Changes

59,602 bytes added , 15:00, 25 August 2007

add image stub

Line 1,141: Line 1,141:

Figure 12. The Anchor

</pre>

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

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

===Figure 13. The Tiller===

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Figure 13. The Tiller

</pre>

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

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

===Table 14. Differential Propositions===

Line 1,667: Line 1,675:

|}

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

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

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

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

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

</pre>

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

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<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>

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

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<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>

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

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<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>

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

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<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>

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

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<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>

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

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<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>

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

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<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>

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

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

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

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

</pre>

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

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

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

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

</pre>

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

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

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

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

</pre>

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

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

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

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Figure 21. Thematization of Disjunction and Equality

</pre>

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

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<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>

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

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

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

===Formula Display 3===

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Figure 31. Operator Diagram (1)

</pre>

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

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

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

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

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Figure 32. Operator Diagram (2)

</pre>

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

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

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

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

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Figure 33-i. Analytic Diagram (1)

</pre>

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

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

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<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>

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

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

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

===Formula Display 4===

Line 4,012: Line 4,104:

Figure 34. Tangent Functor Diagram

</pre>

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

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

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

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

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Figure 35. Conjunction as Transformation

</pre>

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

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<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)===

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

<pre>

Line 4,183: Line 4,285:

</pre>

−

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

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

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

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

<pre>

Line 4,252: Line 4,358:

</pre>

−

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

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

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

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

<pre>

Line 4,292: Line 4,402:

</pre>

−

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

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

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

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

<pre>

Line 4,333: Line 4,447:

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

</pre>

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

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<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)===

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

<pre>

Line 4,547: Line 4,665:

</pre>

−

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

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

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

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

<pre>

Line 4,616: Line 4,738:

</pre>

−

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

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

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

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

<pre>

Line 4,656: Line 4,782:

</pre>

−

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

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

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

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

<pre>

Line 4,697: Line 4,827:

Figure 40-d. Enlargement of J (Digraph)

</pre>

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

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<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)===

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

<pre>

Line 5,007: Line 5,141:

</pre>

−

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

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

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

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

<pre>

Line 5,076: Line 5,214:

</pre>

−

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

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

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

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

<pre>

Line 5,117: Line 5,259:

</pre>

−

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

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

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

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

<pre>

Line 5,155: Line 5,301:

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

</pre>

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

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<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)===

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

<pre>

Line 5,236: Line 5,386:

</pre>

−

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

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

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

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

<pre>

Line 5,305: Line 5,459:

</pre>

−

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

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

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

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

<pre>

Line 5,342: Line 5,500:

</pre>

−

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

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

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

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

<pre>

Line 5,378: Line 5,540:

Figure 46-d. Differential of J (Digraph)

</pre>

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

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<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)===

+

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

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

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

<pre>

Line 5,551: Line 5,721:

</pre>

−

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

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

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

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

<pre>

Line 5,591: Line 5,765:

</pre>

−

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

+

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

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

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

<pre>

Line 5,627: Line 5,805:

Figure 48-d. Remainder of J (Digraph)

</pre>

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

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

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

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

</pre>

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

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<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>

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

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

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

Line 6,386: Line 6,576:

| Transformation, or Mapping

| ['''B'''2] → ['''B'''1]

−

|-

| valign="top" |

Line 6,597: Line 6,786:

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

−

|+ '''Table 54. ~~Cast~~ of ~~Characters~~: ~~Expansive~~ Subtypes ~~of Objects and Operators~~'''

+

|+ '''Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes'''

|- style="background:paleturquoise"

−

! ~~Item~~

+

!  

−

! ~~Notation~~

+

! Operator

−

! ~~Description~~

+

! Proposition

−

! ~~Type~~

+

! Map

|-

−

~~| valign="top"~~ |

+

|

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

−

| W

+

| Tacit

+

|-

+

| Extension

|}

−

~~| valign="top"~~ |

+

|

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

−

| ~~W ~~:

+

| <math>\epsilon</math> :

|-

−

| ''U'' • &~~rarr~~; E''U'' • ,

+

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

|-

−

| ''X'' • → E''X'' •~~&nbsp~~;,

+

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

+

|}

+

|

+

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

+

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

|-

−

| (''U''~~~~ &~~bull~~;~~~~ → ''X''~~ •)~~

+

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

|-

−

| →

+

| '''B'''2 × '''D'''2 → '''B'''

+

|}

+

|

+

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

+

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

+

|-

+

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

|-

−

| (E''U'' &~~bull~~; → E''X''~~ •~~~~) ,~~

+

| ['''B'''2 × '''D'''2] → ['''B'''1]

+

|}

|-

−

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

+

|

+

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

+

| Trope

|-

−

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

+

| Extension

|}

−

~~| valign="top"~~ |

+

|

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

−

| ~~Operator~~

+

| <math>\eta</math> :

−

|}

−

~~| valign="top" |~~

−

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

−

~~|  ~~

|-

−

| [''~~'B'~~''2] → ['''B'''2 &~~times~~; ''~~'D'~~''2] ,

+

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

|-

−

| [''~~'B'~~''1] → [''~~'B'~~''1 &~~times~~; ''~~'D'~~''1]&~~nbsp~~;,

+

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

−

|-

+

|}

−

~~| ([''~~'B'''2~~] &rarr~~; ~~['''B'''~~<~~sup~~>1</~~sup~~>])

+

|

+

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

+

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

|-

−

| →

+

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

|-

−

| (['''B'''2 × '''D'''2] → ['''B'''<~~sup~~>1</~~sup~~> ~~×~~ ''~~'D'~~''~~1])~~

+

| '''B'''2 × '''D'''2 → '''D'''

+

|}

+

|

+

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

+

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

|-

−

|  

+

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

|-

−

|  

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

|}

|-

|

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

−

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

+

| Enlargement

|-

−

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

+

| Operator

+

|}

+

|

+

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

+

| E :

|-

−

| E

+

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

|-

−

| D

+

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

−

|-

−

| d

|}

−

~~| valign="top" |  ~~

+

|

−

~~| colspan="2"~~ |

+

{| 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:60%"

+

| E''J'' :

−

| ~~Tacit Extension Operator || <math>\epsilon</math>~~

|-

−

| ~~Trope Extension Operator || <math>\eta</math>~~

+

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

|-

−

| ~~Enlargement Operator~~ || E

+

| '''B'''2 × '''D'''2 → '''D'''

+

|}

+

|

+

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

+

| E''J'' :

|-

−

| ~~Difference Operator || D~~

+

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

|-

−

| ~~Differential Operator || d~~

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

|}

|-

−

~~| valign="top"~~ |

+

|

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

−

| ~~'''W'''~~

+

| Difference

+

|-

+

| Operator

|}

−

~~| valign="top"~~ |

+

|

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

−

| ~~'''W''' ~~:

+

| D :

|-

−

| ''U'' •~~ ~~→~~ ~~''~~'T'~~''</~~font~~>''U'' •&~~nbsp;=&nbsp~~;E''U'' • ,

+

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

|-

−

| ''X'' •~~ ~~→~~ ~~''~~'T'~~''</~~font~~>''X'' •&~~nbsp;=&nbsp~~;E''X'' •~~&nbsp~~;,

+

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

+

|}

+

|

+

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

+

| D''J'' :

|-

−

| (''U''~~~~ &~~bull~~;~~~~ → ''X''~~ •)~~

+

| 〈''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%"

+

| D''J'' :

|-

−

| ~~(~~'''T'''~~~~''~~U'' • →~~ ~~~~'''T'''~~~~''~~X''~~ &~~bull~~;~~)~~ ,

+

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

|-

−

| ~~for each~~ <~~font face=georgia~~>'''W'''</~~font~~> ~~in the set:~~

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

+

|}

|-

−

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

+

|

−

|}

−

~~| valign="top"~~ |

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

+

| Differential

+

|-

| Operator

|}

−

~~| valign="top"~~ |

+

|

−

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

+

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

−

| ~~ ~~

+

| d :

|-

−

| [''~~'B'~~''2] → ['''B'''2 &~~times~~; ''~~'D'~~''2] ,

+

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

|-

−

| ['''B'''1] → ['''B'''1 &~~times~~; ''~~'D'~~''1~~]&nbsp~~;,

+

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

+

|}

+

|

+

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

+

| d''J'' :

|-

−

| (['''B'''~~2]~~ → ['''B'''~~1])~~

+

| 〈''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%"

+

| d''J'' :

|-

−

| (['''B'''~~2~~ &~~times~~; '''D'''~~2~~] → ~~['''B'''1~~ &~~times~~; ''~~'D'~~''~~1~~])

+

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

|-

−

|  

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

−

|-

−

|  

|}

|-

|

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

−

| ~~'''e'''~~

+

| Remainder

|-

−

| ~~'''E'''~~

+

| Operator

+

|}

+

|

+

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

+

| r :

|-

−

| <~~font face=georgia~~>'''D'''</~~font~~>

+

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

|-

−

| <~~font face=georgia~~>'''T'''</~~font~~>

+

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

|}

−

~~| valign="top" |  ~~

+

|

−

~~| colspan="2"~~ |

+

{| 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:60%"

+

| r''J'' :

