Changes

23,594 bytes added , 15:00, 25 August 2007

add image stub

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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===

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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===

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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===

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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===

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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===

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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===

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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)===

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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===

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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===

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−

===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>

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===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>

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===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)===

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−

===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)===

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−

===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===

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−

===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)===

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===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===

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

</pre>

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

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

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

Line 7,049: Line 7,243:

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

</pre>

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

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

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

Line 7,117: Line 7,315:

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

</pre>

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

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

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

Line 7,185: Line 7,387:

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

</pre>

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

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

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

Line 7,285: Line 7,491:

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

</pre>

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

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

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

Line 7,385: Line 7,595:

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

</pre>

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

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

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

Line 7,485: Line 7,699:

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

</pre>

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[[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===

Line 7,585: Line 7,803:

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

</pre>

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

+

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

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

Line 7,655: Line 7,877:

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

</pre>

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[[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===

Line 7,725: Line 7,951:

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

</pre>

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[[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===

Line 7,795: Line 8,025:

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

</pre>

+

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[[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===

Line 7,865: Line 8,099:

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

</pre>

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[[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===

Line 8,328: Line 8,566:

</pre>

−

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

+

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"

−

|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes

+

|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes'''

−

'''

|- style="background:paleturquoise"

|  

Line 8,337: Line 8,574:

| align="center" | '''Transformation or Mapping'''

|-

−

~~| valign="top"~~ | Operand

+

| Operand

| valign="top" |

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

Line 8,367: Line 8,604:

| <math>\epsilon</math> :

|-

−

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

+

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

|-

−

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

+

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

|}

|

Line 8,404: Line 8,641:

|

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

−

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

+

| <math>\eta</math>''F''''i'' :

|-

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

|-

−

| '''B'''2 × '''D'''2 → '''D'''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

|

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

−

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

+

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

|-

−

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

+

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

|-

−

| ['''B'''2 × '''D'''2] → ['''D'''1]

+

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

|}

|-

Line 8,435: Line 8,672:

|

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

−

| E''J'' :

+

| E''F''''i'' :

|-

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

|-

−

| '''B'''2 × '''D'''2 → '''D'''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

|

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

−

| E''J'' :

+

| E''F'' :

|-

−

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

+

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

|-

−

| ['''B'''2 × '''D'''2] → ['''D'''1]

+

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

|}

|-

Line 8,466: Line 8,703:

|

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

−

| D''J'' :

+

| D''F''''i'' :

|-

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

|-

−

| '''B'''2 × '''D'''2 → '''D'''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

|

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

−

| D''J'' :

+

| D''F'' :

|-

−

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

+

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

|-

−

| ['''B'''2 × '''D'''2] → ['''D'''1]

+

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

|}

|-

Line 8,497: Line 8,734:

|

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

−

| d''J'' :

+

| d''F''''i'' :

|-

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

|-

−

| '''B'''2 × '''D'''2 → '''D'''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

|

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

−

| d''J'' :

+

| d''F'' :

|-

−

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

+

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

|-

−

| ['''B'''2 × '''D'''2] → ['''D'''1]

+

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

|}

|-

Line 8,528: Line 8,765:

|

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

−

| r''J'' :

+

| r''F''''i'' :

|-

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

|-

−

| '''B'''2 × '''D'''2 → '''D'''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

|

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

−

| r''J'' :

+

| r''F'' :

|-

−

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

+

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

|-

−

| ['''B'''2 × '''D'''2] → ['''D'''1]

+

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

|}

|-

Line 8,567: Line 8,804:

|

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

−

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

+

| '''e'''''F'' :

|-

−

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

+

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

|-

−

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

+

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

|}

|-

Line 8,598: Line 8,835:

|

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

−

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

+

| '''E'''''F'' :

|-

−

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

+

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

|-

−

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

+

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

|}

|-

Line 8,629: Line 8,866:

|

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

−

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

+

| '''D'''''F'' :

|-

−

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

+

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

|-

−

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

+

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

|}

|-

Line 8,652: Line 8,889:

|

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

−

| d''J'' :

+

| d''F''''i'' :

|-

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

|-

−

| '''B'''2 × '''D'''2 → '''D'''

