12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Sunday April 06, 2025
Jump to navigationJump to searchadd image stub
Line 1,141:
Line 1,141:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+|-
|}
--------------o+ +| Tacit | !e! : | !e!J : | !e!J : |+| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] |+| | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] |+ +--------------o+ +| Trope | !h! : | !h!J : | !h!J : |+| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |+| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |+ +--------------o+ +| Enlargement | E : | EJ : | EJ : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
−| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
−
−--------------o
−
−| Difference | D : | DJ : | DJ : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
−| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
−
−--------------o
−
−| Differential | d : | dJ : | dJ : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
−| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
−
−--------------o
−
−| Remainder | r : | rJ : | rJ : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
−| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
−
−--------------o
−
−| Radius | $e$ = <!e!, !h!> : | | $e$J : |
−| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
−| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
−
−--------------o
−
−| Secant | $E$ = <!e!, E> : | | $E$J : |
−| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
−| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
−
−--------------o
−
−| Chord | $D$ = <!e!, D> : | | $D$J : |
−| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
−| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
−
−--------------o
−
−| Tangent | $T$ = <!e!, d> : | dJ : | $T$J : |
−| Functor | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] |
−| | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D | [B^2 x D^2]->[B x D] |
−
−--------------o
−</pre>
−
−===Figure 56-a1. Radius Map of the Conjunction J = uv===
−
−<pre>
− +<pre>+o--------------o----------------------o--------------------o----------------------o+o--------------o----------------------o--------------------o----------------------o+| | | | |+o--------------o----------------------o--------------------o----------------------o+| Tacit | !e! : | !e!F_i : | !e!F : |+| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |+o--------------o----------------------o--------------------o----------------------o+| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |+o--------------o----------------------o--------------------o----------------------o+o--------------o----------------------o--------------------o----------------------o+| Difference | D : | DF_i : | DF : |+o--------------o----------------------o--------------------o----------------------o+| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |+o--------------o----------------------o--------------------o----------------------o+| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |+| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |+| | | | |+o--------------o----------------------o--------------------o----------------------o+| | | | |+| Radius | $e$ = <!e!, !h!> : | | $e$F : |+| Operator | | | |+| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |+| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |+| | | | |+| | | | [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>+ +<br><font face="courier new">+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"+|+{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"+| +| ''x''+| =+| ''f''‹''u'', ''v''›+| =
−| ((''u'')(''v''))
−|
−| ''y''
−| =
−| ''g''‹''u'', ''v''›
−| =
−| ((''u'', ''v''))
</font><br>+ +===Formula Display 13===+ +<pre>+o-----------------------------------------------------------o+| |+o-----------------------------------------------------------o+</pre>+ +<br><font face="courier new">
−{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
− +| u | v | f | g |+o-------------o-------------o-------------o-------------o+| 0 | 1 | 1 | 0 |+| | | | |+| 1 | 0 | 1 | 0 |+| | | | |+| 1 | 1 | 1 | 1 |+| | | | |+o-------------o-------------o-------------o-------------o+</pre>+ +===Figure 61. Propositional Transformation===+ +<pre>+ o-----------------------------------------------------o+ | |+ | o-----------o o-----------o |+ | / \ / \ |+ | / o \ |+ | / / \ \ |+ | / / \ \ |+ | o o o o |+ | | u | | v | |+ | | | | | |+ | \ \ / / |+ | \ o / |+ | \ / \ / |+ | o-----------o o-----------o |+ | |+ | |+ o-----------------------------------------------------o+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+ / \ / \+o-------------------------o o-------------------------o+| U | |\U \\\\\\\\\\\\\\\\\\\\\\|+| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|+| //////\ //////\ | |\\\\\/ \\/ \\\\\\|+| |// u //|///|// v //| | |\\| u |\\\| v |\\|+| \///////\////////// | |\\\\ \\/ /\\\|+| \////// \////// | |\\\\\\ /\\ /\\\\\|+| 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|\\\\\\\\\\\\\\\| |
− | o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
− | \///////////////\XXX/\\\\\\\\\\\\\\\/ |
− | \///////////////\X/\\\\\\\\\\\\\\\/ |
− | \///////////////o\\\\\\\\\\\\\\\/ |
− | \////////////// \\\\\\\\\\\\\\/ |
− | o-----------o o-----------o |
− | |
− | |
− o-----------------------------------------------------o
−Figure 61. Propositional Transformation
+<pre>+o-------------------------o o-------------------------o+o-------------------------o o-------------------------o+ \ / \ /+ \ / \ /+ \ / \ /+ \ f / \ g /+ \ / \ /+ \ / \ /+ \ / \ /+ \ / \ /+ \ / \ /+o---------\-----/---------------------\-----/---------o+| X \ / \ / |+| \ / \ / |+| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |+o-----------------------------------------------------o+Figure 62. Propositional Transformation (Short Form)+ +===Figure 63. Transformation of Positions===+ + o-----------------------------------------------------o+ |`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|+ |` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|+ |` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|+ |` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|+ |` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|+ |` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|+ |` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|+ |` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|+ |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|+ |` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|+ |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|+ |` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|+ |` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|+ |` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|+ |` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|+ |` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|+ |` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|+ |` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|+ |` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|+ o-----------\----|---------|---------|----------------o+ " " \ | | | " "+ " " \ | | | " "+ " " \ | | | " "+ " " \| | | " "+o-------------------------o \ | | o-------------------------o+| /'''''''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 |+ | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |+ | |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |+ | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |+ | o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |+ | \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |+ | \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |+ | \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |+ | \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |+ | o-----------o o-----------o |+ | |+ | |+ o-----------------------------------------------------o+Figure 63. Transformation of Positions+ +===Table 64. Transformation of Positions===+ +Table 64. Transformation of Positions+o-----o----------o----------o-------o-------o--------o--------o-------------o+| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |+o-----o----------o----------o-------o-------o--------o--------o-------------o+| | | | | | | | ^ |+| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |+| | | | | | | | |+| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |+| | | | | | | | = |+| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |+| | | | | | | | |+| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |+| | | | | | | | | |+o-----o----------o----------o-------o-------o--------o--------o-------------o+| | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] |+o-----o----------o----------o-------o-------o--------o--------o-------------o+</pre>+ +===Table 65. Induced Transformation on Propositions===+ +<pre>+Table 65. Induced Transformation on Propositions+o------------o---------------------------------o------------o+| X% | <--- F = <f , g> <--- | U% |+o------------o----------o-----------o----------o------------o+| | u = | 1 1 0 0 | = u | |+| | v = | 1 0 1 0 | = v | |+| f_i <x, y> o----------o-----------o----------o f_j <u, v> |+| | x = | 1 1 1 0 | = f<u,v> | |+| | y = | 1 0 0 1 | = g<u,v> | |+o------------o----------o-----------o----------o------------o+| | | | | |+| f_0 | () | 0 0 0 0 | () | f_0 |+| | | | | |+| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |+| | | | | |+| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |+| | | | | |+| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |+| | | | | |+| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |+| | | | | |+| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |+| | | | | |+| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |+| | | | | |+| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |+| | | | | |+o------------o----------o-----------o----------o------------o+| | | | | |+| f_8 | x y | 1 0 0 0 | u v | f_8 |+| | | | | |+| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |+| | | | | |+| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |+| | | | | |+| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |+| | | | | |+| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |+| | | | | |+| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |+| | | | | |+| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |+| | | | | |+| f_15 | (()) | 1 1 1 1 | (()) | f_15 |+| | | | | |+o------------o----------o-----------o----------o------------o+</pre>+ +===Formula Display 14===+ +<pre>+o-------------------------------------------------o+| |+| EG_i = G_i <u + du, v + dv> |+| |+o-------------------------------------------------o+ +<br><font face="courier new">+{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"+</font><br>
−===Formula Display 15===
−<pre>
−o-------------------------------------------------o
−| |
−| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |
−| |
−| = G_i <u, v> + G_i <u + du, v + dv> |
−| |
−o-------------------------------------------------o
−</pre>
−<br><font face="courier new">
−{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| width="4%" | +
−| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
| width="4%" | =
−| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
−| width="4%" | +
−| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
</font><br>
−===Formula Display 16===
−<pre>
−o-------------------------------------------------o
−| |
−| Ef = ((u + du)(v + dv)) |
−| |
−| Eg = ((u + du, v + dv)) |
−| |
−o-------------------------------------------------o
−</pre>
−<br><font face="courier new">
−{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| width="4%" | =
−| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
</font><br>+| width="4%" | =+| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))+| width="4%" | =+| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))+Table 66-i. Computation Summary for f<u, v> = ((u)(v))+| |+| 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 |
−|+ Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))
| = || ''uv'' || <math>\cdot</math> || 1+| + || (''u'')(''v'') || <math>\cdot</math> || 0+|-+| E''f''+
−| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')+| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))+| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''+| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')+| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''+| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))+|-
−| r''f''
−| + || (''u'')''v'' || <math>\cdot</math> || d''u'' d''v''
−| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"+| + || ''u''(''v'') || <math>\cdot</math> || 0+| = || ''uv'' || <math>\cdot</math> || ((d''u'', d''v''))+| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
−| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
−| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
−| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
−| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
−| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
−| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
−| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| = || ''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+| | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 |+| | | | | | | | |+o--------o-------o-------o--------o-------o-------o-------o-------o+| | | | | | | | |+| | | | | | | | |+| | | | | | | | |+| | | | | | | | |+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 |+| | | | | | | | |+</pre>+ +===Table 68. Computation of an Analytic Series in Symbolic Terms===+ +<pre>+Table 68. Computation of an Analytic Series in Symbolic Terms+| u v | f g | Df | Dg | df | dg | rf | rf |+| | | | | | | | |+| | | | | | | | |+o-----o-----o------------o----------o----------o----------o----------o----------o+ + + +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|<font size="3">'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]
Figure 12. The Anchor
Figure 12. The Anchor
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]</p>