−

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

|-

−

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

+

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

|-

−

| ~~Chord Operator || ~~'''D'''</~~font~~> =&~~nbsp~~;~~‹<math>\epsilon</math>,~~ D›

+

| '''B'''2 × '''D'''2 → '''D'''

−

|-

−

~~| Tangent Functor || ~~'''T'''</~~font~~> =&~~nbsp~~;~~‹<math>\epsilon</math>,~~ d›

|}

−

|~~} ~~

+

|

−

+

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

−

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

+

| r''J'' :

−

+

|-

−

~~<pre>~~

+

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

−

o

+

|-

−

~~/X\~~

+

| ['''B'''2 × '''D'''2] → ['''D'''1]

−

~~/XXX\~~

+

|}

−

~~oXXXXXo~~

+

|-

−

/X\~~XXX~~/X\

+

|

−

/~~XXX\X/XXX\~~

+

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

−

~~oXXXXXoXXXXXo~~

+

| Radius

−

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

+

|-

−

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

+

| Operator

−

~~o oXXXXXo o~~

+

|}

−

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

+

|

−

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

+

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

−

~~o o o o o~~

+

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

−

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

+

|-

−

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

+

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

−

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

+

|-

−

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

+

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

−

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

+

|}

−

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

+

|

−

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

+

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

−

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

+

{| 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 o o o~~

+

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

−

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

+

|-

−

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

+

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

−

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

+

|}

−

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

+

|-

−

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

+

|

−

~~o--+--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%"

−

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

+

| Secant

−

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

+

|-

−

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

+

| Operator

−

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

+

|}

−

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

+

|

−

~~. o oXXXXXXXXXXXo o .~~

+

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

−

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

+

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

−

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

+

|-

−

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

+

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

−

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

+

|-

−

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

+

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

−

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

+

|}

−

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

+

|

−

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

+

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

−

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

+

|  

−

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

+

|-

−

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

+

|  

−

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

+

|-

−

\ /

+

|  

−

~~\ /~~

+

|}

−

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

+

|

−

\ /

+

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

−

~~\ /~~

+

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

−

o

+

|-

+

| [''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-a1. Radius Map of the Conjunction J = uv~~

+

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

−

~~</pre>~~

−

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

<pre>

Line 6,818: Line 7,111:

/XXX\

oXXXXXo

−

//\XXX//\

+

/X\XXX/X\

−

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

+

/XXX\X/XXX\

−

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

+

oXXXXXoXXXXXo

−

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

+

/ \XXX/X\XXX/ \

−

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

+

/ \X/XXX\X/ \

−

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

+

o oXXXXXo o

−

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

+

/ \ / \XXX/ \ / \

−

/ \\/ \// \\/ \

+

/ \ / \X/ \ / \

o o o o o

−

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

+

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

−

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

+

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

−

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

+

= | o o o o | =

−

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

+

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

−

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

+

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

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

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

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

o o o o o o o o o o

−

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

+

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

−

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

+

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

−

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

+

| o o o o | | o o o o |

−

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

+

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

−

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

+

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

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

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

Line 6,859: Line 7,152:

. //\XXXXXXXXX/\\ .

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

−

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

+

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

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

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

Line 6,867: Line 7,160:

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

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

−

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

+

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

−

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

+

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

\ /

−

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

+

\ /

−

x = uv \ /

+

x = uv \ / dx = uv

−

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

+

\ /

o

−

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

+

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

</pre>

−

===Figure 56-a3. ~~Chord~~ 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

−

//\

+

/X\

−

////\

+

/XXX\

−

o/////o

+

oXXXXXo

−

/X\////X\

+

//\XXX//\

−

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

+

////\X////\

−

~~oXXXXXoXXXXXo~~

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

/ \\/ \X/ \\/ \

+

/ \\/ \// \\/ \

o o o o o

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

Line 6,901: Line 7,198:

= |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

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

Line 6,927: Line 7,224:

. //\XXXXXXXXX/\\ .

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

−

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

+

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

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

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

Line 6,935: Line 7,232:

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

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

−

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

+

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

−

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

+

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

\ /

−

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

+

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

x = uv \ /

−

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

+

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

\ /

o

−

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

+

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

</pre>

−

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

+

+

[[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-a3. Chord Map of the Conjunction J = uv===

<pre>

Line 6,957: Line 7,258:

/XXX\//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 | =

+

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

−

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

+

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

−

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

+

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

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

//\ | \ / \ / | / \

Line 6,975: Line 7,276:

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

o o o o o o o~~\\\\\~~o~~\\\\\~~o o

+

o o o o o o o o o o

−

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

+

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

−

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

+

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

−

| o o o o | | o o o o |

+

| o o o o | | o o\\\\\o o |

−

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

+

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

−

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

+

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

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

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

Line 6,995: Line 7,296:

. //\XXXXXXXXX/\\ .

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

−

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

+

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

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

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

Line 7,006: Line 7,307:

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

\ /

−

\ /

+

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

−

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

+

x = uv \ /

−

~~\ /~~

+

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

\ /

o

−

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

+

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

</pre>

−

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

+

+

[[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>

+

===Figure 56-a4. Tangent 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//\XXX/\\

−

~~| o~~ / \ ~~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 |

+

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

−

| |

+

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

+

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

−

+

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

−

| 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``|~~

+

\ /

−

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

+

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

−

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

+

\ /

−

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

+

\ /

−

+

o

−

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

−

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

−

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

−

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

+

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

</pre>

−

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

+

+

[[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>

Line 7,125: 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 7,145: 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 7,167: Line 7,444:

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

−

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

+

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

−

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

+

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

−

+

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

| | 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 7,184: Line 7,461:

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

\ / \ / \ /

−

\ EJ / \ J / \ EJ /

+

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

\ / \ / \ /

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

Line 7,212: Line 7,489:

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

−

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

+

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

</pre>

−

===Figure 56-b3. ~~Chord~~ 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 7,263: 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 7,312: Line 7,593:

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

−

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

+

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

</pre>

−

===Figure 56-b4. ~~Tangent~~ Map of 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>

Line 7,325: Line 7,610:

| / \ / \ |

| / o \ |

−

| / du / \ dv \ |

+

| / du /`\ dv \ |

−

| o / \ o |

+

| o /```\ o |

−

| | o o | |

+

| | o`````o | |

−

| | | | | |

+

| | |`````| | |

−

| | o o | |

+

| | o`````o | |

−

| o \ / o |

+

| o \```/ o |

−

| \ \ / / |

+

| \ \`/ / |

| \ o / |

| \ / \ / |

Line 7,347: Line 7,632:

| /````\ / \ | \

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

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

Line 7,369: Line 7,654:

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

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

−

+

−

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

+

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

−

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

+

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

−

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

+

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

−

| | o~~`````~~o`````| | | | o`````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 7,384: Line 7,669:

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

\ / \ / \ /

−

\ dJ / \ J / \ dJ /

+

\ DJ / \ J / \ DJ /

\ / \ / \ /

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

Line 7,412: Line 7,697:

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

−

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

+

Figure 56-b3. Chord 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-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

−

//\ /X\

+

| |

−

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

+

| |

−

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

+

| |

−

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

+

| o--o o--o |

−

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

+

| / \ / \ |

−

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

+

| / o \ |

−

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

+

| / du / \ dv \ |

−

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

+

| o / \ o |

−

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

+

| | o o | |

−

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

+

| | | | | |

−

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

+

| | o o | |

−

o o o o o o o o

+

| o \ / o |

−

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

+

| \ \ / / |

−

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

+

| \ o / |

−

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

+

| \ / \ / |

−

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

+

| o--o o--o |

−

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

+

| |

−

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

+

| |

−

\ / | \ / \ / |

+

| |

−

\ / | du \ / \ / dv |

+

o-----------------------@

−

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

+

\

−

\ / \ /

+

o-----------------------o \

−

\ / \ /

+

| | \

−

o o

+

| | \

−

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

+

| | \

−

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

+

| o--o o--o | \

−

| |

+

| /````\ / \ | \

−

| |

+

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

−

| |

+

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

−

| |

+

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

−

J | | ~~$e$J~~

+

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

−

| |

+

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

−

| |

+

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

−

| |

+

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

−

~~v v~~

+

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

−

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

+

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

−

~~X% $e$ $E$X%~~

+

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

−

o o

+

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

−

//\ /X\

+

| | \ \

−

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

+

| | \ \

−

~~/////~~/\ /~~XXXXX~~\

+

| | \ \

−

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

+

o-----------------------o \ \

−

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

+

\ \

−

////~~////////~~o ~~oXXXXXXXXXXXo~~

+

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

−

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

+

| |\ | \ \ | | |

−

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

+

| | \ | \ @ | | |

−

//~~///////////~~ \ ~~//////~~\~~XXXXX~~/\~~\\\\\~~

+

| | \| \ | | |

−

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

+

−

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

+

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

−

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

+

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

−

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

+

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

\ / \ /

+

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

−

~~\ / \ /~~

+

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

−

~~\ / \ /~~

+

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

−

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

+

−

~~\ / \ /~~

+

| | | | | |

−

o o

+

| | | | | |

−

+

| | | | | |

−

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

+

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

−

</pre>

+

\ / \ / \ /

+

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

+

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

<pre>

Line 7,492: Line 7,815:

