

−

<~~/pre~~>

+

| | | | |

−

+

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

−

==~~=Figure 61.~~ ~~Propositional Transformation===~~

+

| | | | |

+

+

+

| | | | |

+

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

+

| | | | |

+

+

+

| | | | |

+

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

+

| | | | |

+

+

+

+

+

+

| | | | |

+

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

+

| | | | |

+

| W | W : | Operator | |

+

| | U% -> EU%, | | [B^n] -> [B^n x D^n], |

+

| | X% -> EX%, | | [B^k] -> [B^k x D^k], |

+

+

| | for each W among: | | -> |

+

+

| | | | |

+

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

+

| | | |

+

| !e! | | Tacit Extension Operator !e! |

+

| !h! | | Trope Extension Operator !h! |

+

| E | | Enlargement Operator E |

+

| D | | Difference Operator D |

+

| d | | Differential Operator d |

+

| | | |

+

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

+

| | | | |

+

| $W$ | $W$ : | Operator | |

+

+

+

+

| | for each $W$ among: | | -> |

+

+

| | | | |

+

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

+

| | | |

+

| $e$ | | Radius Operator $e$ = <!e!, !h!> |

+

| $E$ | | Secant Operator $E$ = <!e!, E > |

+

| $D$ | | Chord Operator $D$ = <!e!, D > |

+

| $T$ | | Tangent Functor $T$ = <!e!, d > |

+

| | | |

+

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

+

</pre>

−

~~<pre>~~

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"

−

~~o-------------------~~---~~-------------------------------o~~

+

|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''

−

| U |

+

|- style="background:paleturquoise"

−

| |

+

! Item

−

| o-~~----------o o-----------o~~ |

+

! Notation

−

| ~~/ \ / \~~ |

+

! Description

−

| / ~~o \ |~~

+

! Type

−

| ~~/ / \ \~~ |

+

|-

−

| / ~~/ \ \ |~~

+

| valign="top" | ''U'' •

−

| ~~o o o o~~ |

+

| valign="top" | = [''u'', ''v'']

−

| | | | | |

+

| valign="top" | Source Universe

−

| | u | ~~| v |~~ |

+

| valign="top" | ['''B'''''n'']

−

~~| | | | |~~ |

+

|-

−

~~| o o o o~~ |

+

| valign="top" | ''X'' •

−

~~| \ \ / /~~ |

+

| valign="top" |

−

| ~~\ \~~ / / |

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~\ o /~~ |

+

| = [''x'', ''y'']

−

| \ / \ / |

+

|-

−

~~| o-----------o o-----------o~~ |

+

| = [''f'', ''g'']

−

~~| |~~

+

|}

−

| |

+

| valign="top" | Target Universe

−

~~o--------------------~~---------------~~------------------o~~

+

| valign="top" | ['''B'''''k'']

−

/ \ / \

+

|-

−

~~/ \ / \~~

+

| valign="top" | E''U'' •

−

/ \ / \

+

| valign="top" | = [''u'', ''v'', d''u'', d''v'']

−

~~/ \ / \~~

+

| valign="top" | Extended Source Universe

−

~~/ \~~ / \

+

| valign="top" | ['''B'''''n'' × '''D'''''n'']

−

/ ~~\ / \~~

+

|-

−

~~/ \ / \~~

+

| valign="top" | E''X'' •

−

~~/ \ / \~~

+

| valign="top" |

−

/ \ / \

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~/ \ / \~~

+

| = [''x'', ''y'', d''x'', d''y'']

−

~~/ \ / \~~

+

|-

−

/ \ / \

+

| = [''f'', ''g'', d''f'', d''g'']

−

~~o----------------~~---------~~o o--~~------~~-----------------o~~

+

|}

−

| U | |~~\U \\\\\\\\\\\\\\\\\\\\\\~~|

+

| valign="top" | Extended Target Universe

−

| o--~~-o o---o~~ | |\~~\\\\\o~~-~~--o~~\~~\\o~~---~~o\\\\\\~~|

+

| valign="top" | ['''B'''''k'' × '''D'''''k'']

−

| ~~//////\ //////\~~ | |~~\\\\\/ \\~~/ ~~\\\\\\~~|

+

|-

−

| ~~////~~////o/////~~//\ | |\\\\/ o \\\\\~~|

+

| ''F''

−

| //////////\/~~//////\~~ | |~~\\\/ /\\ \\\\~~|

+

| ''F'' = ‹''f'', ''g''› : ''U'' • → ''X'' •

−

| ~~o///////o///o///////o~~ | ~~|\\o o\\\o o\\~~|

+

| Transformation, or Mapping

−

| |~~// u //~~|~~///~~|~~// v //~~| | ~~|\\| u |\\\| v |\\~~|

+

| ['''B'''''n''] → ['''B'''''k'']

−

| o/////~~//o///o///////o~~ | |~~\\o o\\\o o\\~~|

+

|-

−

| \////~~///\//////////~~ | |~~\\\\ \\/ /\\\~~|

+

| valign="top" |

−

| ~~\///////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\\\\\\|~~

+

|-

−

| | |\\\~~\\\\\\\\\\\\\\\\\\\\\\~~|

+

| ''f''

−

~~o----~~-~~--------------------o o-------------------------o~~

+

|-

−

~~\ | | /~~

+

| ''g''

−

~~\ | | /~~

+

|}

−

~~\ | | /~~

+

| valign="top" |

−

~~\ f | | g /~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~\ | | /~~

+

| ''f'', ''g'' : ''U'' → '''B'''

−

~~\ | | /~~

+

|-

−

~~\ | | /~~

+

| ''f'' : ''U'' → [''x''] &sube; ''X'' •

−

~~\ | | /~~

+

|-

−

~~\ | | /~~

+

| ''g'' : ''U'' → [''y''] &sube; ''X'' •

−

~~\ | | /~~

+

|}

−

~~o-------\----|---------------------------|----/-------o~~

+

| valign="top" |

−

~~| X \ | | / |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| \| |/ |~~

+

| Proposition

−

~~| o-----------o o-----------o |~~

+

|}

−

~~| //////////////\ /\\\\\\\\\\\\\\ |~~

+

| valign="top" |

−

~~| ////////////////o\\\\\\\\\\\\\\\\ |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"

−

~~| /////////////////X\\\\\\\\\\\\\\\\\ |~~

+

| '''B'''''n'' → '''B'''

−

~~| /////////////////XXX\\\\\\\\\\\\\\\\\ |~~

+

|-

−

~~| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |~~

+

| ∈ ('''B'''''n'', '''B'''''n'' → '''B''')

−

~~| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |~~

+

|-

−

~~| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |~~

+

| = ('''B'''''n'' +→ '''B''') = ['''B'''''n'']

−

~~| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |~~

+

|}

−

~~| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |~~

+

|-

−

~~| \///////////////\XXX/\\\\\\\\\\\\\\\/ |~~

+

| valign="top" |

−

~~| \///////////////\X/\\\\\\\\\\\\\\\/ |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| \///////////////o\\\\\\\\\\\\\\\/ |~~

+

| W

−

~~| \////////////// \\\\\\\\\\\\\\/ |~~

+

|}

−

~~| o-----------o o-----------o |~~

+

| valign="top" |

−

~~| |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| |~~

+

| W :