+<p><center><font size="+1">'''Figure 12. The Anchor'''</font></center></p>
===Figure 13. The Tiller===
===Figure 13. The Tiller===
Line 1,174:
Line 1,178:
Figure 13. The Tiller
Figure 13. The Tiller
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]</p>
+<p><center><font size="+1">'''Figure 13. The Tiller'''</font></center></p>
===Table 14. Differential Propositions===
===Table 14. Differential Propositions===
Line 1,667:
Line 1,675:
|}
|}
</font><br>
</font><br>
+
+===Figure 16. A Couple of Fourth Gear Orbits===
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]</p>
+<p><center><font size="+1">'''Figure 16. A Couple of Fourth Gear Orbits'''</font></center></p>
===Figure 16-a. A Couple of Fourth Gear Orbits: 1===
===Figure 16-a. A Couple of Fourth Gear Orbits: 1===
Line 2,064:
Line 2,078:
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 18-a. Extension from 1 to 2 Dimensions: Areal'''</font></center></p>
===Figure 18-b. Extension from 1 to 2 Dimensions: Bundle===
===Figure 18-b. Extension from 1 to 2 Dimensions: Bundle===
Line 2,093:
Line 2,111:
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 18-b. Extension from 1 to 2 Dimensions: Bundle'''</font></center></p>
===Figure 18-c. Extension from 1 to 2 Dimensions: Compact===
===Figure 18-c. Extension from 1 to 2 Dimensions: Compact===
Line 2,124:
Line 2,146:
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 18-c. Extension from 1 to 2 Dimensions: Compact'''</font></center></p>
===Figure 18-d. Extension from 1 to 2 Dimensions: Digraph===
===Figure 18-d. Extension from 1 to 2 Dimensions: Digraph===
Line 2,143:
Line 2,169:
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 18-d. Extension from 1 to 2 Dimensions: Digraph'''</font></center></p>
===Figure 19-a. Extension from 2 to 4 Dimensions: Areal===
===Figure 19-a. Extension from 2 to 4 Dimensions: Areal===
Line 2,186:
Line 2,216:
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 19-a. Extension from 2 to 4 Dimensions: Areal'''</font></center></p>
===Figure 19-b. Extension from 2 to 4 Dimensions: Bundle===
===Figure 19-b. Extension from 2 to 4 Dimensions: Bundle===
Line 2,247:
Line 2,281:
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 19-b. Extension from 2 to 4 Dimensions: Bundle'''</font></center></p>
===Figure 19-c. Extension from 2 to 4 Dimensions: Compact===
===Figure 19-c. Extension from 2 to 4 Dimensions: Compact===
Line 2,287:
Line 2,325:
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 19-c. Extension from 2 to 4 Dimensions: Compact'''</font></center></p>
===Figure 19-d. Extension from 2 to 4 Dimensions: Digraph===
===Figure 19-d. Extension from 2 to 4 Dimensions: Digraph===
Line 2,330:
Line 2,372:
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 19-d. Extension from 2 to 4 Dimensions: Digraph'''</font></center></p>
===Figure 20-i. Thematization of Conjunction (Stage 1)===
===Figure 20-i. Thematization of Conjunction (Stage 1)===
Line 2,360:
Line 2,406:
Figure 20-i. Thematization of Conjunction (Stage 1)
Figure 20-i. Thematization of Conjunction (Stage 1)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]</p>
+<p><center><font size="+1">'''Figure 20-i. Thematization of Conjunction (Stage 1)'''</font></center></p>
===Figure 20-ii. Thematization of Conjunction (Stage 2)===
===Figure 20-ii. Thematization of Conjunction (Stage 2)===
Line 2,407:
Line 2,457:
Figure 20-ii. Thematization of Conjunction (Stage 2)
Figure 20-ii. Thematization of Conjunction (Stage 2)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]</p>
+<p><center><font size="+1">'''Figure 20-ii. Thematization of Conjunction (Stage 2)'''</font></center></p>
===Figure 20-iii. Thematization of Conjunction (Stage 3)===
===Figure 20-iii. Thematization of Conjunction (Stage 3)===
Line 2,450:
Line 2,504:
Figure 20-iii. Thematization of Conjunction (Stage 3)
Figure 20-iii. Thematization of Conjunction (Stage 3)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]</p>
+<p><center><font size="+1">'''Figure 20-iii. Thematization of Conjunction (Stage 3)'''</font></center></p>
===Figure 21. Thematization of Disjunction and Equality===
===Figure 21. Thematization of Disjunction and Equality===
Line 2,516:
Line 2,574:
Figure 21. Thematization of Disjunction and Equality
Figure 21. Thematization of Disjunction and Equality
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]</p>
+<p><center><font size="+1">'''Figure 21. Thematization of Disjunction and Equality'''</font></center></p>
===Table 22. Disjunction ''f'' and Equality ''g''===
===Table 22. Disjunction ''f'' and Equality ''g''===
Line 3,673:
Line 3,735:
Figure 30. Generic Frame of a Logical Transformation
Figure 30. Generic Frame of a Logical Transformation
</pre>
</pre>
+
+'''Note.''' The following image was corrupted in transit between software platforms.
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 30 -- Generic Frame of a Logical Transformation.gif|center]]</p>
+<p><center><font size="+1">'''Figure 30. Generic Frame of a Logical Transformation'''</font></center></p>
===Formula Display 3===
===Formula Display 3===
Line 3,729:
Line 3,797:
Figure 31. Operator Diagram (1)
Figure 31. Operator Diagram (1)
</pre>
</pre>
+
+'''Note.''' The following image was corrupted in transit between software platforms.
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 31 -- Operator Diagram (1).gif|center]]</p>
+<p><center><font size="+1">'''Figure 31. Operator Diagram (1)'''</font></center></p>
===Figure 32. Operator Diagram (2)===
===Figure 32. Operator Diagram (2)===
Line 3,754:
Line 3,828:
Figure 32. Operator Diagram (2)
Figure 32. Operator Diagram (2)
</pre>
</pre>
+
+'''Note.''' The following image was corrupted in transit between software platforms.
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 32 -- Operator Diagram (2).gif|center]]</p>
+<p><center><font size="+1">'''Figure 32. Operator Diagram (2)'''</font></center></p>
===Figure 33-i. Analytic Diagram (1)===
===Figure 33-i. Analytic Diagram (1)===
Line 3,774:
Line 3,854:
Figure 33-i. Analytic Diagram (1)
Figure 33-i. Analytic Diagram (1)
</pre>
</pre>
+
+'''Note.''' The following image was corrupted in transit between software platforms.
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 33-i -- Analytic Diagram (1).gif|center]]</p>
+<p><center><font size="+1">'''Figure 33-i. Analytic Diagram (1)'''</font></center></p>
===Figure 33-ii. Analytic Diagram (2)===
===Figure 33-ii. Analytic Diagram (2)===
Line 3,794:
Line 3,880:
Figure 33-ii. Analytic Diagram (2)
Figure 33-ii. Analytic Diagram (2)
</pre>
</pre>
+
+'''Note.''' The following image was corrupted in transit between software platforms.
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 33-ii -- Analytic Diagram (2).gif|center]]</p>
+<p><center><font size="+1">'''Figure 33-ii. Analytic Diagram (2)'''</font></center></p>
===Formula Display 4===
===Formula Display 4===
Line 4,012:
Line 4,104:
Figure 34. Tangent Functor Diagram
Figure 34. Tangent Functor Diagram
</pre>
</pre>
+
+'''Note.''' The following image was corrupted in transit between software platforms.