////\ /XXX\

//////\ oXXXXXo

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

o o o o o o o o

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

\ / | \ / \ / |

Line 7,513: Line 7,836:

\ / \ /

o o

−

U% $E$ $E$U%

+

U% $e$ $E$U%

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

| |

Line 7,519: Line 7,842:

| |

−

J | | $E$J

+

J | | $e$J

| |

Line 7,525: Line 7,848:

v v

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

−

X% $E$ $E$X%

+

X% $e$ $E$X%

o o

//\ /X\

Line 7,552: Line 7,875:

o o

−

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

+

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

</pre>

−

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

+

+

[[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

−

//\ //\

+

//\ /X\

−

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

+

////\ /XXX\

−

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

+

//////\ oXXXXXo

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

o o o o o o o o

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

Line 7,583: Line 7,910:

\ / \ /

o o

−

U% $D$ $E$U%

+

U% $E$ $E$U%

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

| |

Line 7,589: Line 7,916:

| |

−

J | | $D$J

+

J | | $E$J

| |

Line 7,595: Line 7,922:

v v

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

−

X% $D$ $E$X%

+

X% $E$ $E$X%

o o

//\ /X\

Line 7,622: Line 7,949:

o o

−

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

+

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

</pre>

−

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

+

+

[[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>

+

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

<pre>

Line 7,635: Line 7,966:

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

+

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

−

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

+

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

−

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

+

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

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

\ / | \ / \ / |

Line 7,653: Line 7,984:

\ / \ /

o o

−

U% $T$ $E$U%

+

U% $D$ $E$U%

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

| |

Line 7,659: Line 7,990:

| |

−

J | | $T$J

+

J | | $D$J

| |

Line 7,665: Line 7,996:

v v

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

−

X% $T$ $E$X%

+

X% $D$ $E$X%

o o

//\ /X\

Line 7,692: Line 8,023:

o o

−

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

+

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

</pre>

−

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

+

+

[[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>

−

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

+

o o

−

| |

+

//\ //\

−

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

+

////\ ////\

−

| |

+

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

−

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

+

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

−

| |

+

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

−

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

+

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

−

| |

+

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

−

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

+

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

−

</pre>

+

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

−

+

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

−

+

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

−

{| 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, ''F''2›~~

+

\ / | \ / \ / |

−

~~| :~~

+

\ / | du \ / \ / dv |

−

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

+

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

−

~~| →~~

+

\ / \ /

−

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

+

\ / \ /

−

|-

+

o o

−

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

+

U% $T$ $E$U%

−

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

+

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

−

~~| =~~

+

| |

−

~~| align="center" | ''F''1~~

+

| |

−

~~| :~~

+

| |

−

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

+

| |

−

~~| →~~

+

J | | $T$J

−

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

+

| |

−

|-

+

| |

−

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

+

| |

−

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

+

v v

−

~~| =~~

+

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

−

~~| align="center" | ''F''2~~

+

X% $T$ $E$X%

−

~~| :~~

+

o o

−

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

+

//\ /X\

−

~~| →~~

+

////\ /XXX\

−

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

+

//////\ /XXXXX\

−

|}

+

////////\ /XXXXXXX\

−

|}

+

//////////\ /XXXXXXXXX\

−

~~ ~~

+

////////////o oXXXXXXXXXXXo

−

+

///////////// \ //\XXXXXXXXX/\\

−

~~ ~~

+

///////////// \ ////\XXXXXXX/\\\\

−

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

+

///////////// \ //////\XXXXX/\\\\\\

+

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

+

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

+

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

+

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

+

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

+

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

+

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

+

| x \// / | x \// \\/ dx |

+

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

+

\ / \ /

+

\ / \ /

+

\ / \ /

+

\ / \ /

+

\ / \ /

+

o o

+

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

+

</pre>

+

+

[[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

+

| |

+

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

+

| |

+

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

+

| |

+

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

+

| |

+

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

+

</pre>

+

+

{| 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%"

−

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

+

| ''F''

| =

−

| ‹''f'', ''g''›

+

| align="center" | ‹''f'', ''g''›

| =

−

| ‹''F''1, ''F''2›

+

| align="center" | ‹''F''1, ''F''2›

| :

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

Line 7,758: Line 8,132:

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

|-

−

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

+

| colspan="2" | where

−

| ''f''

+

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

| =

−

| ''F''1

+

| align="center" | ''F''1

| :

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

Line 7,767: Line 8,141:

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

|-

−

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

+

| colspan="2" | and

−

| ''g''

+

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

+

| =

+

| align="center" | ''F''2

+

| :

+

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

+

| →

+

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

+

|}

+

|}

+

+

+

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

+

|

+

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

+

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

+

| =

+

| ‹''f'', ''g''›

+

| =

+

| ‹''F''1, ''F''2›

+

| :

+

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

+

| →

+

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

+

|-

+

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

+

| ''f''

+

| =

+

| ''F''1

+

| :

+

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

+

| →

+

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

+

|-

+

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

+

| ''g''

| =

| ''F''2

Line 7,853: Line 8,262:

</pre>

−

===~~Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes=~~==

+

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

−

+

|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''

−

~~<pre>~~

+

|- style="background:paleturquoise"

−

Table 59. ~~Synopsis~~ of ~~Terminology~~: ~~Restrictive~~ and ~~Alternative Subtypes~~

+

! Item

−

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

+

! Notation

−

+

! Description

−

| ~~| or | or | or |~~

+

! Type

−

+

|-

−

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

+

| valign="top" | ''U'' •

−

| ~~| | |~~ |

+

| valign="top" | = [''u'', ''v'']

−

| ~~Operand~~ | ~~F =~~ <~~F_1, F_2~~> ~~| F_i :~~ <~~|u,v|~~> -> ~~B | F : [u, v] -~~> ~~[x, y~~] |

+

| valign="top" | Source Universe

−

| ~~| | | |~~

+

| valign="top" | ['''B'''''n'']

−

+

|-

−

| ~~| | | |~~

+

| valign="top" | ''X'' •

−

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

+

| valign="top" |

−

| | | | |

+

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

−

| ~~Tacit | !e!~~ : ~~| !e!F_i : | !e!F : |~~

+

| = [''x'', ''y'']

−

~~| Extension |~~ U%->~~EU%, X%-~~>~~EX%, |~~ <|u,~~v,du,dv~~|> -> B ~~| [u,v,du,dv]-~~>~~[x, y~~] |

+

|-

−

+

| = [''f'', ''g'']

−

| | | | |

+

|}

−

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

+

| valign="top" | Target Universe

−

| | | | |

+

| valign="top" | ['''B'''''k'']

−

| ~~Trope | !h!~~ : ~~| !h!F_i~~ : ~~| !h!F~~ : |

+

|-

−

+

| valign="top" | E''U'' •

−

+

| valign="top" | = [''u'', ''v'', d''u'', d''v'']

−

| | | | |

+

| valign="top" | Extended Source Universe

−

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

+

| valign="top" | ['''B'''''n'' × '''D'''''n'']

−

~~| | | |~~ |

+

|-

−

| ~~Enlargement | E : | EF_i~~ : ~~| EF~~ : |

+

| valign="top" | E''X'' •

−

| Operator ~~| U%->EU%, X%->EX%, | <~~|~~u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |~~

+

| valign="top" |

−

+

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

−

~~| | | | |~~

+

| = [''x'', ''y'', d''x'', d''y'']

−

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

+

|-

−

~~| | | | |~~

+

| = [''f'', ''g'', d''f'', d''g'']

−

~~| Difference | D : | DF_i : | DF : |~~

+

|}

−

~~| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |~~

+

| valign="top" | Extended Target Universe

−

+

| valign="top" | ['''B'''''k'' × '''D'''''k'']

−

~~| | | | |~~

+

|-

−

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

+

| ''F''

−

~~| | | | |~~

+

| ''F'' = ‹''f'', ''g''› : ''U'' • → ''X'' •

−

~~| Differential | d : | dF_i : | dF : |~~

+

| Transformation, or Mapping

−

~~| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |~~

+

| ['''B'''''n''] → ['''B'''''k'']

−

+

|-

−

~~| | | | |~~

+

| valign="top" |

−

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

+

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

−

~~| | | | |~~

+

|  

−

~~| Remainder | r : | rF_i : | rF : |~~

+

|-

−

~~| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |~~

+

| ''f''

−

+

|-

−

~~| | | | |~~

+

| ''g''

−

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

+

|}

−

~~| | | | |~~

+

| valign="top" |

−

~~| Radius | $e$ = <!e!, !h!> : | | $e$F : |~~

+

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

−

~~| Operator | | | |~~

+

| ''f'', ''g'' : ''U'' → '''B'''