−

~~o-----------------------------------------------------o~~

+

|-

−

~~Figure 61. Propositional Transformation~~

+

| ''U'' • → E''U'' • ,

−

~~</pre>~~

+

|-

−

+

| ''X'' • → E''X'' • ,

−

~~===Figure 62. Propositional Transformation (Short Form)===~~

+

|-

−

+

| (''U'' • → ''X'' •)

−

~~<pre>~~

+

|-

−

~~o-------------------------o o-------------------------o~~

+

| →

−

~~| U | |\U \\\\\\\\\\\\\\\\\\\\\\|~~

+

|-

−

~~| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|~~

+

| (E''U'' • → E''X'' •) ,

−

~~| //////\ //////\ | |\\\\\/ \\/ \\\\\\|~~

+

|-

−

~~| ////////o///////\ | |\\\\/ o \\\\\|~~

+

| for each W in the set:

−

~~| //////////\///////\ | |\\\/ /\\ \\\\|~~

+

|-

−

~~| o///////o///o///////o | |\\o o\\\o o\\|~~

+

| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}

−

~~| |// u //|///|// v //| | |\\| u |\\\| v |\\|~~

+

|}

−

~~| o///////o///o///////o | |\\o o\\\o o\\|~~

+

| valign="top" |

−

~~| \///////\////////// | |\\\\ \\/ /\\\|~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| \///////o//////// | |\\\\\ o /\\\\|~~

+

| Operator

−

~~| \////// \////// | |\\\\\\ /\\ /\\\\\|~~

+

|}

−

~~| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|~~

+

| valign="top" |

−

~~| | |\\\\\\\\\\\\\\\\\\\\\\\\\|~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"

−

~~o-------------------------o o-------------------------o~~

+

|  

−

~~\ / \ /~~

+

|-

−

~~\ / \ /~~

+

| ['''B'''''n''] → ['''B'''''n'' × '''D'''''n''] ,

−

~~\ / \ /~~

+

|-

−

~~\ f / \ g /~~

+

| ['''B'''''k''] → ['''B'''''k'' × '''D'''''k''] ,

−

~~\ / \ /~~

+

|-

−

~~\ / \ /~~

+

| (['''B'''''n''] → ['''B'''''k''])

−

~~\ / \ /~~

+

|-

−

~~\ / \ /~~

+

| →

−

~~\ / \ /~~

+

|-

−

~~o---------\-----/---------------------\-----/---------o~~

+

| (['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k''])

−

~~| X \ / \ / |~~

+

|-

−

~~| \ / \ / |~~

+

|  

−

~~| o-----------o o-----------o |~~

+

|-

−

~~| //////////////\ /\\\\\\\\\\\\\\ |~~

+

|  

−

~~| ////////////////o\\\\\\\\\\\\\\\\ |~~

+

|}

−

~~| /////////////////X\\\\\\\\\\\\\\\\\ |~~

+

|-

−

~~| /////////////////XXX\\\\\\\\\\\\\\\\\ |~~

+

|

−

~~| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |~~

+

| <math>\epsilon</math>

−

~~| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |~~

+

|-

−

~~| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |~~

+

| <math>\eta</math>

−

~~| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |~~

+

|-

−

~~| \///////////////\XXX/\\\\\\\\\\\\\\\/ |~~

+

| E

−

~~| \///////////////\X/\\\\\\\\\\\\\\\/ |~~

+

|-

−

~~| \//~~/~~////////////o\\\\\\\\\\\\\\\/ |~~

+

| D

−

| \/~~///////////// \\\\\\\\\\\\\\/ |~~

+

|-

−

| ~~o-----------o o-----------o |~~

+

| d

−

| |

+

|}

−

~~| |~~

+

| valign="top" |  

−

~~o-----------------------------------------------------o~~

+

| colspan="2" |

−

~~Figure 62. Propositional Transformation (Short Form)~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"

−

<~~/pre~~>

+

| 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%"

+

| '''W'''

+

|}

+

| valign="top" |

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''W''' :

+

|-

+

| ''U'' • → '''T'''''U'' • = E''U'' • ,

+

|-

+

| ''X'' • → '''T'''''X'' • = E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •)

+

|-

+

| →

+

|-

+

| ('''T'''''U'' • → '''T'''''X'' •) ,

+

|-

+

| for each '''W''' in the set:

+

|-

+

| {'''e''', '''E''', '''D''', '''T'''}

+

|}

+

| 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'''''n''] → ['''B'''''n'' × '''D'''''n''] ,

+

|-

+

| ['''B'''''k''] → ['''B'''''k'' × '''D'''''k''] ,

+

|-

+

| (['''B'''''n''] → ['''B'''''k''])

+

|-

+

| →

+

|-

+

| (['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k''])

+

|-

+

|  

+

|-

+

|  

+

|}

+

|-

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''e'''

+

|-

+

| '''E'''

+

|-

+

| '''D'''

+

|-

+

| '''T'''

+

|}

+

| valign="top" |  

+

| colspan="2" |

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"

+

| Radius Operator || '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>›

+

|-

+

| Secant Operator || '''E''' = ‹<math>\epsilon</math>, E›

+

|-

+

| Chord Operator || '''D''' = ‹<math>\epsilon</math>, D›

+

|-

+

| Tangent Functor || '''T''' = ‹<math>\epsilon</math>, d›

+

|}

+

|}

−

===~~Figure 63~~. ~~Transformation~~ of ~~Positions~~===

+

===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===

<pre>

−

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

+

Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes

−

~~|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|~~

+

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

−

~~|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|~~

+

−

~~|` ` ` ` ` `~~ o-----------~~o ` o~~-----------o ~~` ` ` ` ` `~~|

+

| | or | or | or |

−

|~~` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `~~|

+

−

|~~` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `~~|

+

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

−

|~~` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `~~|

+

| | | | |

−

|~~` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `~~|

+

| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |

−

|~~` ` ` `~~o~~' ' ' ' ' ' ' 'o^^^^^~~o~~' ' ' ' ' ' ' 'o` ` ` `|~~

+

| | | | |

−

~~|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|~~

+

−

~~|` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|~~

+

| | | | |

−

~~|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|~~

+

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

−

~~|` `@` `~~o~~' ' ' ' @ ' ' '~~o~~^^@^^~~o~~' ' ' @ ' ' ' 'o` ` ` `~~|

+

| | | | |

−

|~~` ` \ ` \ ' ' '~~ | ~~' ' ' \^|^/ ' ' '~~ | ~~' ' ' / ` ` ` `~~|

+

| Tacit | !e! : | !e!F_i : | !e!F : |

−

|~~` ` `\` `\' ' '~~ | ~~' ' ' '\~~|~~/' ' ' '~~ | ~~' ' '/` ` ` ` `~~|

+

−

|~~` ` ` \ ` \ ' '~~ | ~~' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `~~|