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 34 -- Tangent Functor Diagram.gif|center]]</p>
+<p><center><font size="+1">'''Figure 34. Tangent Functor Diagram'''</font></center></p>
===Figure 35. Conjunction as Transformation===
===Figure 35. Conjunction as Transformation===
Line 4,067:
Line 4,165:
Figure 35. Conjunction as Transformation
Figure 35. Conjunction as Transformation
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]</p>
+<p><center><font size="+1">'''Figure 35. Conjunction as Transformation'''</font></center></p>
===Table 36. Computation of !e!J===
===Table 36. Computation of !e!J===
Line 4,140:
Line 4,242:
</font><br>
</font><br>
−===Figure 37-a. Tacit Extension of J (Areal)===
+===Figure 37-a. Tacit Extension of ''J'' (Areal)===
<pre>
<pre>
Line 4,183:
Line 4,285:
</pre>
</pre>
−===Figure 37-b. Tacit Extension of J (Bundle)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 37-a. Tacit Extension of ''J'' (Areal)'''</font></center></p>
+===Figure 37-b. Tacit Extension of ''J'' (Bundle)===
<pre>
<pre>
Line 4,252:
Line 4,358:
</pre>
</pre>
−===Figure 37-c. Tacit Extension of J (Compact)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 37-b. Tacit Extension of ''J'' (Bundle)'''</font></center></p>
+===Figure 37-c. Tacit Extension of ''J'' (Compact)===
<pre>
<pre>
Line 4,292:
Line 4,402:
</pre>
</pre>
−===Figure 37-d. Tacit Extension of J (Digraph)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 37-c. Tacit Extension of ''J'' (Compact)'''</font></center></p>
+===Figure 37-d. Tacit Extension of ''J'' (Digraph)===
<pre>
<pre>
Line 4,333:
Line 4,447:
Figure 37-d. Tacit Extension of J (Digraph)
Figure 37-d. Tacit Extension of J (Digraph)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 37-d. Tacit Extension of ''J'' (Digraph)'''</font></center></p>
===Table 38. Computation of EJ (Method 1)===
===Table 38. Computation of EJ (Method 1)===
Line 4,504:
Line 4,622:
</font><br>
</font><br>
−===Figure 40-a. Enlargement of J (Areal)===
+===Figure 40-a. Enlargement of ''J'' (Areal)===
<pre>
<pre>
Line 4,547:
Line 4,665:
</pre>
</pre>
−===Figure 40-b. Enlargement of J (Bundle)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 40-a. Enlargement of ''J'' (Areal)'''</font></center></p>
+===Figure 40-b. Enlargement of ''J'' (Bundle)===
<pre>
<pre>
Line 4,616:
Line 4,738:
</pre>
</pre>
−===Figure 40-c. Enlargement of J (Compact)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 40-b. Enlargement of ''J'' (Bundle)'''</font></center></p>
+===Figure 40-c. Enlargement of ''J'' (Compact)===
<pre>
<pre>
Line 4,656:
Line 4,782:
</pre>
</pre>
−===Figure 40-d. Enlargement of J (Digraph)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 40-c. Enlargement of ''J'' (Compact)'''</font></center></p>
+===Figure 40-d. Enlargement of ''J'' (Digraph)===
<pre>
<pre>
Line 4,697:
Line 4,827:
Figure 40-d. Enlargement of J (Digraph)
Figure 40-d. Enlargement of J (Digraph)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 40-d. Enlargement of ''J'' (Digraph)'''</font></center></p>
===Table 41. Computation of DJ (Method 1)===
===Table 41. Computation of DJ (Method 1)===
Line 4,964:
Line 5,098:
</font><br>
</font><br>
−===Figure 44-a. Difference Map of J (Areal)===
+===Figure 44-a. Difference Map of ''J'' (Areal)===
<pre>
<pre>
Line 5,007:
Line 5,141:
</pre>
</pre>
−===Figure 44-b. Difference Map of J (Bundle)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 44-a. Difference Map of ''J'' (Areal)'''</font></center></p>
+===Figure 44-b. Difference Map of ''J'' (Bundle)===
<pre>
<pre>
Line 5,076:
Line 5,214:
</pre>
</pre>
−===Figure 44-c. Difference Map of J (Compact)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 44-b. Difference Map of ''J'' (Bundle)'''</font></center></p>
+===Figure 44-c. Difference Map of ''J'' (Compact)===
<pre>
<pre>
Line 5,117:
Line 5,259:
</pre>
</pre>
−===Figure 44-d. Difference Map of J (Digraph)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 44-c. Difference Map of ''J'' (Compact)'''</font></center></p>
+===Figure 44-d. Difference Map of ''J'' (Digraph)===
<pre>
<pre>
Line 5,155:
Line 5,301:
Figure 44-d. Difference Map of J (Digraph)
Figure 44-d. Difference Map of J (Digraph)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 44-d. Difference Map of ''J'' (Digraph)'''</font></center></p>
===Table 45. Computation of dJ===
===Table 45. Computation of dJ===
Line 5,193:
Line 5,343:
</font><br>
</font><br>
−===Figure 46-a. Differential of J (Areal)===
+===Figure 46-a. Differential of ''J'' (Areal)===
<pre>
<pre>
Line 5,236:
Line 5,386:
</pre>
</pre>
−===Figure 46-b. Differential of J (Bundle)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 46-a. Differential of ''J'' (Areal)'''</font></center></p>
+===Figure 46-b. Differential of ''J'' (Bundle)===
<pre>
<pre>
Line 5,305:
Line 5,459:
</pre>
</pre>
−===Figure 46-c. Differential of J (Compact)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 46-b. Differential of ''J'' (Bundle)'''</font></center></p>
+===Figure 46-c. Differential of ''J'' (Compact)===
<pre>
<pre>
Line 5,342:
Line 5,500:
</pre>
</pre>
−===Figure 46-d. Differential of J (Digraph)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 46-c. Differential of ''J'' (Compact)'''</font></center></p>
+===Figure 46-d. Differential of ''J'' (Digraph)===
<pre>
<pre>
Line 5,378:
Line 5,540:
Figure 46-d. Differential of J (Digraph)
Figure 46-d. Differential of J (Digraph)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 46-d. Differential of ''J'' (Digraph)'''</font></center></p>
===Table 47. Computation of rJ===
===Table 47. Computation of rJ===
Line 5,439:
Line 5,605:
</font><br>
</font><br>
−===Figure 48-a. Remainder of J (Areal)===
+===Figure 48-a. Remainder of ''J'' (Areal)===
<pre>
<pre>
Line 5,482:
Line 5,648:
</pre>
</pre>
−===Figure 48-b. Remainder of J (Bundle)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 48-a. Remainder of ''J'' (Areal)'''</font></center></p>
+===Figure 48-b. Remainder of ''J'' (Bundle)===
<pre>
<pre>
Line 5,551:
Line 5,721:
</pre>
</pre>
−===Figure 48-c. Remainder of J (Compact)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 48-b. Remainder of ''J'' (Bundle)'''</font></center></p>
+===Figure 48-c. Remainder of ''J'' (Compact)===
<pre>
<pre>
Line 5,591:
Line 5,765:
</pre>
</pre>
−===Figure 48-d. Remainder of J (Digraph)===
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 48-c. Remainder of ''J'' (Compact)'''</font></center></p>
+===Figure 48-d. Remainder of ''J'' (Digraph)===
<pre>
<pre>
Line 5,627:
Line 5,805:
Figure 48-d. Remainder of J (Digraph)
Figure 48-d. Remainder of J (Digraph)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 48-d. Remainder of ''J'' (Digraph)'''</font></center></p>
===Table 49. Computation Summary for J===
===Table 49. Computation Summary for J===
Line 6,228:
Line 6,410:
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]</p>
+<p><center><font size="+1">'''Figure 52. Decomposition of E''J'''''</font></center></p>
===Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===
===Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===
Line 6,279:
Line 6,465:
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]</p>
+<p><center><font size="+1">'''Figure 53. Decomposition of D''J'''''</font></center></p>
===Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators===
===Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators===
Line 6,386:
Line 6,576:
| Transformation, or Mapping
| Transformation, or Mapping
| ['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