−

~~| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |~~

+

|-

−

~~| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |~~

+

| ''f'' : ''U'' → [''x''] &sube; ''X'' •

−

~~| | | | |~~

+

|-

−

~~| | | | [B^n x D^n] -> |~~

+

| ''g'' : ''U'' → [''y''] &sube; ''X'' •

−

~~| | | | [B^k x D^k] |~~

+

|}

−

~~| | | | |~~

+

| valign="top" |

−

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

+

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

−

~~| | | | |~~

+

| Proposition

−

~~| Secant | $E$ = <!e!, E> : | | $E$F : |~~

+

|}

−

~~| Operator | | | |~~

+

| valign="top" |

−

~~| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |~~

+

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

−

~~| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |~~

+

| '''B'''''n'' → '''B'''

−

~~| | | | |~~

+

|-

−

~~| | | | [B^n x D^n] -> |~~

+

| ∈ ('''B'''''n'', '''B'''''n'' → '''B''')

−

~~| | | | [B^k x D^k] |~~

+

|-

−

~~| | | | |~~

+

| = ('''B'''''n'' +→ '''B''') = ['''B'''''n'']

−

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

+

|}

−

~~| | | | |~~

+

|-

−

~~| Chord | $D$ = <!e!, D> : | | $D$F : |~~

+

| valign="top" |

−

~~| Operator | | | |~~

+

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

−

~~| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |~~

+

| W

−

~~| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |~~

+

|}

−

~~| | | | |~~

+

| valign="top" |

−

~~| | | | [B^n x D^n] -> |~~

+

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

−

~~| | | | [B^k x D^k] |~~

+

| W :

−

~~| | | | |~~

+

|-

−

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

+

| ''U'' • → E''U'' • ,

−

~~| | | | |~~

+

|-

−

~~| Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |~~

+

| ''X'' • → E''X'' • ,

−

~~| Functor | | | |~~

+

|-

−

~~| | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |~~

+

| (''U'' • → ''X'' •)

−

~~| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |~~

+

|-

−

~~| | | | |~~

+

| →

−

~~| | | B^n x D^n -> D | [B^n x D^n] -> |~~

+

|-

−

~~| | | | [B^k x D^k] |~~

+

| (E''U'' • → E''X'' •) ,

−

~~| | | | |~~

+

|-

−

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

+

| for each W in the set:

−

~~</pre>~~

+

|-

−

+

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

−

~~===Formula Display 12===~~

+

|}

−

+

| valign="top" |

−

~~<pre>~~

+

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

−

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

+

| Operator

−

~~| |~~

+

|}

−

~~| x = f(u, v) = ((u)(v)) |~~

+

| valign="top" |

−

~~| |~~

+

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

−

~~| y = g(u, v) = ((u, v)) |~~

−

~~| |~~

−

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

−

~~</pre>~~

−

~~ ~~

−

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

−

|

−

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

|  

−

| ''x''

+

|-

−

| =

+

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

−

| ''f''‹''u'', ''v''›

+

|-

−

| =

+

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

−

| ((''u'')(''v''))

+

|-

+

| (['''B'''''n''] → ['''B'''''k''])

+

|-

+

| →

+

|-

+

| (['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k''])

+

|-

|  

|-

|  

−

~~| ''y''~~

−

~~| =~~

−

~~| ''g''‹''u'', ''v''›~~

−

~~| =~~

−

~~| ((''u'', ''v''))~~

−

~~|  ~~

|}

+

|-

+

|

+

{| 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" |

−

~~===Formula Display 13===~~

+

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

−

+

| '''W'''

−

~~<pre>~~

−

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

−

~~| |~~

−

~~| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |~~

−

| |

−

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

−

~~</pre>~~

−

~~ ~~

−

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

−

|

−

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

−

| ~~‹''x'', ''y''›~~

−

| =

−

| ''F''‹''~~u'', ''v''›~~

−

~~| =~~

−

~~| ‹((''u'')(''v'')), ((''u'', ''v''))›~~

|}

−

|}

+

| valign="top" |

−

+

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

−

+

| '''W''' :

−

+

|-

−

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

+

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

−

|

+

|-

−

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

+

| ''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''])

+

|-

|  

−

| ~~‹''x'', ''y''›~~

+

|-

−

~~| =~~

−

~~| ''F''‹''u'', ''v''›~~

−

~~| =~~

−

~~| ‹((''u'')(''v'')), ((''u'', ''v''))›~~

|  

|}

+

|-

+

|

+

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

+

| '''e'''

+

|-

+

| '''E'''

+

|-

+

| '''D'''

+

|-

+

| '''T'''

|}

−

~~ ~~

+

| valign="top" |  

−

+

| colspan="2" |

−

=~~==Table 60. Propositional Transformation===~~

+

{| 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>›

−

~~<pre>~~

+

|-

−

~~Table 60. Propositional Transformation~~

+

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

−

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

+

|-

−

~~| u | v | f | g~~ |

+

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

−

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

+

|-

−

| ~~| | |~~ |

+

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

−

| 0 | 0 ~~| 0 | 1 |~~

+

|}

−

| | ~~| |~~ |

+

|}

−

| ~~0 | 1 | 1 | 0 |~~

−

| | ~~| |~~ |

−

| ~~1 | 0 | 1 | 0 |~~

−

| | ~~| | |~~

−

~~| 1 | 1 | 1 | 1~~ |

−

| ~~| | | |~~

−

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

−

| | | ~~((u)(v)) | ((u~~, ~~v))~~ |

−

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

−

<~~/pre~~>

−

===~~Figure 61~~. ~~Propositional Transformation~~===

+

===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----------------------o

−

~~| |~~

+

| | | | |

−

~~| |~~

+

| Trope | !h! : | !h!F_i : | !h!F : |

−

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

+

−

~~/ \ / \~~

+

−

~~/ \ / \~~

+

| | | | |

−

~~/ \ / \~~

+

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

−

~~/ \ / \~~

+

| | | | |

−

~~/ \ / \~~

+

| Enlargement | E : | EF_i : | EF : |

−

/ ~~\ / \~~

+

−

~~/ \ / \~~

+

−

/ ~~\ / \~~

+

| | | | |

−

~~/ \ / \~~

+

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

−

~~/ \ / \~~

+

| | | | |

−

~~/ \ / \~~

+

| Difference | D : | DF_i : | DF : |

−

~~/ \ / \~~

+

−

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

+

−

~~| U | |\U \\\\\\\\\\\\\\\\\\\\\\|~~

+

| | | | |

−

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

+

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

−

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

+

| | | | |

−

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

+

| Differential | d : | dF_i : | dF : |

−

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

+

−

| ~~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 /\\\\|~~

+

| Remainder | r : | rF_i : | rF : |

−

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

+

−

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

+

−

~~| | |\\\\\\\\\\\\\\\\\\\\\\\\\|~~

+

| | | | |

−

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

+

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

−

\ | | /

+

| | | | |

−

\ | | /

+

| Radius | $e$ = <!e!, !h!> : | | $e$F : |

−

\ | | /

+

−

~~\ f~~ | | ~~g /~~

+

−

\ | | /

+

−

\ | | /

+

| | | | |

−

\ | | /

+

−

\ | | /

+

−

\ | | /

+

| | | | |

−

\ | | /

+

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

−

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

+

| | | | |

−

~~| X \ | | / |~~

+

| Secant | $E$ = <!e!, E> : | | $E$F : |

−

~~| \| |/ |~~

+

−

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

+

−

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

+

−

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

+

| | | | |

−

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

+

−

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

+

−

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

+

| | | | |

−

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

+

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

−

| |~~//////~~ x ~~//////~~|~~XXXXX|\\\\\\ y \\\\\\~~| |

+

| | | | |

−

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

+

| Chord | $D$ = <!e!, D> : | | $D$F : |

−

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

+

−

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

+

−

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

+

−

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

+

| | | | |

−

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

+

−

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

+

−

~~| |~~

+

| | | | |

−

~~| |~~

+

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

−

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

+

| | | | |

−

~~Figure 61. Propositional Transformation~~

+

| Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |

+

+

+

+

| | | | |

+

+

+

| | | | |

+

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

</pre>

−

===~~Figure 62. Propositional Transformation (Short Form)~~===

+

{| 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"

−

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

+

|  

−

| U | |~~\U \\\\\\\\\\\\\\\\\\\\\\~~|

+

| align="center" | '''Operator or Operand'''

−

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

+

| align="center" | '''Proposition or Component'''

−

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

+

| align="center" | '''Transformation or Mapping'''

−

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

+

|-

−

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

+

| Operand

−

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

+

| valign="top" |

−

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

+

{| 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'' = ‹''F''1, ''F''2›

−

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

+

|-

−

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

+

| ''F'' = ‹''f'', ''g''› : ''U'' → ''X''

−

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

+

|}

−

| 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~~

+

| ''F''''i'' : 〈''u'', ''v''〉 → '''B'''

−

\ / \ /

+

|-

−

~~\ / \ /~~

+

| ''F''''i'' : '''B'''''n'' → '''B'''

−

\ / \ /

+

|}

−

~~\ f / \ g /~~

+

| valign="top" |

−

\ / \ /

+

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

−

~~\ / \ /~~

+

| ''F'' : [''u'', ''v''] → [''x'', ''y'']

−

~~\ / \ /~~

+

|-

−

~~\ / \ /~~

+

| ''F'' : '''B'''''n'' → '''B'''''k''