+

−

|~~` ` ` `\` `\' '~~ | ~~' ' ' '/~~|~~\' ' ' '~~ | ~~' '/` ` ` ` ` `~~|

+

| | | | |

−

|~~` ` ` ` \ ` o~~---|-------o ~~| o~~-------|---~~o ` ` ` ` ` `|~~

+

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

−

~~|` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|~~

+

| | | | |

−

~~|` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|~~

+

| Trope | !h! : | !h!F_i : | !h!F : |

−

o-----------\----|---------|---------|----------------o

+

−

~~" " \~~ | | | ~~" "~~

+

−

~~" " \~~ | | | ~~" "~~

+

| | | | |

−

~~" " \~~ | | | ~~" "~~

+

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

−

~~" " \~~| ~~| | " "~~

+

| | | | |

−

o-------------------------~~o \ | | o~~-------------------------o

+

| Enlargement | E : | EF_i : | EF : |

−

~~| U | |\ | | |`U```````````````````````|~~

+

−

~~| o~~---~~o o~~---o ~~| | \ | | |``````o~~---~~o```o~~---~~o``````|~~

+

−

~~| /'''''\ /'''''\ |~~ | \ | | |~~`````/ \`/ \`````~~|

+

| | | | |

−

| ~~/'''''''o'''''''\~~ | | \ | ~~| |````/ o \````~~|

+

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

−

| ~~/'''''''/'\'''''''\~~ | ~~| \~~ | | ~~|```/ /`\ \```~~|

+

| | | | |

−

| ~~o'''''''o'''o'''''''o~~ | | ~~\ |~~ | ~~|``o o```o o``~~|

+

| Difference | D : | DF_i : | DF : |

−

| |~~'''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``````~~|

+

| Differential | d : | dF_i : | dF : |

−

| | | | ~~\ * |`````````````````````````~~|

+

−

o-------------------------o ~~| | \ / o~~-------------------------o

+

−

~~\ | |~~ | ~~\ /~~ | /

+

| | | | |

−

~~\ ((u)(v))~~ | | | \/ | ~~((u, v)) /~~

+

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

−

\ | | | /\ | /

+

| | | | |

−

\ | | | ~~/ \~~ | /

+

| Remainder | r : | rF_i : | rF : |

−

\ | | | ~~/ \~~ | /

+

−

\ | | | / * | /

+

−

\ | | ~~| /~~ | | /

+

| | | | |

−

~~\ | | |/ | | /~~

+

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

−

~~\ | | / | | /~~

+

| | | | |

−

~~\ | | /| | | /~~

+

| Radius | $e$ = <!e!, !h!> : | | $e$F : |

−

o-------\----|---|-------/-|---------|---|----/-------o

+

−

| ~~X \~~ | | / | | | / |

+

−

| \| | / | | |~~/ |~~

+

−

| ~~o--~~-|----/--o | o--~~-----|---o~~ |

+

| | | | |

−

~~| /' ' | ' / ' '\|/` ` ` ` | ` `\~~ |

+

−

| ~~/ ' '~~ | ~~'/' ' '~~ | ~~` ` ` ` | ` ` \~~ |

+

−

~~| /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |~~

+

| | | | |

−

~~| / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |~~

+

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

−

~~| @~~ o~~' ' ' ' @ ' ' 'o^^@^^~~o~~` ` ` @ ` ` ` `o |~~

+

| | | | |

−

| |~~' ' ' ' ' ' ' '~~|~~^^^^^~~|~~` ` ` ` ` ` ` `~~| |

+

| Secant | $E$ = <!e!, E> : | | $E$F : |

−

| |~~' ' ' ' f ' ' '~~|~~^^^^^~~|~~` ` ` g ` ` ` `~~| |

+

−

| |~~' ' ' ' ' ' ' '~~|~~^^^^^~~|~~` ` ` ` ` ` ` `~~| |

+

−

| ~~o' ' ' ' ' ' ' 'o~~^^^^^~~o` ` ` ` ` ` ` `o~~ |

+

−

| ~~\ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` /~~ |

+

| | | | |

−

| ~~\' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/~~ |

+

−

~~| \ ' ' ' ' ' ' '~~ o ~~` ` ` ` ` ` ` / |~~

+

−

~~| \' ' ' ' ' ' '/ \` ` ` ` ` ` `/~~ |

+

| | | | |

−

| o---------~~--o o~~-----------o |

+

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

−

~~| |~~

+

| | | | |

−

~~| |~~

+

| Chord | $D$ = <!e!, D> : | | $D$F : |

−

~~o-------------------------------~~----------------------o

+

−

~~Figure 63. Transformation of Positions~~

+

+

+

| | | | |

+

+

+

| | | | |

+

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

+

| | | | |

+

| Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |

+

+

+

+

| | | | |

+

+

+

| | | | |

+

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

</pre>

−

===~~Table 64. Transformation of Positions=~~==

+

{| 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'''

−

~~<pre>~~

+

|- style="background:paleturquoise"

−

Table 64. ~~Transformation~~ of ~~Positions~~

+

|  

−

~~o-----o----------o----------o-------o-------o-~~--~~-----o--------o-------------o~~

+

| align="center" | '''Operator or Operand'''

−

| ~~u v | x | y | x y | x(y) | (x)y | (x)(y) | X%~~ = ~~[x, y] |~~

+

| align="center" | '''Proposition or Component'''

−

~~o-----o----------o----------o-------o-------o-~~---~~----o--------o-------------o~~

+

| align="center" | '''Transformation or Mapping'''

−

| ~~| | | | | | | ^~~ |

+

|-

−

| 0 0 ~~| 0 | 1 | 0 | 0 | 1 | 0 | | |~~

+

| Operand

−

| ~~| | | | |~~ | | |

+

| valign="top" |

−

| ~~0 1~~ | 1 | 0 | 0 ~~| 1~~ | 0 | 0 | F |

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| | ~~| | | | | |~~ = |

+

| ''F'' = ‹''F''1, ''F''2›

−

~~| 1~~ 0 ~~| 1 | 0 |~~ 0 | ~~1 | 0 | 0 |~~ <~~f , g~~> |

+

|-

−

| ~~| | | | | | | |~~

+

| ''F'' = ‹''f'', ''g''› : ''U'' → ''X''

−

| ~~1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |~~

+

|}

−

~~| | | | | | | | | |~~

+

| valign="top" |

−

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%"

−

| | ~~((u)(v))~~ | ~~((u, v)) | u v | (u,v) | (u)(v) |~~ 0 ~~| U%~~ = ~~[u, v] |~~

+

| ''F''''i'' : 〈''u'', ''v''〉 → '''B'''

−

~~o-----o~~------~~----o----------o-------o-------o--------o--------o-------------o~~

+

|-

−

</~~pre~~>

+

| ''F''''i'' : '''B'''''n'' → '''B'''

−

+

|}

−

~~===Table 65. Induced Transformation on Propositions===~~

+

| valign="top" |

−

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"

−

<~~pre~~>

+

| ''F'' : [''u'', ''v''] → [''x'', ''y'']

−

~~Table 65. Induced Transformation on Propositions~~

+

|-

−

~~o------------o-------------------------~~----~~----o------------o~~

+

| ''F'' : '''B'''''n'' → '''B'''''k''

−

| ~~X% |~~ <~~--- F = <f , g~~> <--- | U~~% |~~

+

|}

−

~~o------------o-------~~-~~--o-----------o----------o------------o~~

+

|-

−

| | u = ~~| 1 1~~ 0 ~~0 |~~ = ~~u | |~~

+

|

−

~~| | v~~ = ~~| 1~~ 0 ~~1 0 |~~ = ~~v | |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~f_i~~ <~~x, y~~> o-~~---------o-----------o----------o f_j <~~u, v> |