| ['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
−|-
|-
| valign="top" |
| valign="top" |
Line 6,614:
Line 6,803:
| <math>\epsilon</math> :
| <math>\epsilon</math> :
|-
|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup>, ''X''<sup> •</sup> → E''X''<sup> •</sup>,
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
Line 6,622:
Line 6,811:
| <math>\epsilon</math>''J'' :
| <math>\epsilon</math>''J'' :
|-
|-
−| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
|-
|-
−| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''B'''
+| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''B'''
|}
|}
|
|
Line 6,630:
Line 6,819:
| <math>\epsilon</math>''J'' :
| <math>\epsilon</math>''J'' :
|-
|-
−| [''u'', ''v'', d''u'', d''v''] → [''x'']
+| [''u'', ''v'', d''u'', d''v''] → [''x'']
|-
|-
−| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
|}
|}
|-
|-
Line 6,645:
Line 6,834:
| <math>\eta</math> :
| <math>\eta</math> :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
Line 6,653:
Line 6,842:
| <math>\eta</math>''J'' :
| <math>\eta</math>''J'' :
|-
|-
−|
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
−|
+| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
Line 6,661:
Line 6,850:
| <math>\eta</math>''J'' :
| <math>\eta</math>''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
Line 6,676:
Line 6,865:
| E :
| E :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
Line 6,684:
Line 6,873:
| E''J'' :
| E''J'' :
|-
|-
−|
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
−|
+| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
Line 6,692:
Line 6,881:
| E''J'' :
| E''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
Line 6,707:
Line 6,896:
| D :
| D :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
Line 6,715:
Line 6,904:
| D''J'' :
| D''J'' :
|-
|-
−|
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
−|
+| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
Line 6,723:
Line 6,912:
| D''J'' :
| D''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
Line 6,738:
Line 6,927:
| d :
| d :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
Line 6,746:
Line 6,935:
| d''J'' :
| d''J'' :
|-
|-
−|
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
−|
+| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
Line 6,754:
Line 6,943:
| d''J'' :
| d''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
Line 6,769:
Line 6,958:
| r :
| r :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
Line 6,777:
Line 6,966:
| r''J'' :
| r''J'' :
|-
|-
−|
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
−|
+| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
Line 6,785:
Line 6,974:
| r''J'' :
| r''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
Line 6,800:
Line 6,989:
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
Line 6,816:
Line 7,005:
| <font face=georgia>'''e'''</font>''J'' :
| <font face=georgia>'''e'''</font>''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}
|-
|-
Line 6,831:
Line 7,020:
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
Line 6,847:
Line 7,036:
| <font face=georgia>'''E'''</font>''J'' :
| <font face=georgia>'''E'''</font>''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}
|-
|-
Line 6,862:
Line 7,051:
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
Line 6,878:
Line 7,067:
| <font face=georgia>'''D'''</font>''J'' :
| <font face=georgia>'''D'''</font>''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}
−|-
|-
|
|
Line 6,894:
Line 7,082:
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
|-
|-
−|
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
−|
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
Line 6,902:
Line 7,090:
| d''J'' :
| d''J'' :
|-
|-
−|
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
−|
+| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
Line 6,910:
Line 7,098:
| <font face=georgia>'''T'''</font>''J'' :
| <font face=georgia>'''T'''</font>''J'' :
|-
|-
−|
+| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
−|
+| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
−|}
|}
+|}<br>
+
+===Figure 56-a1. Radius Map of the Conjunction J = uv===
<pre>
<pre>
− o
− /X\
− /XXX\
− oXXXXXo
− /X\XXX/X\
− /XXX\X/XXX\
− oXXXXXoXXXXXo
− / \XXX/X\XXX/ \
− / \X/XXX\X/ \
− o oXXXXXo o
− / \ / \XXX/ \ / \
− / \ / \X/ \ / \
− o o o o o
− =|\ / \ / \ / \ /|=
− o
− /X\
− /XXX\
− oXXXXXo
− /X\XXX/X\
− /XXX\X/XXX\
− oXXXXXoXXXXXo
− / \XXX/X\XXX/ \
− / \X/XXX\X/ \
− o oXXXXXo o
− / \ / \XXX/ \ / \
− / \ / \X/ \ / \
− o o o o o
− =|\ / \ / \ / \ /|=
= | \ / \ / \ / \ / | =
= | \ / \ / \ / \ / | =
= | o o o o | =
= | o o o o | =
Line 7,047:
Line 7,171:
Figure 56-a1. Radius Map of the Conjunction J = uv
Figure 56-a1. Radius Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-a1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-a2. Secant Map of the Conjunction J = uv===
===Figure 56-a2. Secant Map of the Conjunction J = uv===
Line 7,115:
Line 7,243:
Figure 56-a2. Secant Map of the Conjunction J = uv
Figure 56-a2. Secant Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-a2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-a3. Chord Map of the Conjunction J = uv===
===Figure 56-a3. Chord Map of the Conjunction J = uv===
Line 7,183:
Line 7,315:
Figure 56-a3. Chord Map of the Conjunction J = uv
Figure 56-a3. Chord Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-a3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-a4. Tangent Map of the Conjunction J = uv===
===Figure 56-a4. Tangent Map of the Conjunction J = uv===
Line 7,251:
Line 7,387:
Figure 56-a4. Tangent Map of the Conjunction J = uv
Figure 56-a4. Tangent Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-a4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b1. Radius Map of the Conjunction J = uv===
===Figure 56-b1. Radius Map of the Conjunction J = uv===
Line 7,351:
Line 7,491:
Figure 56-b1. Radius Map of the Conjunction J = uv
Figure 56-b1. Radius Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-b1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b2. Secant Map of the Conjunction J = uv===
===Figure 56-b2. Secant Map of the Conjunction J = uv===
Line 7,451:
Line 7,595:
Figure 56-b2. Secant Map of the Conjunction J = uv
Figure 56-b2. Secant Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-b2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b3. Chord Map of the Conjunction J = uv===
===Figure 56-b3. Chord Map of the Conjunction J = uv===
Line 7,551:
Line 7,699:
Figure 56-b3. Chord Map of the Conjunction J = uv
Figure 56-b3. Chord Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-b3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b4. Tangent Map of the Conjunction J = uv===
===Figure 56-b4. Tangent Map of the Conjunction J = uv===
Line 7,651:
Line 7,803:
Figure 56-b4. Tangent Map of the Conjunction J = uv
Figure 56-b4. Tangent Map of the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===
===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===
Line 7,721:
Line 7,877:
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 57-1. Radius Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===
===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===
Line 7,791:
Line 7,951:
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 57-2. Secant Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===
===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===
Line 7,861:
Line 8,025:
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 57-3. Chord Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===
===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===