−

\ ~~/ \~~ /

+

|}

−

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

+

|-

−

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

+

|

−

~~| \ / \ /~~ |

+

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

−

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

+

| Tacit

−

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

+

|-

−

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

+

| Extension

−

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

+

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

−

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

+

|-

−

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

+

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

−

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

+

|}

−

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

+

|

−

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

+

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

−

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

+

| <math>\epsilon</math>''F''''i'' :

−

| |

+

|-

−

| |

+

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

−

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

+

|-

−

~~Figure 62. Propositional Transformation (Short Form)~~

+

| '''B'''''n'' × '''D'''''n'' → '''B'''

−

</~~pre~~>

+

|}

−

+

|

−

~~===Figure 63. Transformation of Positions===~~

+

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

−

+

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

−

<~~pre~~>

+

|-

−

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

+

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

−

|`U~~` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|~~

+

|-

−

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

+

| ['''B'''''n'' × '''D'''''n''] → ['''B'''''k'']

−

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

+

|}

−

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

+

|-

−

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

+

|

−

|~~` ` ` ` `/' ' ' ' '~~ ' ' '~~/^\~~' ' ' ' ' ~~' ' '\` ` ` ` `~~|

+

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

−

|~~` ` ` ` /~~ ' ' ' ' ' ' ' ~~/^^^\~~ ' ' ' ' ' ' ' ~~\ ` ` ` `|~~

+

| Trope

−

~~|` ` ` `o~~' ' ' ' ' ' ' '~~o^^^^^o~~' ' ' ' ' ' ' '~~o` ` ` `|~~

+

|-

−

~~|` ` ` `|~~' ' ' ' ' ' ' '~~|^^^^^|' ' '~~ ' ' ' ' '~~|` ` ` `|~~

+

| Extension

−

~~|` ` ` `|~~' ' ~~' ' u~~ ' ' '|~~^^^^^|' ' ' v ' ' ' '|` ` ` `~~|

+

|}

−

|~~` ` ` `|' ' ' '~~ ' ' ' '|~~^^^^^~~|' ' ' ' ' ' ' '~~|` ` ` `|~~

+

|

−

~~|` `@` `o~~' ' ' ' @ ' ' '~~o^^@^^o~~' ' ' @ ' ' ' '~~o` ` ` `~~|

+

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

−

|~~` ` \ ` \~~ ' ' ' | ' ' ' ~~\^|^~~/ ' ' ' | ' ' ' ~~/ ` ` ` `|~~

+

| <math>\eta</math> :

−

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

+

|-

−

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

+

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

−

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

+

|-

−

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

+

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

−

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

+

|}

−

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

+

|

−

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

+

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

−

~~" " \~~ | ~~| | " "~~

+

| <math>\eta</math>''F''''i'' :

−

~~" " \~~ | ~~| |~~ " "

+

|-

−

" " ~~\ | | |~~ " "

+

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

−

" " ~~\| | |~~ " "

+

|-

−

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

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

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

+

|}

−

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

+

|

−

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

+

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

−

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

+

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

−

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

+

|-

−

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

+

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

−

| |~~'''u'''~~|~~'''~~|~~'''v'''~~| ~~| | \ | | |``| u |```| v |``~~|

+

|-

−

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

+

| ['''B'''''n'' × '''D'''''n''] → ['''D'''''k'']

−

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

+

|}

−

| \''''''~~'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%"

−

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

+

| Enlargement

−

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

+

|-

−

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

+

| Operator

−

~~\ ((u)(v))~~ | ~~| | \/ | ((u, v)) /~~

+

|}

−

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

+

|

−

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

+

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

−

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

+

| E :

−

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

+

|-

−

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

+

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

−

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

+

|-

−

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

+

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

−

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

+

|}

−

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

+

|

−

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

+

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

−

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

+

| E''F''''i'' :

−

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

+

|-

−

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

+

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

−

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

+

|-

−

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

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

| / ' ' ' |/' ' ' /~~^|^\ ` ` ` | ` ` ` \ |~~

+

|}

−

~~| @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |~~

+

|

−

~~| |~~' ' ' ' ' ' ' '|~~^^^^^|` ` ` ` ` ` ` `| |~~

+

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

−

~~| |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `|~~ |

+

| E''F'' :

−

~~| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `|~~ |

+

|-

−

| o' ' ' ' ' ' ' '~~o^^^^^o` ` ` ` ` ` ` `o~~ |

+

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

−

| \ ' ' ' ' ' ' ' ~~\^^^/ ` ` ` ` ` ` ` / |~~

+

|-

−

~~| \~~' ' ' ' ' ' ' '~~\^/` ` ` ` ` ` ` `/ |~~

+

| ['''B'''''n'' × '''D'''''n''] → ['''D'''''k'']

−

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

+

|}

−

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

+

|-

−

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

+

|

−

~~| |~~

+

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

−

~~| |~~

+

| Difference

−

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

+

|-

−

~~Figure 63. Transformation of Positions~~

+

| Operator

−

</~~pre~~>

+

|}

−

+

|

−

~~===Table 64. Transformation of Positions===~~

+

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

−

+

| D :

−

<~~pre~~>

+

|-

−

~~Table 64. Transformation of Positions~~

+

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

−

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

+

|-

−

~~| u v~~ | x | y | ~~x y~~ | ~~x(y)~~ | ~~(x)y~~ | ~~(x)(y)~~ | ~~X% = [x, y]~~ |

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~o-----o----------o----------o-------o-------o--------o--~~---~~---o-------------o~~

+

|}

−

~~| | | | | | | | ^ |~~

+

|

−

~~| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| | | | | | | | |~~

+

| D''F''''i'' :

−

~~| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |~~

+

|-

−

~~| | | | | | | | = |~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g~~> |

+

|-

−

~~| | | | | | | | |~~

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

~~| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |~~

+

|}

−

~~| | | | | | | | | |~~

+

|

−

~~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] |~~

+

| D''F'' :

−

~~o-----o----------o----------o-------o-------o--------o--------o-------------o~~

+

|-

−

~~</pre>~~

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']

−

+

|-

−

~~===Table 65. Induced Transformation on Propositions===~~

+

| ['''B'''''n'' × '''D'''''n''] → ['''D'''''k'']

−

+

|}

−

~~<pre>~~

+

|-

−

~~Table 65. Induced Transformation on Propositions~~

+

|

−

~~o------------o---------------------------------o------------o~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| X% | <--- F = <f , g> <--- | U% |~~

+

| Differential

−

~~o------------o----------o-----------o----------o------------o~~

+

|-

−

~~| | u = | 1 1 0 0 | = u | |~~

+

| Operator

−

~~| | 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> | |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| | y = | 1 0 0 1 | = g<u,v> | |~~

+

| d :

−

~~o------------o----------o-----------o----------o------------o~~

+

|-

−

~~| | | | | |~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~| f_0 | () | 0 0 0 0 | () | f_0 |~~

+

|-

−

~~| | | | | |~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |~~

+

|}

−

~~| | | | | |~~

+

|

−

~~| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| | | | | |~~

+

| d''F''''i'' :

−

~~| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |~~

+

|-

−

~~| | | | | |~~

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |~~

+

|-

−

~~| | | | | |~~

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

~~| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |~~

+

|}

−

~~| | | | | |~~

+

|

−

~~| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| | | | | |~~

+

| d''F'' :

−

~~| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |~~

+

|-

−

~~| | | | | |~~

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']

−

~~o------------o----------o-----------o----------o------------o~~

+

|-

−

~~| | | | | |~~

+

| ['''B'''''n'' × '''D'''''n''] → ['''D'''''k'']

−

~~| f_8 | x y | 1 0 0 0 | u v | f_8 |~~

+

|}

−

~~| | | | | |~~

+

|-

−

~~| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |~~

+

|

−

~~| | | | | |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |~~

+

| Remainder

−

~~| | | | | |~~

+

|-

−

~~| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |~~

+

| Operator

−

~~| | | | | |~~

+

|}

−

~~| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |~~

+

|

−

~~| | | | | |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |~~

+

| r :

−

~~| | | | | |~~

+

|-

−

~~| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |~~

+

| ''U'' • → E''U'' • ,  ''X'' • → E''X'' • ,

−

~~| | | | | |~~

+

|-

−

~~| f_15 | (()) | 1 1 1 1 | (()) | f_15 |~~

+

| (''U'' • → ''X'' •) → (E''U'' • → d''X'' •)

−

~~| | | | | |~~

+

|}

−

~~o------------o----------o-----------o----------o------------o~~

+

|

−

~~</pre>~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

+

| r''F''''i'' :

−

~~===Formula Display 14===~~

+

|-

−

+

| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''

−

~~<pre>~~

+

|-

−

~~o-------------------------------------------------o~~

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

−

~~| |~~

+

|}

−

~~| EG_i = G_i~~ <~~u + du, v + dv~~> |

+

|

−

~~| |~~

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

−

~~o-------------------------------------------------o~~

+

| r''F'' :

−

</~~pre~~>

+

|-

−

+

| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']

−

~~ ~~

+

|-

−

{| ~~align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font~~-~~weight:bold; text-align:left; width:96%"~~

+

| ['''B'''''n'' × '''D'''''n''] → ['''D'''''k'']