+

| '''B'''''n'' × '''D'''''n'' → '''D'''

|}

|

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

−

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

+

| '''T'''''F'' :

|-

−

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

+

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

|-

−

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

+

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

|}

|}

+

===Formula Display 12===

<pre>

−

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

+

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

−

+

| |

−

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

+

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

−

+

| |

−

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

+

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

−

~~| | | | |~~

+

| |

−

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

+

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

−

~~| | | | |~~

+

</pre>

−

+

−

~~| | | | |~~

+

−

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

+

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

−

~~| | | | |~~

+

|

−

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

+

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

−

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

+

|  

−

+

| ''x''

−

~~| | | | |~~

+

| =

−

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

+

| ''f''‹''u'', ''v''›

−

~~| | | | |~~

+

| =

−

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

+

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

−

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

+

|  

−

+

|-

−

~~| | | | |~~

+

|  

−

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

+

| ''y''

−

~~| | | | |~~

+

| =

−

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

+

| ''g''‹''u'', ''v''›

−

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

+

| =

−

+

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

−

~~| | | | |~~

+

|  

−

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

+

|}

−

~~| | | | |~~

+

|}

−

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

+

−

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

+

−

+

===Formula Display 13===

−

~~| | | | |~~

+

−

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

+

<pre>

−

~~| | | | |~~

+

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

−

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

+

| |

−

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

+

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

−

+

| |

−

~~| | | | |~~

−

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

−

~~| | | | |~~

−

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

−

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

−

−

~~| | | | |~~

−

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

−

~~| | | | |~~

−

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

−

~~| Operator | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

~~| | | | |~~

−

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

−

~~| Operator | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

~~| | | | |~~

−

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

−

~~| Operator | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

~~| | | | |~~

−

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

−

~~| Functor | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

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

−

~~| | | | |~~

−

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

−

~~</pre>~~

−

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

−

~~<pre>~~

−

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

−

| |

−

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

−

| |

−

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

−

| |

−

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

−

</pre>

−

−

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

−

|

−

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

−

|  

−

| ''x''

−

| =

−

| ''f''‹''u'', ''v''›

−

| =

−

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

−

|  

−

|-

−

|  

−

| ''y''

−

| =

−

| ''g''‹''u'', ''v''›

−

| =

−

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

−

|  

−

|}

−

|}

−

−

===Formula Display 13===

−

<pre>

−

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

−

| |

−

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

−

| |

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

</pre>

Line 8,852: Line 8,999:

</pre>

−

===Figure 61. Propositional Transformation===

+

−

+

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

−

<pre>

+

|+ '''Table 60. Propositional Transformation'''

−

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

+

|- style="background:paleturquoise"

−

| U |

+

| width="25%" | ''u''

−

| |

+

| width="25%" | ''v''

−

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

+

| width="25%" | ''f''

−

| / \ / \ |

+

| width="25%" | ''g''

−

| / o \ |

+

|-

−

| / / \ \ |

+

| width="25%" |

−

| / / \ \ |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

−

| o o o o |

+

| 0

−

| | | | | |

+

|-

+

| 0

+

|-

+

| 1

+

|-

+

| 1

+

|}

+

| width="25%" |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 0

+

|-

+

| 1

+

|-

+

| 0

+

|-

+

| 1

+

|}

+

| width="25%" |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 0

+

|-

+

| 1

+

|-

+

| 1

+

|-

+

| 1

+

|}

+

| width="25%" |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

+

| 1

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 1

+

|}

+

|-

+

| width="25%" |  

+

| width="25%" |  

+

| width="25%" | ((''u'')(''v''))

+

| width="25%" | ((''u'', ''v''))

+

|}

+

+

===Figure 61. Propositional Transformation===

+

<pre>

+

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

+

| U |

+

| |

+

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

+

| / \ / \ |

+

| / o \ |

+

| / / \ \ |

+

| / / \ \ |

+

| o o o o |

+

| | | | | |

| | u | | v | |

| | | | | |

Line 8,936: Line 9,140:

Figure 61. Propositional Transformation

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]

+

<center>'''Figure 61. Propositional Transformation'''</center>

===Figure 62. Propositional Transformation (Short Form)===

Line 8,987: Line 9,195:

Figure 62. Propositional Transformation (Short Form)

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]

+

<center>'''Figure 62. Propositional Transformation (Short Form)'''</center>

===Figure 63. Transformation of Positions===

Line 9,064: Line 9,276:

Figure 63. Transformation of Positions

</pre>

+

+

[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]

+

<center>'''Figure 63. Transformation of Positions'''</center>

===Table 64. Transformation of Positions===

Line 9,086: Line 9,302:

</pre>

−

===Table 65. Induced Transformation on Propositions===

+

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

−

+

|+ '''Table 64. Transformation of Positions'''

−

<pre>

+

|- style="background:paleturquoise"

−

Table 65. Induced Transformation on Propositions

+

| ''u''  ''v''

−

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

+

| ''x''

−

| X% | <--- F = <f , g> <--- | U% |

+

| ''y''

−

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

+

| ''x'' ''y''

−

| | u = | 1 1 0 0 | = u | |

+

| ''x'' (''y'')

−

| | v = | 1 0 1 0 | = v | |

+

| (''x'') ''y''

−

| f_i <x, y> o----------o-----------o----------o f_j <u, v> |

+

| (''x'')(''y'')

−

| | x = | 1 1 1 0 | = f<u,v> | |

+

| ''X'' • = [''x'', ''y'' ]

−

| | y = | 1 0 0 1 | = g<u,v> | |

+

|-

−

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

+

| width="12%" |

−

~~| | | | | |~~

+

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

−

~~| f_0 | () | 0 0 0 0 | () | f_0 |~~

+

| 0  0

−

~~| | | | | |~~

+

|-

−

~~| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |~~

+

| 0  1

−

~~| | | | | |~~

+

|-

−

~~| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |~~

+

| 1  0

−

~~| | | | | |~~

+

|-

−

~~| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |~~

+

| 1  1

−

~~| | | | | |~~

+

|}

−

~~| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |~~

+

| width="12%" |

−

~~| | | | | |~~

+

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

−

~~| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |~~

+

| 0

−

~~| | | | | |~~

+

|-

−

~~| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |~~

+

| 1

−

~~| | | | | |~~

+

|-

−

~~| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |~~

+

| 1

−

~~| | | | | |~~

+

|-

−

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

+

| 1

−

~~| | | | | |~~

+

|}

−

~~| f_8 | x y | 1 0 0 0 | u v | f_8 |~~

+

| width="12%" |

−

~~| | | | | |~~

+

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

−

~~| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |~~

+

| 1

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 1

+

|}

+

| width="12%" |

+

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

+

| 0

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 1

+

|}

+

| width="12%" |

+

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

+

| 0

+

|-

+

| 1

+

|-

+

| 1

+

|-

+

| 0

+

|}

+

| width="12%" |

+

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

+

| 1

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 0

+

|}

+

| width="12%" |

+

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

+

| 0

+

|-

+

| 0

+

|-

+

| 0

+

|-

+

| 0

+

|}

+

| width="12%" |

+

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

+

| ↑

+

|-

+

| ''F''

+

|-

+

| ‹''f'', ''g'' ›

+

|-

+

| ↑

+

|}

+

|-

+

|  

+

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

+

| ((''u'', ''v''))

+

| ''u'' ''v''

+

| (''u'', ''v'')

+

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

+

| ( )

+

| ''U'' • = [''u'', ''v'' ]

+

|}

+

+

===Table 65. Induced Transformation on Propositions===

+

<pre>

+

Table 65. Induced Transformation on Propositions

+

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

+

| X% | <--- F = <f , g> <--- | U% |

+

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

+

| | u = | 1 1 0 0 | = u | |

+

| | v = | 1 0 1 0 | = v | |

+

| f_i <x, y> o----------o-----------o----------o f_j <u, v> |

+

| | x = | 1 1 1 0 | = f<u,v> | |

+

| | y = | 1 0 0 1 | = g<u,v> | |

+

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

| | | | | |

−

| ~~f_10~~ ~~| y~~ | 1 0 1 0 | (~~(u, v)~~) | ~~f_9~~ |

+

| f_0 | () | 0 0 0 0 | () | f_0 |

| | | | | |

−

| ~~f_11~~ | (x (y)) | 1 0 1 1 | (~~(u, v)~~) | ~~f_9~~ |

+

| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |

| | | | | |

−

| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |

+

| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |

+

| | | | | |

+

| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |

+

| | | | | |

+

| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |

+

| | | | | |

+

| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |

+

| | | | | |

+

| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |

+

| | | | | |

+

| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |

+

| | | | | |

+

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

+

| | | | | |

+

| f_8 | x y | 1 0 0 0 | u v | f_8 |

+

| | | | | |

+

| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |

+

| | | | | |

+

| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |

+

| | | | | |

+

| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |

+

| | | | | |

+

| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |

| | | | | |

| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |

Line 9,135: Line 9,455:

| | | | | |

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

−

~~</pre>~~

−

~~===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="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