+

| Tacit

−

| ~~| x = | 1 1 1 0 | = f~~<~~u,v~~> | |

+

|-

−

| | y = ~~| 1~~ 0 0 ~~1 |~~ = ~~g<u,v>~~ | |

+

| Extension

−

~~o------------o----------o----~~----~~---o----------o------------o~~

+

|}

−

| ~~| | | | |~~

+

|

−

~~| f_0 | () | 0 0 0 0 | () | f_0 |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| | | | | |~~

+

| <math>\epsilon</math> :

−

~~| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |~~

+

|-

−

~~| | | | | |~~

+

| ''U'' • → E''U'' • , ''X'' • → E''X'' • ,

−

~~| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |~~

+

|-

−

~~| | | | | |~~

+

| (''U'' • → ''X'' •) → (E''U'' • → ''X'' •)

−

~~| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |~~

+

|}

−

~~| | | | | |~~

+

|

−

~~| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| | | | | |~~

+

| <math>\epsilon</math>''F''''i'' :

−

~~| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |~~

+

|-

−

~~| | | | | |~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''

−

~~| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |~~

+

|-

−

~~| | | | | |~~

+

| '''B'''''n'' × '''D'''''n'' → '''B'''

−

~~| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |~~

+

|}

−

~~| | | | | |~~

+

|

−

~~o------------o----------o-----------o----------o------------o~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| | | | | |~~

+

| <math>\epsilon</math>''F'' :

−

~~| f_8 | x y | 1 0 0 0 | u v | f_8 |~~

+

|-

−

~~| | | | | |~~

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'']

−

~~| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |~~

+

|-

−

~~| | | | | |~~

+

| ['''B'''''n'' × '''D'''''n''] → ['''B'''''k'']

−

~~| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |~~

+

|}

−

~~| | | | | |~~

+

|-

−

~~| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |~~

+

|

−

~~| | | | | |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |~~

+

| Trope

−

~~| | | | | |~~

+

|-

−

~~| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |~~

+

| Extension

−

~~| | | | | |~~

+

|}

−

~~| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |~~

+

|

−

~~| | | | | |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| f_15 | (()) | 1 1 1 1 | (()) | f_15 |~~

+

| <math>\eta</math> :

−

~~| | | | | |~~

+

|-

−

~~o------------o----------o-----------o----------o------------o~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~</pre>~~

+

|-

−

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~===Formula Display 14===~~

+

|}

−

+

|

−

~~<pre>~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~o-------------------------------------------------o~~

+

| <math>\eta</math>''F''''i'' :

−

~~| |~~

+

|-

−

~~| EG_i = G_i |~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~| |~~

+

|-

−

~~o-------------------------------------------------o~~

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

~~</pre>~~

+

|}

−

+

|

−

~~ ~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"~~

+

| <math>\eta</math>''F'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']

+

|-

+

| ['''B'''''n'' × '''D'''''n''] → ['''D'''''k'']

+

|}

+

|-

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| Enlargement

+

|-

+

| Operator

+

|}

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan~~; font-weight:bold~~; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~width="8%"~~ | E''G''<~~sub~~>''i''</~~sub~~>

+

| E :

−

~~| width="4%" | =~~

+

|-

−

~~| width="88%" |~~ ''G''<~~sub~~>''i''</~~sub~~>‹''u'' ~~+ d~~''u'', ''v'' + d''v''›

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| E''F''''i'' :

+

|-

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

+

|-

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

−

~~ ~~

−

~~===Formula Display 15===~~

−

~~<pre>~~

−

~~o-------------------------------------------------o~~

−

~~| |~~

−

~~| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |~~

−

~~| |~~

−

~~| = G_i <u, v> + G_i |~~

−

~~| |~~

−

~~o-------------------------------------------------o~~

−

~~</pre>~~

−

~~ ~~

−

~~{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"~~

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan~~; font-weight:bold~~; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~width="8%" | D~~''G''~~''i''~~

+

| E''F'' :

−

| ~~width="4%" | =~~

+

|-

−

| ~~width="20%" |~~ ''G''~~~~''i''~~‹~~''u'', ''v''›

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']

−

| ~~width="4%" | +~~

+

|-

−

| ~~width="64%" | E~~''G''<~~sub~~>''i''</~~sub~~>‹''u'', ''v''~~, d~~''u''~~, d~~''v''›

+

| ['''B'''''n'' × '''D'''''n''] → ['''D'''''k'']

+

|}

+

|-

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| Difference

|-

−

| ~~width="8%" |  ~~

+

| Operator

−

~~| width="4%" | =~~

−

~~| width="20%" | ''G''''i''‹''u'', ''v''›~~

−

~~| width="4%" | +~~

−

~~| width="64%" | ''G''''i''‹''u'' + d''u'', ''v'' + d''v''›~~

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| D :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

|}

−

~~ ~~

−

~~===Formula Display 16===~~

−

~~<pre>~~

−

~~o-------------------------------------------------o~~

−

~~| |~~

−

~~| Ef = ((u + du)(v + dv)) |~~

−

~~| |~~

−

~~| Eg = ((u + du, v + dv)) |~~

−

~~| |~~

−

~~o-------------------------------------------------o~~

−

~~</pre>~~

−

~~ ~~

−

~~{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"~~

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan~~; font-weight:bold~~; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~width="8%" | E~~''f''

+

| D''F''''i'' :

−

| ~~width="4%" | =~~

+

|-

−

| ~~width="88%" | ((~~''u'' + d''u'')(''v'' ~~+ d~~''v''))

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

|-

−

| ~~width~~="8%" | E''g''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

| ~~width~~="4%" | =

+

|}

−

| ~~width~~="88%" | ((''u'' ~~+ d~~''u'', ''v'' + d''v''))

+

|

+

{| 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'''''n'' × '''D'''''n''] → ['''D'''''k'']

+

|}

+

|-

+

|

+

{| 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'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

|}

−

|}

−

~~ ~~

−

~~===Formula Display 17===~~

−

~~<pre>~~

−

~~o-------------------------------------------------o~~

−

~~| |~~

−

~~| Df = ((u)(v)) + ((u + du)(v + dv)) |~~

−

~~| |~~

−

~~| Dg = ((u, v)) + ((u + du, v + dv)) |~~

−

~~| |~~

−

~~o-------------------------------------------------o~~

−

~~</pre>~~

−

~~ ~~

−

~~{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"~~

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan~~; font-weight:bold~~; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~width="8%" | D~~''f''

+

| d''F''''i'' :

−

| ~~width="4%" | =~~

+

|-

−

| ~~width="20%" | ((~~''u'')(''v''))

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~| width="4%" | +~~

−

~~| width="64%" | ((~~''u'' + d''u'')(''v'' ~~+ d''v~~''))

|-

−

| ~~width="8%" | D~~''g''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

~~| width="4%" | =~~

−

~~| width="20%" | ((~~''u'', ''v''))