+

|}

+

|-

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| Radius

+

|-

+

| Operator

+

|}

+

|

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

+

| '''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'''''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="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~~>

+

| Secant

−

~~| width="4%" | =~~

+

|-

−

~~| width="88%" |~~ ''G''<~~sub~~>''i''</~~sub~~>‹''u'' ~~+ d~~''u'', ''v'' ~~+ d~~''v''›

+

| 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'' •)

|}

−

|}

−

~~ ~~

−

~~===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''~~

+

|  

−

| ~~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''›~~

|}

+

|

+

{| 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'']

|}

−

~~ ~~

+

|-

−

~~===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''~~

+

| Chord

−

~~| width="4%" | =~~

−

~~| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))~~

|-

−

| ~~width="8%" | E''g''~~

+

| Operator

−

~~| 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''' = ‹<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%"

+

|  

+

|-

+

|  

+

|-

+

|  

|}

−

~~ ~~

−

~~===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'' :

−

| ~~width="4%" | =~~

+

|-

−

| ~~width="20%" | ((~~''u'')(''v''))

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']

−

~~| width="4%" | +~~

−

~~| width="64%" | ((~~''u'' + d''u'')(''v'' + d''v''))

|-

−

| ~~width="8%" |~~ D''g''

+

| ['''B'''''n'' × '''D'''''n''] → ['''B'''''k'' × '''D'''''k'']

−

~~| 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%"

+

| 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'' •)

|}

−

~~ ~~

−

~~===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''

+

| d''F''''i'' :

−

~~| = || ''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'', ''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~~''))

|-

−

| D''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''~~

+

{| 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''))

+

| '''T'''''F'' :

|-

−

| ~~d''f''~~

+

| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']

−

~~| = || ''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''] → ['''B'''''k'' × '''D'''''k'']

−

~~| = ||~~ ''~~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))~~===

+

===Formula Display 12===

<pre>

−

~~Table 66-ii. Computation Summary for g<u, v> = ((u, v))~~

+

o-----------------------------------------------------------o

−

o~~---------------------~~-----------------------------------------------------------o

+

| |

−

| |

+

| x = f(u, v) = ((u)(v)) |

−

| ~~!e!g~~ = ~~uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |~~

+

| |

−

~~| |~~

+

| y = g(u, v) = ((u, v)) |

−

| ~~Eg = uv.((du, dv)) + u(v). (du, dv) +~~ (u~~)v. (du~~, ~~dv) + (u)(~~v)~~.((du, dv)) |~~

+

| |

−

~~| |~~

+

o-----------------------------------------------------------o

−

| 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="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

−

| ~~<math>\epsilon</math>~~''g''

+

|  

−

| = || ''uv'' ~~|| <math>\cdot</math> || 1~~

+

| ''x''

−

~~| + ||~~ ''u''(''v'') |~~| <math>\cdot</math> || 0~~

+

| =

−

~~| + |~~| (~~''u'')''v'' || <math>\cdot</math> || 0~~

+

| ''f''‹''u'', ''v''›

−

~~| + ||~~ (''u'')(''v'') |~~| <math>\cdot</math> || 1~~

+

| =

+

| ((''u'')(''v''))

+

|  

|-

−

| E''g''

+

|  

−

| = || ''uv'' ~~|| <math>\cdot</math> || ((d~~''u'', d''v''))

+

| ''y''

−

| ~~+ |~~| ''u''(''v'') || <~~math~~>~~\cdot~~<~~/math~~> || ~~(d''u'', d''v'')~~

+

| =

−

| ~~+ || (''u'')''v'' ||~~ <~~math~~>~~\cdot~~<~~/math> || (d''~~u'', ~~d''~~v~~'')~~

+

| ''g''‹''u'', ''v''›

−

~~| + ||~~ (''u'')(''v'') ~~|| <math>\cdot</math> ||~~ ((~~d''~~u'', ~~d''~~v''))

+

| =

−

|-

+

| ((''u'', ''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'')

+

−

|-

+

===Formula Display 13===

−

| ~~d''g''~~

+

−

~~| = || ''uv'' ||~~ <~~math~~>~~\cdot~~<~~/math~~> ~~|| (d''u'', d''v'')~~

+

<pre>

−

~~| + || ''u''(''v'') ||~~ <~~math~~>~~\cdot~~<~~/math~~> || ~~(d''u'', d''v'')~~

+

o-----------------------------------------------------------o

−

| + || ~~(''u'')''v'' || <math>\cdot</math> || (d~~''u'', d''v'')

+

| |

−

| + |~~| ('~~'u'')('~~'v'') || <math>\cdot</math> || (d~~''u'', d''v'')

+

| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |

−

|-

+

| |

−

~~| r''g''~~

+

o-----------------------------------------------------------o

−

| = ~~|| ''uv'' || <math>\cdot</math> || 0~~

+

</pre>

−

| ~~+ ||~~ ''u''(''v'') ~~|| <math>\cdot</math> || 0~~

+

−

~~| + ||~~ (''u'')''v~~'' || <math>\cdot</math> || 0~~

+

−

~~| + || (''u~~'')~~(''v''~~) |~~| <math>\cdot</math> || 0~~

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

+

|

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| ‹''x'', ''y''›

+

| =

+

| ''F''‹''u'', ''v''›

+

| =

+

| ‹((''u'')(''v'')), ((''u'', ''v''))›

+

|}

+

|}

+

+

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

+

|

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

|  

+

| ‹''x'', ''y''›

+

| =

+

| ''F''‹''u'', ''v''›

+

| =

+

| ‹((''u'')(''v'')), ((''u'', ''v''))›

+

|  

|}

−

===Table 67. ~~Computation of an Analytic Series in Terms of Coordinates~~===

+

===Table 60. Propositional Transformation===

<pre>

−

Table 67. ~~Computation of an Analytic Series in Terms of Coordinates~~

+

Table 60. Propositional Transformation

−

o------~~--o~~-------o-----~~--o~~--------o-------o-------o-------o-------o

+

o-------------o-------------o-------------o-------------o

−

+

| u | v | f | g |

−

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 | 0 | 0 | 1 |

−

| ~~| |~~ | | | ~~| |~~ |

+

| | | | |

−

| ~~| 0 1 | 0 1 | | 1~~ 0 | 1 1 | 1 1 | 0 0 |

+

| 0 | 1 | 1 | 0 |

−

| ~~| |~~ | | | ~~| |~~ |

+

| | | | |

−

| | 1 0 | 1 0 | | 1 ~~0 | 1 1~~ | ~~1 1 | 0~~ 0 |

+

| 1 | 0 | 1 | 0 |

−

| ~~| |~~ | | | ~~| |~~ |

+

| | | | |

−

| ~~| 1 1 |~~ 1 ~~1 |~~ | 1 1 | 1 ~~0 | 0 0~~ | 1 0 |

+

| 1 | 1 | 1 | 1 |

−

| ~~| |~~ | | | ~~| |~~ |

+

| | | | |

−

o------~~--o~~-------o-----~~--o~~--------o-------o-------o-------o-------o

+

o-------------o-------------o-------------o-------------o

−

| ~~| | | | | | |~~ |

+

| | | ((u)(v)) | ((u, v)) |

−

| ~~0 1~~ | 0 ~~0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |~~

+

o-------------o-------------o-------------o-------------o

−

~~| | | | | | | | |~~

−

~~| | 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===~~

+

−

+

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

−

~~<pre>~~

+

|+ '''Table 60. Propositional Transformation'''

−

~~Table 68. Computation of an Analytic Series in Symbolic Terms~~

+

|- style="background:paleturquoise"

−

~~o-----o-----o------------o----------o----------o----------o----------o----------o~~

+

| width="25%" | ''u''

−

~~| u v | f g | Df | Dg | df | dg | rf | rf |~~

+

| width="25%" | ''v''

−

~~o-----o-----o------------o----------o----------o----------o----------o----------o~~

+

| width="25%" | ''f''

−

~~| | | | | | | | |~~

+

| width="25%" | ''g''

−

~~| 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~~%"

−

| ~~ ~~

|-

−

| ~~D''f''~~

+

| width="25%" |

−

| = |~~| ''uv'' || <math>\cdot</math> || d''u'' d''v''~~

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

−

| ~~+ || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')~~

+

| 0

−

~~| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''~~

−

| ~~+ || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))~~

|-

−

| ~~ ~~

+

| 0

|-

−

| ~~D''g''~~

+

| 1

−

~~| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')~~

|-

−

| ~~ ~~

+

| 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 69. ~~Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›~~===

+

===Figure 61. Propositional Transformation===

<pre>

−

o-----------------------------------~~o o~~-----------------------------------o

+

o-----------------------------------------------------o

−

| ~~U | |`U`````````````````````````````````~~|

+

| U |

−

| ~~| |```````````````````````````````````~~|

+

| |

−

| ^ | |~~```````````````````````````````````~~|

+

| 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``~~|

+

| | 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~~

+

/ \ / \

−

\ / \ /

+

/ \ / \

−

\ / \ /

+

/ \ / \

−

~~\ ((u)(v))~~ / ~~\ ((u, v))~~ /

+

/ \ / \

−

\ / \ /

+

/ \ / \

−

\ / \ /

+

/ \ / \

−

~~o----------~~\~~-------------~~/~~-----------------------~~\~~-------------~~/~~----------o~~