+

|+ Table 65. Induced Transformation on Propositions

+

|- style="background:paleturquoise"

+

| ''X'' •

+

| colspan="3" |

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"

+

| ←

+

| ''F'' = ‹''f'' , ''g''›

+

| ←

+

|}

+

| ''U'' •

+

|- style="background:paleturquoise"

+

| rowspan="2" | ''f''''i''‹''x'', ''y''›

+

|

+

{| align="right" style="background:paleturquoise; text-align:right"

+

| ''u'' =

+

|-

+

| ''v'' =

+

|}

|

−

{| align="~~left" border="0" cellpadding="12" cellspacing="0~~" style="background:~~lightcyan; font-weight:bold~~; text-align:~~left; width:100%~~"

+

{| align="center" style="background:paleturquoise; text-align:center"

−

| ~~width="8%" | E''G''''i''~~

+

| 1 1 0 0

−

| ~~width="4%" | =~~

+

|-

−

| ~~width="88%" | ''G''''i''‹''u'' + d''u'', ''v'' + d''v''›~~

+

| 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'' =

|}

−

~~ ~~

−

~~===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="center" style="background:paleturquoise; text-align:center"

−

| ~~width="8%" | D''G''''i''~~

+

| 1 1 1 0

−

~~| width="4%" | =~~

−

~~| width="20%" | ''G''''i''‹''u'', ''v''›~~

−

~~| width="4%" | +~~

−

~~| width="64%" | E''G''''i''‹''u'', ''v'', d''u'', d''v''›~~

|-

−

| ~~width="8%" |  ~~

+

| 1 0 0 1

−

~~| width="4%" | =~~

−

~~| width="20%" | ''G''''i''‹''u'', ''v''›~~

−

~~| width="4%" | +~~

−

~~| width="64%" | ''G''''i''‹''u'' + d''u'', ''v'' + d''v''›~~

|}

+

|

+

{| align="left" style="background:paleturquoise; text-align:left"

+

| = ''f''‹''u'', ''v''›

+

|-

+

| = ''g''‹''u'', ''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%~~"

+

{| cellpadding="2" style="background:lightcyan"

−

| ~~width="8%"~~ | E''f''

+

| ''f''0

−

| ~~width="4%"~~ | =

+

|-

−

| ~~width="88%"~~ | ((''u'' ~~+ d~~''u'')(''v'' ~~+ d~~''v''))

+

| ''f''1

+

|-

+

| ''f''2

+

|-

+

| ''f''3

+

|-

+

| ''f''4

+

|-

+

| ''f''5

+

|-

+

| ''f''6

|-

−

| ~~width="8%" | E~~''g''

+

| ''f''7

−

~~| width="4%" | =~~

−

~~| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))~~

|}

+

|

+

{| cellpadding="2" style="background:lightcyan"

+

| ()

+

|-

+

|  (''x'')(''y'') 

+

|-

+

|  (''x'') ''y''  

+

|-

+

|  (''x'')    

+

|-

+

|   ''x'' (''y'') 

+

|-

+

|     (''y'') 

+

|-

+

|  (''x'', ''y'') 

+

|-

+

|  (''x''  ''y'') 

|}

−

~~ ~~

−

~~===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%"~~

+

{| cellpadding="2" style="background:lightcyan"

−

| ~~width="8%"~~ | ~~D''f''~~

+

| 0 0 0 0

−

| ~~width="4%"~~ | =

+

|-

−

| ~~width="20%"~~ | ~~((''u'')(''v''))~~

+

| 0 0 0 1

−

| ~~width="4%"~~ | +

+

|-

−

| ~~width="64%" | ((''u'' + d''u'')(''v'' + d''v''))~~

+

| 0 0 1 0

+

|-

+

| 0 0 1 1

+

|-

+

| 0 1 0 0

+

|-

+

| 0 1 0 1

+

|-

+

| 0 1 1 0

|-

−

| ~~width~~="8%" | D''g''

+

| 0 1 1 1

−

| ~~width="4%"~~ | =

+

|}

−

| ~~width="20%"~~ | ((''u'', ''v''))

+

|

−

| ~~width="4%"~~ | +

+

{| cellpadding="2" style="background:lightcyan"

−

| ~~width="64%"~~ | ((''u'' ~~+ d~~''u'', ''v'' ~~+ d~~''v''))

+

| ()

+

|-

+

| ()