−

~~| width="4%" | +~~

−

~~| width="64%" | ((~~''u'' ~~+ d~~''u'', ''~~v'' + d~~''v''))

|}

+

|

+

{| 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'''''n'' × '''D'''''n''] → ['''D'''''k'']

+

|}

+

|-

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| Remainder

+

|-

+

| Operator

|}

−

~~ ~~

−

~~===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>~~

−

~~~~

−

~~{| 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%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| ~~<math>\epsilon</math>''f''~~

+

| r :

−

~~| = || ''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~~''

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~| = ||~~ ''~~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''~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~| = ||~~ ''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="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| + || (''u'')(''v''~~) || <math>\cdot~~</~~math~~> ~~|| ((d''u'')(d''v''))~~

+

| r''F''''i'' :

|-

−

| ~~d''f''~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~| = || ''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''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

~~| = ||~~ ''~~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''

|}

+

|

+

{| 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'''''n'' × '''D'''''n''] → ['''D'''''k'']

+

|}

+

|-

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| Radius

+

|-

+

| Operator

|}

−

~~ ~~

−

~~===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>~~

−

~~~~

−

~~{| 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%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| <~~math>\epsilon</math>''g''~~

+

| '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>› :

−

| = ~~|| ''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~~''

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~| = ||~~ ''~~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''~~

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

−

~~| = ||~~ ''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="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| + || (''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

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

| + || (''u'')(''v'') || <~~math~~>~~\cdot~~</~~math~~> || 0

+

| '''e'''''F'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']

+

|-

+

| ['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k'']

+

|}

+

|-

+

|

+

{| 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%"

+

| '''E''' = ‹<math>\epsilon</math>, E› :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

|  

+

|-

+

|  

+

|-

+

|  

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''E'''''F'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']

+

|-

+

| ['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k'']

+

|}

+

|-

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| Chord

+

|-

+

| Operator

|}

−

|}

−

~~ ~~

−

~~===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>~~

−

~~===Table 68. Computation of an Analytic Series in Symbolic Terms===~~

−

~~<pre>~~

−

~~Table 68. Computation of an Analytic Series in Symbolic Terms~~

−

~~o-----o-----o------------o----------o----------o----------o----------o----------o~~

−

~~| u v | f g | Df | Dg | df | dg | rf | rf |~~

−

~~o-----o-----o------------o----------o----------o----------o----------o----------o~~

−

~~| | | | | | | | |~~

−

~~| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |~~

−

~~| | | | | | | | |~~

−

~~| 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |~~

−

~~| | | | | | | | |~~

−

~~| 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |~~

−

~~| | | | | | | | |~~

−

~~| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |~~

−

~~| | | | | | | | |~~

−

~~o-----o-----o------------o----------o----------o----------o----------o----------o~~

−

~~</pre>~~

−

~~===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>~~

−

~~ ~~

−

~~{| 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%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''D''' = ‹<math>\epsilon</math>, D› :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; 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''))~~

|-

|  

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''D'''''F'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']

+

|-

+

| ['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k'']

+

|}

+

|-

+

|

+

{| 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%"

+

| '''T''' = ‹<math>\epsilon</math>, d› :

+

|-

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

+

|-

+

| (''U'' • → ''X'' •) → (E''U'' • → E''X'' •)

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| d''F''''i'' :

|-

−

| ~~D''g''~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~| = || ''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~~'')

|-

−

|  

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''T'''''F'' :

+

|-

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']

+

|-

+

| ['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k'']