Line 7,931:
Line 8,099:
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]</p>
+<p><center><font size="+1">'''Figure 57-4. Tangent Functor Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Formula Display 11===
===Formula Display 11===
Line 8,090:
Line 8,262:
</pre>
</pre>
−===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''
−|- style="background:paleturquoise"
−Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
+! Item
−! Notation
−| | Operator | Proposition | Transformation |
+! Description
−| | or | or | or |
+! Type
−| | Operand | Component | Mapping |
+|-
−| valign="top" | ''U''<sup> •</sup>
−| valign="top" | <font face="courier new">= </font>[''u'', ''v'']
−| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
+| valign="top" | Source Universe
−| | | | |
+| valign="top" | ['''B'''<sup>''n''</sup>]
−| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
+|-
−| | | | |
+| valign="top" | ''X''<sup> •</sup>
−| valign="top" |
−| | | | |
+{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <font face="courier new">= </font>[''x'', ''y'']
−|-
−| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
+| <font face="courier new">= </font>[''f'', ''g'']
−| | | | |
+|}
−| valign="top" | Target Universe
−| | | | |
+| valign="top" | ['''B'''<sup>''k''</sup>]
−| Trope | !h! : | !h!F_i : | !h!F : |
+|-
−| valign="top" | E''U''<sup> •</sup>
−| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
+| valign="top" | <font face="courier new">= </font>[''u'', ''v'', d''u'', d''v'']
−| | | | |
+| valign="top" | Extended Source Universe
−| valign="top" | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
−| | | | |
+|-
−| Enlargement | E : | EF_i : | EF : |
+| valign="top" | E''X''<sup> •</sup>
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
+| valign="top" |
−| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
+{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| | | | |
+| <font face="courier new">= </font>[''x'', ''y'', d''x'', d''y'']
−|-
−| | | | |
+| <font face="courier new">= </font>[''f'', ''g'', d''f'', d''g'']
−|}
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
+| valign="top" | Extended Target Universe
−| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
+| valign="top" | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
−| | | | |
+|-
−| ''F''
−| | | | |
+| ''F'' = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup>
−| 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'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
−|-
−| | | | |
+| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| | | | |
+|
−| Remainder | r : | rF_i : | rF : |
+|-
−| ''f''
−|-
−| ''g''
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| ''f'', ''g'' : ''U'' → '''B'''
−|-
−| ''f'' : ''U'' → [''x''] ⊆ ''X''<sup> •</sup>
−|-
−| ''g'' : ''U'' → [''y''] ⊆ ''X''<sup> •</sup>
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| Proposition
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
−| '''B'''<sup>''n''</sup> → '''B'''
−|-
−| ∈ ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> → '''B''')
−|-
−| = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
−|}
−|-
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| W
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| W :
−|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup> ,
−|-
−| ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
−|-
−| →
−|-
−| (E''U''<sup> •</sup> → E''X''<sup> •</sup>) ,
−|-
−| for each W in the set:
−|-
−| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| Operator
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
−|
−|-
−| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
−|-
−| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
−|-
−| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
−|-
−| →
−|-
−| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
−|-
−|
−|
|-
|-
−|
|
|}
|}
+|-
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| <math>\epsilon</math>
+|-
+| <math>\eta</math>
+|-
+| E
+|-
+| D
+|-
+| d
|}
|}
−| valign="top" |
−| colspan="2" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
−| Tacit Extension Operator || <math>\epsilon</math>
−|-
−| Trope Extension Operator || <math>\eta</math>
−|-
−| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
+| Enlargement Operator || E
−| |
+|-
−| Difference Operator || D
−|-
−| Differential Operator || d
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−|
−| ‹''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%"
+| <font face=georgia>'''W'''</font>
|}
|}
−</font><br>
+| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−<br><font face="courier new">
+| <font face=georgia>'''W'''</font> :
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|-
−|
+| ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> ,
−{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+|-
−|
+| ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> ,
−| ‹''x'', ''y''›
+|-
−| =
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
−| ''F''‹''u'', ''v''›
+|-
−| =
+| →
−| ‹((''u'')(''v'')), ((''u'', ''v''))›
+|-
+| (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) ,
+|-
+| for each <font face=georgia>'''W'''</font> in the set:
+|-
+| {<font face=georgia>'''e'''</font>, <font face=georgia>'''E'''</font>, <font face=georgia>'''D'''</font>, <font face=georgia>'''T'''</font>}
+|}
+| valign="top" |
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| Operator
+|}
+| valign="top" |
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
+|
+|-
+| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
+|-
+| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
+|-
+| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
+|-
+| →
+|-
+| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
+|-
+|
+|-
|
|
|}
|}
+|-
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| <font face=georgia>'''e'''</font>
+|-
+| <font face=georgia>'''E'''</font>
+|-
+| <font face=georgia>'''D'''</font>
+|-
+| <font face=georgia>'''T'''</font>
|}
|}
−</font><br>
+| valign="top" |
+| colspan="2" |
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
+| Radius Operator || <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>›
+|-
+| Secant Operator || <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›
+|-
+| Chord Operator || <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D›
+|-
+| Tangent Functor || <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d›
+|}
+|}<br>
−===Table 60. Propositional Transformation===
+===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
<pre>
<pre>
−Table 60. Propositional Transformation
+Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
−o-------------o-------------o-------------o-------------o
+o--------------o----------------------o--------------------o----------------------o
−| | Operator | Proposition | Transformation |
−| | or | or | or |
−| | | | |
+| | Operand | Component | Mapping |
−| 0 | 0 | 0 | 1 |
+o--------------o----------------------o--------------------o----------------------o
−| | | | |
+| | | | |
−| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
−| | | | |
−| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
−| | | | |
−o--------------o----------------------o--------------------o----------------------o
−| | | | |
−o-------------o-------------o-------------o-------------o
+| Tacit | !e! : | !e!F_i : | !e!F : |
−| | | ((u)(v)) | ((u, v)) |
+| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
−| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
−| | | | |
−o--------------o----------------------o--------------------o----------------------o