+

|-

+

|  (''u'')(''v'') 

+

|-

+

|  (''u'')(''v'') 

+

|-

+

|  (''u'', ''v'') 

+

|-

+

|  (''u'', ''v'') 

+

|-

+

|  (''u''  ''v'') 

+

|-

+

|  (''u''  ''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%~~"

+

{| cellpadding="2" style="background:lightcyan"

−

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

+

| ''f''0

−

~~| = || ''uv'' ||~~ <~~math~~>~~\cdot~~</~~math~~> |~~| 1~~

+

|-

−

~~| + |~~| ''~~u''(''v~~''~~) ||~~ <~~math~~>~~\cdot~~</~~math~~> |~~| 1~~

+

| ''f''0

−

| ~~+ || (~~''u'')'~~'v'' ||~~ <~~math~~>~~\cdot~~</~~math~~> |~~| 1~~

+

|-

−

| ~~+ || (~~''u''~~)(''v'') ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| 0~~

+

| ''f''1

+

|-

+

| ''f''1

|-

−

| E''f''

+

| ''f''6

−

~~| = || ''uv'' ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| (d''u'' d''v'')~~

−

~~| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')~~

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))~~

|-

−

| D''f''

+

| ''f''6

−

~~| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''~~

−

~~| + || ''u''(''v'') ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| d''u'' (d''v'')~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''~~

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))~~

|-

−

| d''f''

+

| ''f''7

−

~~| = || ''uv'' || <math>\cdot</math> || 0~~

−

~~| + || ''u''(''v'') ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| d''u''~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || d''v''~~

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')~~

|-

−

| r''f''

+

| ''f''7

−

~~| = || ''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%~~"

+

{| cellpadding="2" style="background:lightcyan"

−

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

+

| ''f''8

−

~~| = |~~| ''uv'' || <~~math~~>~~\cdot~~</~~math~~> |~~| 1~~

+

|-

−

| ~~+ || ''u''(~~''v''~~) ||~~ <~~math~~>~~\cdot~~</~~math~~> |~~| 0~~

+

| ''f''9

−

| ~~+ || (''u'')~~''v'' || <~~math~~>~~\cdot~~</~~math~~> |~~| 0~~

+

|-

−

| ~~+ || (''u'')(~~''v''~~) ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| 1~~

+

| ''f''10

+

|-

+

| ''f''11

|-

−

| E''g''

+

| ''f''12

−

~~| = || ''uv'' || <math>\cdot</math> || ((d''u'', d''v''))~~

−

~~| + || ''u''(''v'') ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| (d''u'', d''v'')~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))~~

|-

−

| D''g''

+

| ''f''13

−

~~| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || ''u''(''v'') ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| (d''u'', d''v'')~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')~~

|-

−

| d''g''

+

| ''f''14

−

~~| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || ''u''(''v'') ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| (d''u'', d''v'')~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')~~

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')~~

|-

−

| r''g''

+

| ''f''15

−

~~| = || ''uv'' || <math>\cdot</math> || 0~~

−

~~| + || ''u''(''v'') || <math>\cdot</math> || 0~~

−

~~| + || (''u'')''v'' || <math>\cdot</math> || 0~~

−

~~| + || (''u'')(''v'') ||~~ <~~math~~>~~\cdot~~</~~math~~> ~~|| 0~~

|}

−

|}

+

|

−

+

{| cellpadding="2" style="background:lightcyan"

−

+

|   ''x''  ''y''  

−

===~~Table 67. Computation of an Analytic Series in Terms of Coordinates~~===

+

|-

+

| ((''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>

−

~~Table 67. Computation of an Analytic Series in Terms of Coordinates~~

+

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

−

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

+

| |

−

| ~~u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |~~

+

| EG_i = G_i |

−

~~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>

−

===~~Table 68.~~ ~~Computation of an Analytic Series in Symbolic Terms~~===

+

+

{| 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>

−

~~Table 68. Computation of an Analytic Series in Symbolic Terms~~

+

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

−

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

+

| |

−

| ~~u v~~ | ~~f g~~ | Df | Dg | df | dg | ~~rf | rf |~~

+

| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |

−

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

+

| |

−

~~| | | | | | | | |~~

+

| = G_i <u, v> + G_i |

−

| ~~0 0 | 0~~ 1 | ~~((du)(dv)) | (du, dv)~~ | ~~(du, dv)~~ | ~~(du, dv) | du dv~~ | ~~() |~~