|}

−

~~~~

+

|}

−

===~~Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›~~===

+

===Formula Display 12===

<pre>

−

o------------------------~~-----------o o~~-----------------------------------o

+

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

−

| ~~U | |`U`````````````````````````````````|~~

+

| |

−

~~| | |```````````````````````````````````|~~

+

| x = f(u, v) = ((u)(v)) |

−

~~| ^ | |```````````````````````````````````|~~

+

| |

−

~~| | | |```````````````````````````````````|~~

+

| y = g(u, v) = ((u, v)) |

−

~~| o-------o | o-------o | |```````o-------o```o-------o```````|~~

+

| |

−

~~| ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |~~

+

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

−

~~| \ /```````````|```````````\ / | |``\``/ \ o / \``/``~~|

+

</pre>

−

| ~~\/`````~~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>

−

~~===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>~~

−

{| 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%"

+

{| 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''))

|  

−

|-

−

~~| 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'')~~

|-

|  

−

|-

+

| ''y''

−

~~| d~~''g''

+

| =

−

| = || ''uv'' ~~|| <math>\cdot</math> || (d~~''u'', d''v'')

+

| ''g''‹''u'', ''v''›

−

| ~~+ || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')~~

+

| =

−

~~| + |~~| (~~''u'')''v'' || <math>\cdot</math> ||~~ (d''u'', ~~d''v'')~~

+

| ((''u'', ''v''))

−

~~| + || (''u'')(~~''v'') ~~|| <math>\cdot</math> || (d''u'', d''v''~~)

−

|-

|  

|}

Line 8,595: Line 8,940:

−

===~~Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›~~===

+

===Formula Display 13===

<pre>

−

o o

+

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

−

~~/ \ / \~~

+

| |

−

/ ~~\ /~~ \

+

| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |

−

~~/ \ / O \~~

+

| |

−

~~/ \ o /@\ o~~

+

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

−

~~/ \ / \ / \~~

+

</pre>

−

~~/ \ /~~ \ ~~/ \~~

+

−

~~/ O \ / O \ / O \~~

+

−

~~o /@\ o o /@\ o /@\ o~~

+

{| 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%"

−

~~/ \ / \ / O \ / O \ / O \~~

+

| ‹''x'', ''y''›

−

~~/ \ / \~~ o ~~/@ o /@\ o /@~~ o

+

| =

−

/ ~~\ / \ / \ \ / \ / \ \ / \~~

+

| ''F''‹''u'', ''v''›

−

~~/ \ / \ / \ / \ / \ / \~~

+

| =

−

~~/ 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~~ |

+

−

| ~~\ / \ / | | |\ / \ / \ / \ / \ /| |~~

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

−

| ~~\ / \ / | | | \ / \ / \ / | |~~

+

|

−

| u ~~\ / O \ /~~ v | | u | \ / ~~O \ / O \ /~~ ~~| v |~~

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

−

o-------o @\ o-------o o---+---~~o @\ o @\ o~~---+---o

+

|  

−

~~\ /~~ | ~~\ / \ / \ / \ /~~ |

+

| ‹''x'', ''y''›

−

~~\ /~~ | ~~\ / \ /~~ |

+

| =

−

~~\ / | du \ / O \ / dv~~ |

+

| ''F''‹''u'', ''v''›

−

~~\ /~~ o-------~~o @\ o~~-------o

+

| =

−

~~\ / \ /~~

+

| ‹((''u'')(''v'')), ((''u'', ''v''))›

−

~~\ / \ /~~

+

|  

−

~~\ / \ /~~

+

|}

−

~~o o~~

+

|}

−

~~U% $T$ $E$U%~~

+

−

o------------------>o

+

−

~~| |~~

+

===Table 60. Propositional Transformation===

−

~~| |~~

+

−

~~| |~~

+

<pre>

−

~~| |~~

+

Table 60. Propositional Transformation

−

~~F | | $T$F~~

+

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

−

~~| |~~

+

| u | v | f | g |

−

~~| |~~

+

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

−

~~| |~~

+

| | | | |

−

~~v v~~

+

| 0 | 0 | 0 | 1 |

−

o------------------>o

+

| | | | |

−

~~X% $T$ $E$X%~~

+

| 0 | 1 | 1 | 0 |

−

~~o o~~

+

| | | | |

−

~~/ \ / \~~

+

| 1 | 0 | 1 | 0 |

−

~~/ \ / \~~

+

| | | | |

−

~~/ \ / O \~~

+

| 1 | 1 | 1 | 1 |

−

~~/ \ 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 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>

−

===~~Figure 70~~-b. ~~Tangent Functor Ferris Wheel for F‹u, v›~~ = ‹((u)(v)), ((u, v))›===

+

+

{| 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''

+

| 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%"

+

| 0

+

|-

+

| 0

+

|-

+

| 1

+

|-

+

| 1

+

|}

+

| width="25%" |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 0

+

|-

+

| 1

+

|-

+

| 0

+

|-

+

| 1

+

|}

+

| width="25%" |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 0

+

|-

+

| 1

+

|-

+

| 1

+

|-

+

| 1

+

|}

+

| width="25%" |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 1

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 1

+

|}

+

|-

+

| width="25%" |  

+

| width="25%" |  

+

| width="25%" | ((''u'')(''v''))

+

| width="25%" | ((''u'', ''v''))

+

|}

+

+

===Figure 61. Propositional Transformation===

<pre>

−

o-----------------------~~o o~~-----------------------o o-----------------------o

+

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

−

| dU | | dU | | dU |

+

| U |

−

+

| |

−

| ~~/////\ /////\~~ | | ~~/XXXX\ /XXXX\~~ | | /\\\\\ /\\\\\ |

+

| o-----------o o-----------o |

−

| ~~///////~~o~~//////\~~ | | ~~/XXXXXXoXXXXXX\~~ | | ~~/\\\\\\~~o~~\\\\\\\ |~~

+

| / \ / \ |

−

~~| //////// \//////\ | |~~ /~~XXXXXX~~/ \~~XXXXXX~~\ ~~| |~~ /\\\\\\/ \\\\\\\\ |

+

| / o \ |

−

~~| o~~/////// \~~//////~~o ~~| | oXXXXXX/ \XXXXXXo | |~~ o~~\\\\\\/ \\\\\\\~~o |

+

| / / \ \ |

−

| ~~|/////o o/////| | | |XXXXXo oXXXXX| | |~~ |\\\\\~~o o~~\\\\\| |

+

| / / \ \ |

−

| ~~|/du//| |//dv/| | | |XXXXX| |XXXXX| | |~~ |\du\\~~| |~~\\dv\| |

+

| o o o o |

−

| |/////~~o o~~/////~~| | | |XXXXXo~~ ~~oXXXXX~~| ~~| |~~ |\\\\\o o\\\\\| |

+

| | | | | |

−

| o//////\ ///////~~o | | oXXXXXX~~\ ~~/XXXXXXo~~ | | ~~o\\\~~\\\\ /\\\\\\o |

+

| | u | | v | |

−

| \//////\ //////// ~~| | \XXXXXX~~\ ~~/XXXXXX/~~ | | ~~\\\\\~~\\\ /\\\\\\/ |

+

| | | | | |

−

| \//////o/////// | | ~~\XXXXXXoXXXXXX/ |~~ | \\\\\\\o\\~~\\\\/~~ |

+

| o o o o |

−

| \///// \///// | | \~~XXXX/~~ \~~XXXX/~~ | | \\\\\~~/ \\\\\/~~ |

+

| \ \ / / |

−

+

| \ \ / / |

−

| | | | | |

+

| \ o / |

−

o------~~-----------------o o--------------------~~---o o---~~--------------------o~~

+

| \ / \ / |

−

~~= du' @ (u)(v) o--~~---------------------o ~~dv' @ (u)(v) =~~

+

| o-----------o o-----------o |

−

~~= | dU' | =~~

+

| |

−

~~= | o~~--~~o 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

−

+

| U | |\U \\\\\\\\\\\\\\\\\\\\\\|

−

o~~-----------------------~~o ~~o-----------------------o o-----------------------o~~

+

| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|

−

| dU | | dU | | dU |

+

| //////\ //////\ | |\\\\\/ \\/ \\\\\\|

−

+

| ////////o///////\ | |\\\\/ o \\\\\|

−

| / \ /////~~\ | |~~ /\\\\\ ~~/XXXX~~\ ~~| | /~~\\\\\ /\\\\\ |

+

| //////////\///////\ | |\\\/ /\\ \\\\|

−

| / o//////~~\ | |~~ /\\\\\\~~oXXXXXX~~\ ~~| | /~~\\\\\\o\\\~~\\\\~~ |

+

| o///////o///o///////o | |\\o o\\\o o\\|

−

| / //\//////\ ~~| |~~ /\\\\\\//\~~XXXXXX~~\ ~~| | /~~\\\\\\/ ~~\\\\\\\\~~ |

+

| |// u //|///|// v //| | |\\| u |\\\| v |\\|

−

| o ////\//////~~o | | o\\\\\\~~////\~~XXXXXXo | | o~~\\\\\\/ \\\\\\\o |

+

| o///////o///o///////o | |\\o o\\\o o\\|

−

| ~~| o~~/////o/////~~| | | |~~\\\\\~~o/////oXXXXX| | | |~~\\\\\~~o o~~\\\\\| |

+

| \///////\////////// | |\\\\ \\/ /\\\|

−

| ~~| du |/////|//dv/~~| | | |~~\\\\\~~|~~/////|XXXXX| | | |\du\\| |\\dv\| |~~

+

| \///////o//////// | |\\\\\ o /\\\\|

−

~~| | o/////~~o/~~////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |~~

+

| \////// \////// | |\\\\\\ /\\ /\\\\\|

−

~~| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |~~

+

| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|

−

~~| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |~~

+

| | |\\\\\\\\\\\\\\\\\\\\\\\\\|

−

~~| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |~~

+

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

−

~~| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |~~

+

\ | | /

−

+

\ | | /

−

~~| | | | | |~~

+

\ | | /

−

~~o-----------------------o o-----------------------o o-----------------------o~~

+

\ f | | g /

−

~~= du' @ (u) v o-----------------------o dv' @ (u) v =~~

+

\ | | /

−

~~= | dU' | =~~

+

\ | | /

−

~~= | o--o o--o | =~~

+

\ | | /

−

~~= | /////\ /\\\\\ | =~~

+

\ | | /

−

~~= | ///////o\\\\\\\ | =~~

+

\ | | /

−

~~= | ////////X\\\\\\\\ | =~~

+

\ | | /

−

~~= | o///////XXX\\\\\\\o | =~~

+

o-------\----|---------------------------|----/-------o

−

~~= | |/////oXXXXXo\\\\\| | =~~

+

| X \ | | / |

−

~~= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =~~

+

| \| |/ |

−

~~| |/////oXXXXXo\\\\\| |~~

+

| o-----------o o-----------o |

−

~~| o//////\XXX/\\\\\\o |~~

+

| //////////////\ /\\\\\\\\\\\\\\ |

−

~~| \//////\X/\\\\\\/ |~~

+

| ////////////////o\\\\\\\\\\\\\\\\ |

−

~~| \//////o\\\\\\/ |~~

+