+

/ \ / \

−

| X \ / \ / |

+

/ \ / \

−

| \ / \ / |

+

/ \ / \

−

| \ / \ / |

+

o-------------------------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\\|

−

| / / ~~\ \~~ |

+

| |// u //|///|// v //| | |\\| u |\\\| v |\\|

−

| ~~/ /~~ \ \ |

+

| o///////o///o///////o | |\\o o\\\o o\\|

−

| o o o o |

+

| \///////\////////// | |\\\\ \\/ /\\\|

−

| ~~| | | |~~ |

+

| \///////o//////// | |\\\\\ o /\\\\|

−

| ~~| | | |~~ |

+

| \////// \////// | |\\\\\\ /\\ /\\\\\|

−

| ~~| f | | g~~ | |

+

| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|

−

~~| | | | |~~ |

+

| | |\\\\\\\\\\\\\\\\\\\\\\\\\|

−

| ~~| | |~~ | |

+

o-------------------------o o-------------------------o

−

~~| o o o o |~~

+

\ | | /

−

| \ \ ~~/ / |~~

+

\ | | /

−

| \ \ / ~~/ |~~

+

\ | | /

−

| \ \ / ~~/ |~~

+

\ f | | g /

−

| \ \ ~~/ / |~~

+

\ | | /

−

| \ \ ~~/ /~~ |

+

\ | | /

−

| \ o ~~/ |~~

+

\ | | /

−

| \ / \ ~~/ |~~

+

\ | | /

−

| o---~~----------~~---o ~~o----------------o~~ |

+

\ | | /

−

| |

+

\ | | /

−

| |

+

o-------\----|---------------------------|----/-------o

−

| |

+

| X \ | | / |

−

o~~--------------------~~-----------------------------------------------------o

+

| \| |/ |

−

Figure 69. ~~Difference Map of F = <f, g> = <((u)(v)), ((u, v))>~~

+

| o-----------o o-----------o |

+

| //////////////\ /\\\\\\\\\\\\\\ |

+

| ////////////////o\\\\\\\\\\\\\\\\ |

+

| /////////////////X\\\\\\\\\\\\\\\\\ |

+

| /////////////////XXX\\\\\\\\\\\\\\\\\ |

+

| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |

+

| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |

+

| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |

+

| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |

+

| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |

+

| \///////////////\XXX/\\\\\\\\\\\\\\\/ |

+

| \///////////////\X/\\\\\\\\\\\\\\\/ |

+

| \///////////////o\\\\\\\\\\\\\\\/ |

+

| \////////////// \\\\\\\\\\\\\\/ |

+

| o-----------o o-----------o |

+

| |

+

| |

+

o-----------------------------------------------------o

+

Figure 61. Propositional Transformation

</pre>

−

===~~Formula Display 19~~===

+

+

[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]

+

<center>'''Figure 61. Propositional Transformation'''</center>

+

===Figure 62. Propositional Transformation (Short Form)===

<pre>

−

o-------------------------------------------------------------------------------o

+

o-------------------------o o-------------------------o

−

~~| |~~

+

| U | |\U \\\\\\\\\\\\\\\\\\\\\\|

−

~~| df = uv.~~ 0 ~~+ u(v). du~~ ~~+ (u)v. dv~~ ~~+ (u)(v).(du, dv)~~ |

+

| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|

−

| |

+

| //////\ //////\ | |\\\\\/ \\/ \\\\\\|

−

| ~~dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv)~~ |

+

| ////////o///////\ | |\\\\/ o \\\\\|

−

| |

+

| //////////\///////\ | |\\\/ /\\ \\\\|

−

o----------------------~~----~~-----------------------------------------------------o

+

| o///////o///o///////o | |\\o o\\\o o\\|

+

| |// u //|///|// v //| | |\\| u |\\\| v |\\|

+

| o///////o///o///////o | |\\o o\\\o o\\|

+

| \///////\////////// | |\\\\ \\/ /\\\|

+

| \///////o//////// | |\\\\\ o /\\\\|

+

| \////// \////// | |\\\\\\ /\\ /\\\\\|

+

| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|

+

| | |\\\\\\\\\\\\\\\\\\\\\\\\\|

+

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>

−

+

−

~~{| align="~~center~~" border~~=~~"1" cellpadding~~=~~"8" cellspacing~~=~~"0" style~~=~~"background:lightcyan; font~~-~~weight:bold; text~~-~~align:center; width:96%"~~

+

[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]

−

|

+

<center>'''Figure 62. Propositional Transformation (Short Form)'''</center>

−

~~{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font~~-~~weight:bold; text~~-~~align:center; width:100%"~~

+

−

~~|  ~~

+

===Figure 63. Transformation of Positions===

−

|-

+

−

~~| d''f''~~

+

<pre>

−

~~| = || ''uv'' || <math>\cdot</math> || 0~~

+

o-----------------------------------------------------o

−

~~| +~~ || ~~''u''(''v'')~~ |~~| <math>\cdot</math> || d''u''~~

+

|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|

−

| ~~+ || (~~''u'')''v'' ~~|| <math>~~\~~cdot<~~/~~math> || d~~''v''

+

|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|

−

| ~~+ || (~~''u'')(''v'') || ~~<math>\cdot<~~/~~math> || (d~~''u''~~, d~~''v'')

+

|` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|

−

|-

+

|` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|

−

~~|  ~~

+

|` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|

−

|-

+

|` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|

−

~~| d~~''g''

+

|` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|

−

| ~~= ||~~ ''uv'' ~~|| <math>\cdot</math> || (d~~''u''~~, d~~''v'')

+

|` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|

−

| ~~+ ||~~ ''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'')

+

|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|

−

|-

+

|` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|

−

| ~~ ~~

+

|` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|

−

|}

+

|` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|

−

|}

+

|` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|

−

</~~font> ~~

+

|` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|

−

+

|` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|

−

~~===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

−

~~/ \ / \ /~~ \

+

| U | |\ | | |`U```````````````````````|

−

~~/ \ / \ /~~ \

+

−

~~/ 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 | | \ / o-------------------------o

−

| ~~\ / \ / |~~ | ~~\ / O \ / O \ / O \ / |~~

+

\ | | | \ / | /

−

| \ / ~~\ /~~ | | o /@ ~~o @~~\ ~~o /@ o~~ |

+

\ ((u)(v)) | | | \/ | ((u, v)) /

−

~~| \ / \ /~~ | | |~~\ / \ / \ / \ / \ /~~| |

+

\ | | | /\ | /

−

~~| \~~ ~~/ \ /~~ | ~~| | \ / \ / \ / |~~ |

+

\ | | | / \ | /

−

~~| 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

−

\ / \ /

+

| X \ | | / | | | / |

−

~~\ / \ /~~

+

| \| | / | | |/ |

−

~~o o~~

+

| o---|----/--o | o-------|---o |

−

~~U% $T$ $E$U%~~

+

| /' ' | ' / ' '\|/` ` ` ` | ` `\ |

−

o------------------>o

+

| / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |

−

| |

+

| /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |

−

~~| |~~

+

| / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |

−

~~| |~~

+

| @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |

−

~~| |~~

+

| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |

−

~~F | | $T$F~~

+

| |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |

−

~~| |~~

+

| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |

−

~~| |~~

+

| o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |

−

~~| |~~

+

| \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |

−

~~v v~~

+

| \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |

−

o---------------~~--->o~~

+

| \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |

−

~~X% $T$ $E$X%~~

+

| \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |

−

~~o o~~

+

| o-----------o o-----------o |

−

/ \ ~~/ \~~

+

| |

−

/ \ / \

+

| |

−

/ \ / ~~O \~~

+

o-----------------------------------------------------o

−

/ \ o /@\ o

+

Figure 63. Transformation of Positions

−

/ \ / \ / \

−

/ \ / \ / \

−

/ O \ / ~~O \~~ / ~~O \~~

−

o /@\ ~~o o~~ /~~@\ o~~ /~~@\ o~~

−

/ \ / \ / \ ~~\ / \~~ / / \

−

/ \ / \ / \ / ~~\ / \~~

−

/ \ / ~~\ / O \~~ / ~~O \ / O \~~

−

/ \ / ~~\ o~~ /@ 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))›~~===

+

+

[[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>

−

o-----------------------o o-----------------------o o-----------------------o

+

Table 64. Transformation of Positions

−

| dU | | dU | | dU |

+

o-----o----------o----------o-------o-------o--------o--------o-------------o

−

+

| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |

−

| ~~/////\ /////\~~ | | ~~/XXXX\ /XXXX\~~ | | ~~/\\\\\ /\\\\\~~ |

+

o-----o----------o----------o-------o-------o--------o--------o-------------o

−

| ~~///////o//////\~~ | | ~~/XXXXXXoXXXXXX\~~ | | ~~/\\\\\\o\\\\\\\~~ |

+

| | | | | | | | ^ |

−

| ~~//////// \//////\~~ | | ~~/XXXXXX/ \XXXXXX\~~ | | ~~/\\\\\\/ \\\\\\\\ |~~

+

| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |

−

| ~~o///////~~ ~~\//////o~~ | | ~~oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o~~ |