+

| |

−

| | | | ~~| | |~~ | |

+

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

−

| ~~0 1~~ | ~~1 0 | (du) dv | (du~~, ~~dv)~~ | ~~dv | (du, dv)~~ | ~~du dv~~ | ~~() |~~

+

</pre>

−

| ~~| | | | | | |~~ |

+

−

| ~~1 0~~ | ~~1 0~~ | ~~du (dv)~~ | ~~(du, dv)~~ | du | ~~(du~~, ~~dv)~~ | ~~du dv | ()~~ |

+

−

~~| | | | | | | | |~~

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"

−

~~| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |~~

+

|

−

~~| | | | | | | | |~~

+

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

−

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

+

| width="8%" | D''G''''i''

−

~~</pre>~~

+

| width="4%" | =

−

+

| width="20%" | ''G''''i''‹''u'', ''v''›

−

==~~=Formula Display 18===~~

+

| width="4%" | +

−

+

| width="64%" | E''G''''i''‹''u'', ''v'', d''u'', d''v''›

−

~~<pre>~~

+

|-

−

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

+

| width="8%" |  

−

| |

+

| width="4%" | =

−

~~| Df~~ = ~~uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |~~

+

| width="20%" | ''G''''i''‹''u'', ''v''›

−

~~| |~~

+

| width="4%" | +

−

~~| Dg~~ = ~~uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv)~~ |

+

| width="64%" | ''G''''i''‹''u'' + d''u'', ''v'' + d''v''›

−

| |

+

|}

−

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

+

|}

−

~~</pre>~~

+

−

+

−

+

===Formula Display 16===

−

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

+

−

|

+

<pre>

−

~~{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font~~-~~weight:bold; text~~-~~align:center; width:100%"~~

+

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

−

~~|  ~~

+

| |

−

|-

+

| Ef = ((u + du)(v + dv)) |

−

| ~~D''f''~~

+

| |

−

~~| =~~ || ~~''uv'' || <math>\cdot</math> || d''~~u~~'' d''~~v''

+

| Eg = ((u + du, v + dv)) |

−

| + ~~|| ''~~u''(''v'') || ~~<math>\cdot</math>~~ || ~~d''~~u'' (~~d''~~v'')

+

| |

−

| ~~+ |~~| ~~(''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''~~

+

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

−

~~| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))~~

+

</pre>

−

|-

+

−

~~|  ~~

+

−

|-

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"

−

~~| D''g''~~

+

|

−

~~| = || ''uv'' ||~~ <~~math~~>~~\cdot~~<~~/math~~> ~~|| (d''u'', d''v'')~~

+

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

−

| ~~+ || ''u''(''v'') || <math>\cdot</math> |~~| ~~(d''u'', d''v'')~~

+

| width="8%" | E''f''

−

| ~~+ || (''u'')~~''v'' || ~~<math>\cdot</math> |~~| (d''u''~~, d''v'')~~

+

| width="4%" | =

−

~~| + ||~~ (~~''u'')~~(~~''v'') || <math>\cdot</math> || (d''~~u'', ~~d''v'')~~

+

| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))

−

|-

+

|-

−

|  

+

| width="8%" | E''g''

−

|}

+

| width="4%" | =

−

|}

+

| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))

−

~~ ~~

+

|}

−

+

|}

−

~~===Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u,~~ v))›===

+

−

+

−

<pre>

+

===Formula Display 17===

−

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

+

−

| U | |`U`````````````````````````````````|

+

<pre>

−

| | |```````````````````````````````````|

+

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

−

| ^ | |```````````````````````````````````|

+

| |

−

| | | |```````````````````````````````````|

+

| Df = ((u)(v)) + ((u + du)(v + dv)) |

−

| o-------o | o-------o | |```````o-------o```o-------o```````|

+

| |

−

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

+

| Dg = ((u, v)) + ((u + du, v + dv)) |

−

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

+

| |

−

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

+

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

−

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

+

</pre>

−

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

+

+

{| 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``|

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

| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|

Line 9,532: Line 10,515:

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===

Line 9,570: Line 10,557:

−

===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===

+

===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===

<pre>

Line 9,655: Line 10,642:

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))›===

−

~~[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›''']]~~

<pre>

Line 9,838: Line 10,827:

Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>

</pre>

+

[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›''']]

Jon Awbrey

12,080

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User:Jon Awbrey/ATLAS (view source)

Revision as of 15:00, 25 August 2007

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