| /////////////////X\\\\\\\\\\\\\\\\\ |

−

~~| \///// \\\\\/ |~~

+

| /////////////////XXX\\\\\\\\\\\\\\\\\ |

−

~~| o--o o--o |~~

+

| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |

−

~~| |~~

+

| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |

−

~~o-----------------------o~~

+

| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |

+

| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |

+

| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |

+

| \///////////////\XXX/\\\\\\\\\\\\\\\/ |

+

| \///////////////\X/\\\\\\\\\\\\\\\/ |

+

| \///////////////o\\\\\\\\\\\\\\\/ |

+

| \////////////// \\\\\\\\\\\\\\/ |

+

| o-----------o o-----------o |

+

| |

+

| |

+

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

+

Figure 61. Propositional Transformation

+

</pre>

−

~~o-----------------------o o-----------------------o o-----------------------o~~

+

−

~~| dU | | dU | | dU |~~

+

[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]

−

+

<center>'''Figure 61. Propositional Transformation'''</center>

−

~~| /////\ / \ | | /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~~

−

= 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

+

===Figure 62. Propositional Transformation (Short Form)===

−

| dU | | dU | | dU |

+

−

+

<pre>

−

| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |

+

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

−

| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |

+

| U | |\U \\\\\\\\\\\\\\\\\\\\\\|

−

| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |

+

| 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 |

+

| |// 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 62. Propositional Transformation (Short Form)

+

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]

+

<center>'''Figure 62. Propositional Transformation (Short Form)'''</center>

+

===Figure 63. Transformation of Positions===

+

<pre>

+

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

+

|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|

+

|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|

+

|` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|

+

|` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|

+

|` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|

+

|` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|

+

|` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|

+

|` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|

+

|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|

+

|` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|

+

|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|

+

|` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|

+

|` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|

+

|` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|

+

|` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|

+

|` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|

+

|` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|

+

|` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|

+

|` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|

+

o-----------\----|---------|---------|----------------o

+

" " \ | | | " "

+

" " \ | | | " "

+

" " \ | | | " "

+

" " \| | | " "

+

o-------------------------o \ | | o-------------------------o

+

| U | |\ | | |`U```````````````````````|

+

+

| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|

+

| /'''''''o'''''''\ | | \ | | |````/ o \````|

+

| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|

+

+

| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|

+

+

| \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|

+

| \'''''''o'''''''/ | | \ | |````\ o /````|

+

| \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|

+

+

| | | | \ * |`````````````````````````|

+

o-------------------------o | | \ / o-------------------------o

+

\ | | | \ / | /

+

\ ((u)(v)) | | | \/ | ((u, v)) /

+

\ | | | /\ | /

+

\ | | | / \ | /

+

\ | | | / \ | /

+

\ | | | / * | /

+

\ | | | / | | /

+

\ | | |/ | | /

+

\ | | / | | /

+

\ | | /| | | /

+

o-------\----|---|-------/-|---------|---|----/-------o

+

| X \ | | / | | | / |

+

| \| | / | | |/ |

+

| 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>

+

+

[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]

+

<center>'''Figure 63. Transformation of Positions'''</center>

+

===Table 64. Transformation of Positions===

+

<pre>

+

Table 64. Transformation of Positions

+

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

+

| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |

+

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

+

| | | | | | | | ^ |

+

| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |

+

| | | | | | | | |

+

| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |

+

| | | | | | | | = |

+

| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |

+

| | | | | | | | |

+

| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |

+

| | | | | | | | | |

+

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>

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

+

|+ '''Table 64. Transformation of Positions'''

+

|- style="background:paleturquoise"

+

| ''u''  ''v''

+

| ''x''

+

| ''y''

+

| ''x'' ''y''

+

| ''x'' (''y'')

+

| (''x'') ''y''

+

| (''x'')(''y'')

+

| ''X'' • = [''x'', ''y'' ]

+

|-

+

| 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

+

|-

+

| 1  0

+

|-

+

| 1  1

+

|}

+

| width="12%" |

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 0

+

|-

+

| 1

+

|-

+

| 1

+

|-

+

| 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%"

+

| 0

+

|-

+

| 1

+

|-

+

| 1

+

|-

+

| 0

+

|}

+

| width="12%" |

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 1

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 0

+

|}

+

| width="12%" |

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 0

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 0

+

|}

+

| width="12%" |

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| ↑

+

|-

+

| ''F''

+

|-

+

| ‹''f'', ''g'' ›

+

|-

+

| ↑

+

|}

+

|-

+

|  

+

| ((''u'')(''v''))

+

| ((''u'', ''v''))

+

| ''u'' ''v''

+

| (''u'', ''v'')

+

| (''u'')(''v'')

+

| ( )

+

| ''U'' • = [''u'', ''v'' ]

+

|}

+

+

===Table 65. Induced Transformation on Propositions===

+

<pre>

+

Table 65. Induced Transformation on Propositions

+

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

+

| X% | <--- F = <f , g> <--- | U% |

+

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

+

| | u = | 1 1 0 0 | = u | |

+

| | v = | 1 0 1 0 | = v | |

+

| f_i <x, y> o----------o-----------o----------o f_j <u, v> |

+

| | x = | 1 1 1 0 | = f<u,v> | |

+

| | y = | 1 0 0 1 | = g<u,v> | |

+

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

+

| | | | | |

+

| f_0 | () | 0 0 0 0 | () | f_0 |

+

| | | | | |

+

| 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 |

+

| | | | | |

+

| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |

+

| | | | | |

+

| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |

+

| | | | | |

+

| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |

+

| | | | | |

+

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

+

| | | | | |

+

| f_8 | x y | 1 0 0 0 | u v | f_8 |

+

| | | | | |

+

| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |

+

| | | | | |

+

| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |

+

| | | | | |

+

| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |

+

| | | | | |

+

| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |

+

| | | | | |

+

| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |

+

| | | | | |

+

| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |

+

| | | | | |

+

| f_15 | (()) | 1 1 1 1 | (()) | f_15 |

+

| | | | | |

+

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

+

</pre>

+

+

{| 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'' •

+

| 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'' •

+

|- style="background:paleturquoise"

+

| rowspan="2" | ''f''''i''‹''x'', ''y''›

+

|

+

{| align="right" style="background:paleturquoise; text-align:right"

+

| ''u'' =

+

|-

+

| ''v'' =

+

|}

+

|

+

{| align="center" style="background:paleturquoise; text-align:center"

+

| 1 1 0 0

+

|-

+

| 1 0 1 0

+

|}

+

|

+

{| align="left" style="background:paleturquoise; text-align:left"

+

| = ''u''

+

|-

+

| = ''v''

+

|}

+

| rowspan="2" | ''f''''j''‹''u'', ''v''›

+

|- style="background:paleturquoise"

+

|

+

{| 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"

+

| = ''f''‹''u'', ''v''›

+

|-

+

| = ''g''‹''u'', ''v''›

+

|}

+

|-

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