−| | | | |
−| Trope | !h! : | !h!F_i : | !h!F : |
−| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
−| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
− | U |
+| | | | |
−o--------------o----------------------o--------------------o----------------------o
−| | | | |
−| Enlargement | E : | EF_i : | EF : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
−| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
−| | | | |
−o--------------o----------------------o--------------------o----------------------o
− | | | | | |
+| | | | |
−| Difference | D : | DF_i : | DF : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
− | o o o o |
+| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
− | \ \ / / |
+| | | | |
−o--------------o----------------------o--------------------o----------------------o
−| | | | |
−| Differential | d : | dF_i : | dF : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
−| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
−| | | | |
−o--------------o----------------------o--------------------o----------------------o
−| | | | |
−| Remainder | r : | rF_i : | rF : |
−| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
−| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
−| | | | |
−o--------------o----------------------o--------------------o----------------------o
−| | | | |
−| Radius | $e$ = <!e!, !h!> : | | $e$F : |
−| Operator | | | |
−| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
−| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
−| | | | |
−| | | | [B^n x D^n] -> |
−| | | | [B^k x D^k] |
−| | | | |
−o--------------o----------------------o--------------------o----------------------o
−| ////////o///////\ | |\\\\/ o \\\\\|
+| | | | |
−| //////////\///////\ | |\\\/ /\\ \\\\|
+| Secant | $E$ = <!e!, E> : | | $E$F : |
−| o///////o///o///////o | |\\o o\\\o o\\|
+| Operator | | | |
−| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
−| o///////o///o///////o | |\\o o\\\o o\\|
+| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
−| | | | |
−| \///////o//////// | |\\\\\ o /\\\\|
+| | | | [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
− | |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
− | |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
−</pre>
</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'''
−|- style="background:paleturquoise"
−|
−| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
+| align="center" | '''Operator<br>or<br>Operand'''
−| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
+| align="center" | '''Proposition<br>or<br>Component'''
−| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
+| align="center" | '''Transformation<br>or<br>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''<sub>1</sub>, ''F''<sub>2</sub>›
−| \///////\////////// | |\\\\ \\/ /\\\|
+|-
−| \///////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%"
−| ''F''<sub>''i''</sub> : 〈''u'', ''v''〉 → '''B'''
−|-
−| ''F''<sub>''i''</sub> : '''B'''<sup>''n''</sup> → '''B'''
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
−| ''F'' : [''u'', ''v''] → [''x'', ''y'']
−|-
−| ''F'' : '''B'''<sup>''n''</sup> → '''B'''<sup>''k''</sup>
−|}
−|-
−|
−{| 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 \\\\\\| |
+|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
+|-
−| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
−| \///////////////\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''<sub>''i''</sub> :
−| |
+|-
−| |
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
−|-
−| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B'''
−</pre>
+|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <math>\epsilon</math>''F'' :
−<pre>
+|-
−| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'']
−|-
−| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
−|}
−|-
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| Trope
−|-
−| Extension
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <math>\eta</math> :
−|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <math>\eta</math>''F''<sub>''i''</sub> :
−|-
−| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
−|-
−| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''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%"
−| <math>\eta</math>''F'' :
−| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
+|-
−| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
−| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
+|-
−| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
+| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
−|}
−| \'''''''o'''''''/ | | \ | |````\ o /````|
+|-
−|
−| o---o o---o | | | \ | |``````o---o```o---o``````|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| | | | \ * |`````````````````````````|
+| Enlargement
−|-
−| Operator
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| E :
−|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| E''F''<sub>''i''</sub> :
−|-
−| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
−|-
−| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| E''F'' :
−|-
−| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
−|-
−| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
−|}
−|-
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| Difference
−|-
−| Operator
−</pre>
+|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| D :
−<pre>
+|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| D''F''<sub>''i''</sub> :
−|-
−| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
−|-
−| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| D''F'' :
−|-
−| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
−|-
−| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
−|}
−|-
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| Differential
−|-
−| Operator
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| d :
−|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| d''F''<sub>''i''</sub> :
−|-
−| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
−|-
−| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| d''F'' :
−|-
−| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
−|-
−| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
−|}
−|-
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| Remainder
−|-
−| Operator
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| r :
−|-
−| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| r''F''<sub>''i''</sub> :
−|-
−| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
−|-
−| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
−|}
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| r''F'' :
−</pre>
+|-
−| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
−|-
−| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
+|}
+|-
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| Radius
+|-
+| Operator
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
+|-
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
+|-
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+|
+|-
+|
+|-
+|
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| <font face=georgia>'''e'''</font>''F'' :
+|-
+| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
+|-
+| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
+|}
+|-
|
|
−{| align="left" border="0" cellpadding="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%"
+| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
+|-
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
+|-
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+|
+|-
+|
+|-
+|
|}
|}
−|
|