+

| | | | | | | | |

−

| ~~|/////o~~ ~~o/////~~| | | |~~XXXXXo oXXXXX~~| | | ~~|\\\\\o o\\\\\|~~ |

+

| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |

−

| |~~/du//~~| |~~//dv/~~| | | |~~XXXXX~~| |~~XXXXX~~| | | |~~\du\\~~| |~~\\dv\|~~ |

+

| | | | | | | | = |

−

| |~~/////o~~ ~~o/////~~| | | |~~XXXXXo oXXXXX~~| | | |~~\\\\\o~~ ~~o\\\\\~~| |

+

| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |

−

| ~~o//////\ ///////o~~ | | ~~oXXXXXX\ /XXXXXXo~~ | | ~~o\\\\\\\ /\\\\\\o~~ |

+

| | | | | | | | |

−

| ~~\//////\ ////////~~ | | ~~\XXXXXX\ /XXXXXX/~~ | | ~~\\\\\\\\ /\\\\\\/~~ |

+

| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |

−

~~| \//////o///////~~ | ~~| \XXXXXXoXXXXXX/~~ | ~~| \\\\\\\o\\\\\\/~~ |

+

| | | | | | | | | |

−

| ~~\///// \/////~~ | | ~~\XXXX/ \XXXX/~~ | | ~~\\\\\/ \\\\\/~~ |

+

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

−

o--------------------~~---o~~ o-----------------------o o-----------------------o

+

</pre>

−

~~= 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~~

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

−

| ~~dU |~~ ~~| dU | | dU |~~

+

|+ '''Table 64. Transformation of Positions'''

−

+

|- style="background:paleturquoise"

−

| ~~/ \ /////\~~ | | /~~\\\\\ /XXXX\~~ | | ~~/\\\\\ /\\\\\~~ |

+

| ''u''  ''v''

−

| ~~/ o//////\~~ | | ~~/\\\\\\oXXXXXX\~~ | | ~~/\\\\\\o\\\\\\\~~ |

+

| ''x''

−

| ~~/ //\//////\~~ | | ~~/\\\\\\//\XXXXXX\~~ | | ~~/\\\\\\/ \\\\\\\\~~ |

+

| ''y''

−

| ~~o ////\//////o~~ | | ~~o\\\\\\////\XXXXXXo~~ | | ~~o\\\\\\/ \\\\\\\o~~ |

+

| ''x'' ''y''

−

| | ~~o/////o/////~~| | | |~~\\\\\o/////oXXXXX~~| | | |~~\\\\\o o\\\\\~~| |

+

| ''x'' (''y'')

−

| | du |~~/////~~|~~//dv/~~| | | |~~\\\\\~~|~~/////~~|~~XXXXX~~| | | |~~\du\\~~| |~~\\dv\~~| |

+

| (''x'') ''y''

−

| ~~| o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\|~~ |

+

| (''x'')(''y'')

−

| ~~o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |~~

+

| ''X'' • = [''x'', ''y'' ]

−

| ~~\ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |~~

+

|-

−

| ~~\ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |~~

+

| width="12%" |

−

| ~~\ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/~~ |

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

−

+

| 0  0

−

| | | | | |

+

|-

−

~~o-----------------------o o-----------------------o o-----------------------o~~

+

| 0  1

−

= ~~du' @ (u) v o-----------------------o dv' @ (u) v~~ =

+

|-

−

= ~~| dU' |~~ =

+

| 1  0

−

= ~~| o~~--~~o o--o~~ | =

+

|-

−

= | ~~/////\ /\\\\\~~ | =

+

| 1  1

−

= | ~~///////o\\\\\\\~~ | =

+

|}

−

= | ~~////////X\\\\\\\\~~ | =

+

| width="12%" |

−

= | ~~o///////XXX\\\\\\\o~~ | =

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

−

= | ~~|/////oXXXXXo\\\\\| | =~~

+

| 0

−

~~= = = = = =~~ = = = = =|~~/du~~'/|~~XXXXX~~|~~\dv~~'\|~~= = = = = = = = = = =~~

+

|-

−

| |~~/////oXXXXXo\\\\\~~| |

+

| 1

−

| ~~o//////\XXX/\\\\\\o~~ |

+

|-

−

| ~~\//////\X/\\\\\\/~~ |

+

| 1

−

| ~~\//////o\\\\\\/~~ |

+

|-

−

| \/~~//// \\\\\/~~ |

+

| 1

−

~~| o--o o--o |~~

+

|}

−

~~| |~~

+

| width="12%" |

−

~~o-----------------------o~~

+

{| 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===

−

o-----------------------o o-----------------------o o-----------------------o

+

<pre>

−

| dU | | dU | | dU |

+

Table 65. Induced Transformation on Propositions

−

+

o------------o---------------------------------o------------o

−

| ~~/////\ / \~~ | | ~~/XXXX\ /\\\\\~~ | | ~~/\\\\\ /\\\\\~~ |

+

| X% | <--- F = <f , g> <--- | U% |

−

| ~~///////o \~~ | | ~~/XXXXXXo\\\\\\\~~ | | ~~/\\\\\\o\\\\\\\~~ |

+

o------------o----------o-----------o----------o------------o

−

| ~~/////////\ \~~ | | ~~/XXXXXX//\\\\\\\\~~ | | ~~/\\\\\\/ \\\\\\\\~~ |

+

| | u = | 1 1 0 0 | = u | |

−

| ~~o//////////\ o~~ | ~~| oXXXXXX////\\\\\\\o~~ | | ~~o\\\\\\/ \\\\\\\o~~ |

+

| | v = | 1 0 1 0 | = v | |

−

| |~~/////o/////o~~ | | | |~~XXXXXo/////o\\\\\~~| | | |~~\\\\\o~~ ~~o\\\\\~~| |

+

| f_i <x, y> o----------o-----------o----------o f_j <u, v> |

−

| |~~/du//~~|~~/////~~| dv | | | |~~XXXXX~~|~~/////~~|~~\\\\\~~| | | |~~\du\\~~| |~~\\dv\~~| |

+

| | x = | 1 1 1 0 | = f<u,v> | |

−

| |~~/////o/////o~~ | | | |~~XXXXXo/////o\\\\\~~| | | |~~\\\\\o~~ ~~o\\\\\~~| |

+

| | y = | 1 0 0 1 | = g<u,v> | |

−

| ~~o//////\//// o~~ | ~~| oXXXXXX\////\\\\\\o~~ | ~~| o\\\\\\\ /\\\\\\o~~ |

+

o------------o----------o-----------o----------o------------o

−

| ~~\//////\// /~~ | | ~~\XXXXXX\//\\\\\\/~~ | | ~~\\\\\\\\ /\\\\\\/~~ |

+

| | | | | |

−

| ~~\//////o /~~ | | ~~\XXXXXXo\\\\\\/~~ | | ~~\\\\\\\o\\\\\\/~~ |

+

| f_0 | () | 0 0 0 0 | () | f_0 |

−

| ~~\///// \ /~~ | | ~~\XXXX/ \\\\\/~~ | | ~~\\\\\/ \\\\\/~~ |

+

| | | | | |

−

+

| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |

−

| | | | | |

+

| | | | | |

−

o-----------------------o o-----------------------o ~~o-----------~~------------o

+

| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |

−

~~= du' @~~ u (v) ~~o-----------------------o dv' @~~ u (v) =

+

| | | | | |

−

= | ~~dU'~~ | =

+

| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |

−

= | ~~o--o~~ ~~o--o~~ | =

+

| | | | | |

−

= | ~~/////\ /\\\\\~~ | =

+

| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |

−

= | ~~///////o\\\\\\\~~ | =

+

| | | | | |

−

= | ~~////////X\\\\\\\\~~ | =

+

| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |

−

= | ~~o///////XXX\\\\\\\o~~ | =

+

| | | | | |

−

= | |~~/////oXXXXXo\\\\\~~| | =

+

| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |

−

~~= = = = = = = = = = =~~|~~/du'/~~|~~XXXXX~~|~~\dv'\~~|~~= = = = = = = = = = =~~

+

| | | | | |

−

| |~~/////oXXXXXo\\\\\~~| |

+

| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |

−

| ~~o//////\XXX/\\\\\\o~~ |

+

| | | | | |

−

| ~~\//////\X/\\\\\\/~~ |

+

o------------o----------o-----------o----------o------------o

−

| ~~\//////o\\\\\\/~~ |

+

| | | | | |

−

| ~~\///// \\\\\/~~ |

+

| f_8 | x y | 1 0 0 0 | u v | f_8 |

−

| o--o o--~~o |~~

+

| | | | | |

−

~~| |~~

+

| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |

−

o-----------------------o

+

| | | | | |

+

| 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>

−

o-----------------------o o-----------------------o o-----------------------o

+

−

| dU | | dU | | dU |

+

{| 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"

−

| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |

+

| ''X'' •

−

| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |

+

| colspan="3" |

−

| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"

−

| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |

+

| ←

−

| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |

+

| ''F'' = ‹''f'' , ''g''›

−

| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |

+

| ←

−

| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |

+

|}

−

| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |

+

| ''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

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