| ''f''0

+

|-

+

| ''f''1

+

|-

+

| ''f''2

+

|-

+

| ''f''3

+

|-

+

| ''f''4

+

|-

+

| ''f''5

+

|-

+

| ''f''6

+

|-

+

| ''f''7

+

|}

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

| ()

+

|-

+

|  (''x'')(''y'') 

+

|-

+

|  (''x'') ''y''  

+

|-

+

|  (''x'')    

+

|-

+

|   ''x'' (''y'') 

+

|-

+

|     (''y'') 

+

|-

+

|  (''x'', ''y'') 

+

|-

+

|  (''x''  ''y'') 

+

|}

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

| 0 0 0 0

+

|-

+

| 0 0 0 1

+

|-

+

| 0 0 1 0

+

|-

+

| 0 0 1 1

+

|-

+

| 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'') 

+

|-

+

|  (''u''  ''v'') 

+

|-

+

|  (''u''  ''v'') 

+

|}

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

| ''f''0

+

|-

+

| ''f''0

+

|-

+

| ''f''1

+

|-

+

| ''f''1

+

|-

+

| ''f''6

+

|-

+

| ''f''6

+

|-

+

| ''f''7

+

|-

+

| ''f''7

+

|}

+

|-

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

| ''f''8

+

|-

+

| ''f''9

+

|-

+

| ''f''10

+

|-

+

| ''f''11

+

|-

+

| ''f''12

+

|-

+

| ''f''13

+

|-

+

| ''f''14

+

|-

+

| ''f''15

+

|}

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

|   ''x''  ''y''  

+

|-

+

| ((''x'', ''y''))

+

|-

+

|      ''y''  

+

|-

+

|  (''x'' (''y''))

+

|-

+

|   ''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''))

+

|-

+

| (())

+

|-

+

| (())

+

|}

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

| ''f''8

+

|-

+

| ''f''8

+

|-

+

| ''f''9

+

|-

+

| ''f''9

+

|-

+

| ''f''14

+

|-

+

| ''f''14

+

|-

+

| ''f''15

+

|-

+

| ''f''15

+

|}

+

|}

+

+

===Formula Display 14===

+

<pre>

+

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

+

| |

+

| EG_i = G_i |

+

| |

+

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

+

</pre>

+

+

{| 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''''i''

+

| width="4%" | =

+

| width="88%" | ''G''''i''‹''u'' + d''u'', ''v'' + d''v''›

+

|}

+

|}

+

+

===Formula Display 15===

+

<pre>

+

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

+

| |

+

| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |

+

| |

+

| = G_i <u, v> + G_i |

+

| |

+

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

+

</pre>

+

+

{| 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''''i''

+

| width="4%" | =

+

| width="20%" | ''G''''i''‹''u'', ''v''›

+

| width="4%" | +

+

| width="64%" | E''G''''i''‹''u'', ''v'', d''u'', d''v''›

+

|-

+

| width="8%" |  

+

| width="4%" | =

+

| width="20%" | ''G''''i''‹''u'', ''v''›

+

| width="4%" | +

+

| width="64%" | ''G''''i''‹''u'' + d''u'', ''v'' + d''v''›

+

|}

+

|}

+

+

===Formula Display 16===

+

<pre>

+

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

+

| |

+

| Ef = ((u + du)(v + dv)) |

+

| |

+

| Eg = ((u + du, v + dv)) |

+

| |

+

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

+

</pre>

+

+

{| 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''f''

+

| width="4%" | =

+

| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))

+

|-

+

| width="8%" | E''g''

+

| width="4%" | =

+

| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))

+

|}

+

|}

+

+

===Formula Display 17===

+

<pre>

+

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

+

| |

+

| Df = ((u)(v)) + ((u + du)(v + dv)) |

+

| |

+

| Dg = ((u, v)) + ((u + du, v + dv)) |

+

| |

+

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

+

</pre>

+

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"

+

|

+

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

+

| width="8%" | D''f''

+

| width="4%" | =

+

| width="20%" | ((''u'')(''v''))

+

| width="4%" | +

+

| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))

+

|-

+

| width="8%" | D''g''

+

| width="4%" | =

+

| width="20%" | ((''u'', ''v''))

+

| width="4%" | +

+

| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))

+

|}

+

|}

+

+

===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>

+

+

{| 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''

+

|}

+

|}

+

+

===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>

+

+

{| 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

+

|}

+

|}

+

+

===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

+

+

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''’

+

| ''v''’

+

|}

+

|-

+

| 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%"

+

| d2''f''

+

| d2''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

+

|}

+

|}

+

|}

+

+

===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 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''

+

| d2''f''

+

| d2''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%"

+

| ( )

+

|-

+

| ( )

+

|-

+

| ( )

+

|-

+

| ( )

+

|}

+

|}

+

+

===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>

+

+

{| 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'')

+

|-

+

|  

+

|}

+

|}

+

+

===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>

+

+

[[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]

+

<center>'''Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›'''</center>

+

===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>

+

+

{| 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'')

+

|-

+

|  

+

|}

+

|}

+

+

===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>

+

+

[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]

+

<center>'''Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›'''</center>

+

===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 |

+

+

| /////\ /////\ | | /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

+

= 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 |

+

+

| / \ /////\ | | /\\\\\ /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

+

= 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 |

+

+

| /////\ / \ | | /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

+

= 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\\\\\| |

+

| | 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

−

= 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

+

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

−

| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|

+

| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|

−

+

−

| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|

+

| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|

−

| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|

+

| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|

−

| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|

+

| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|

−

| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|

+

| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|

−

| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|

+

| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|

−

| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|

+

| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|

−

| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|

+

| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|

−

| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|

+

| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|

−

| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|

+

| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|

−

| \//////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 v' =

−

= | U' | =

+

= | U' | =

−

= | o--o o--o | =

+

= | o--o o--o | =

−

= | /////\ /\\\\\ | =

+

= | /////\ /\\\\\ | =

−

= | ///////o\\\\\\\ | =

+

= | ///////o\\\\\\\ | =

−

= | ////////X\\\\\\\\ | =

+

= | ////////X\\\\\\\\ | =

−

= | o///////XXX\\\\\\\o | =

+

= | o///////XXX\\\\\\\o | =

−

= | |/////oXXXXXo\\\\\| | =

+

= | |/////oXXXXXo\\\\\| | =

−

= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =

+

= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =

−

| |/////oXXXXXo\\\\\| |

+

| |/////oXXXXXo\\\\\| |

−

| o//////\XXX/\\\\\\o |

+

| o//////\XXX/\\\\\\o |

−

| \//////\X/\\\\\\/ |

+

| \//////\X/\\\\\\/ |

−

| \//////o\\\\\\/ |

+

| \//////o\\\\\\/ |

−

| \///// \\\\\/ |

+

| \///// \\\\\/ |

−

| o--o o--o |

+

| o--o o--o |

−

| |

+

| |

−

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

+

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

+

Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>

+

</pre>

−

Figure 70-b. Tangent Functor Ferris Wheel for ~~F<u~~, v> = <((u)(v)), ((u, v))>

+

[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›''']]

−

</~~pre~~>

Jon Awbrey

12,089

edits

Changes

User:Jon Awbrey/ATLAS (view source)

Revision as of 15:00, 25 August 2007