−{| align="left" border="0" cellpadding="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''<sub>''i''</sub>
+| <font face=georgia>'''E'''</font>''F'' :
−| width="4%" | =
+|-
−| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
+| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
−|-
|-
−| width="8%" |
+| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
−|}
|}
+|-
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| Chord
+|-
+| Operator
|}
|}
−|
|
−{| align="left" border="0" cellpadding="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''
+| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
−|-
|-
−| width="8%" | E''g''
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
−| width="4%" | =
+|-
−| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+|
+|-
+|
+|-
+|
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| <font face=georgia>'''D'''</font>''F'' :
+|-
+| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
+|-
+| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
+|}
+|-
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| Tangent
+|-
+| Functor
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
+|-
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
+|-
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
+|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| d''F''<sub>''i''</sub> :
+|-
+| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
+|-
+| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|}
+|
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| <font face=georgia>'''T'''</font>''F'' :
+|-
+| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
+|-
+| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|}
−|}<br>
−===Formula Display 17===
+===Formula Display 12===
<pre>
<pre>
−o-------------------------------------------------o
+o-----------------------------------------------------------o
−| |
+| |
−| Df = ((u)(v)) + ((u + du)(v + dv)) |
+| x = f(u, v) = ((u)(v)) |
−| |
+| |
−| Dg = ((u, v)) + ((u + du, v + dv)) |
+| y = g(u, v) = ((u, v)) |
−| |
+| |
−o-------------------------------------------------o
+o-----------------------------------------------------------o
</pre>
</pre>
<br><font face="courier new">
<br><font face="courier new">
−{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
|
−{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−| width="8%" | D''f''
+|
−| ''x''
−| width="20%" | ((''u'')(''v''))
+| =
−| width="4%" | +
+| ''f''‹''u'', ''v''›
−| =
+| ((''u'')(''v''))
+|
|-
|-
−| width="8%" | D''g''
+|
−| ''y''
−| width="20%" | ((''u'', ''v''))
+| =
−| width="4%" | +
+| ''g''‹''u'', ''v''›
−| =
+| ((''u'', ''v''))
+|
|}
|}
|}
|}
</font><br>
</font><br>
−===Table 66-i. Computation Summary for f‹u, v› = ((u)(v))===
+===Formula Display 13===
<pre>
<pre>
−o-----------------------------------------------------------o
−o--------------------------------------------------------------------------------o
+| |
−| |
+| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
−| !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
+| |
−o-----------------------------------------------------------o
−| |
−o--------------------------------------------------------------------------------o
</pre>
</pre>
−<font face="courier new">
+<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−|
|
−{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−| <math>\epsilon</math>''f''
+| ‹''x'', ''y''›
−| =
−| + || ''u''(''v'') || <math>\cdot</math> || 1
+| ''F''‹''u'', ''v''›
−| + || (''u'')''v'' || <math>\cdot</math> || 1
+| =
−| ‹((''u'')(''v'')), ((''u'', ''v''))›
−|}
−|}
−| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')
+</font><br>
−| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))
+<br><font face="courier new">
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−|-
+|
−| D''f''
+{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−|
−| ‹''x'', ''y''›
−| =
−| ''F''‹''u'', ''v''›
−|-
+| =
−| d''f''
+| ‹((''u'')(''v'')), ((''u'', ''v''))›
−| = || ''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'')
−| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
−| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''
−|}
|}
|}
|}
</font><br>
</font><br>
−===Table 66-ii. Computation Summary for g‹u, v› = ((u, v))===
+===Table 60. Propositional Transformation===
<pre>
<pre>
−Table 66-ii. Computation Summary for g<u, v> = ((u, v))
+Table 60. Propositional Transformation
−o--------------------------------------------------------------------------------o
+o-------------o-------------o-------------o-------------o
−| |
+| u | v | f | g |
−| !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |
+o-------------o-------------o-------------o-------------o
−| |
+| | | | |
−| Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) |
+| 0 | 0 | 0 | 1 |
−| |
+| | | | |
−| Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
+| 0 | 1 | 1 | 0 |
−| |
+| | | | |
−| dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
+| 1 | 0 | 1 | 0 |
−| |
+| | | | |
−| rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 |
+| 1 | 1 | 1 | 1 |
−| |
+| | | | |
−o--------------------------------------------------------------------------------o
+o-------------o-------------o-------------o-------------o
−</pre>
+| | | ((u)(v)) | ((u, v)) |
+o-------------o-------------o-------------o-------------o
+</pre>
<font face="courier new">
<font face="courier new">
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−|+ Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))
+|+ '''Table 60. Propositional Transformation'''
−|
+|- style="background:paleturquoise"
−| width="25%" | ''u''
−| <math>\epsilon</math>''g''
+| width="25%" | ''v''
−| = || ''uv'' || <math>\cdot</math> || 1
+| width="25%" | ''f''
−| width="25%" | ''g''
−| + || (''u'')''v'' || <math>\cdot</math> || 0
−| + || (''u'')(''v'') || <math>\cdot</math> || 1
|-
|-
−| E''g''
+| width="25%" |
−{| 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''))
|-
|-
−| D''g''
+| 0
−|-
|-
−| d''g''
+| 1
−|-
|-
−| r''g''
+| 1
−|}
|}
+| width="25%" |
+{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0
+|-
+| 1
+|-
+| 0
+|-
+| 1
+|}
+| width="25%" |
+{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0
+|-
+| 1
+|-
+| 1
+|-
+| 1
+|}
+| width="25%" |
+{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1
+|-
+| 0
+|-
+| 0
+|-
+| 1
+|}
+|-
+| width="25%" |
+| width="25%" |
+| width="25%" | ((''u'')(''v''))
+| width="25%" | ((''u'', ''v''))
|}
|}
</font><br>
</font><br>
−===Table 67. Computation of an Analytic Series in Terms of Coordinates===
+===Figure 61. Propositional Transformation===
<pre>
<pre>
− o-----------------------------------------------------o
−o--------o-------o-------o--------o-------o-------o-------o-------o
+ | U |
−| u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |
+ | |
−o--------o-------o-------o--------o-------o-------o-------o-------o
+ | o-----------o o-----------o |
−| | | | | | | | |
+ | / \ / \ |
−| 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 |
+ | / o \ |
−| | | | | | | | |
+ | / / \ \ |
−| | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
+ | / / \ \ |
−| | | | | | | | |
+ | o o o o |
−| | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
+ | | | | | |
−| | | | | | | | |
+ | | u | | v | |
− | | | | | |
− | o o o o |
− | \ \ / / |
− | \ \ / / |
−| 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
+ | \ o / |
− | \ / \ / |
−| | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
+ | o-----------o o-----------o |
−| | | | | | | | |
+ | |
−| | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
+ | |
−| | | | | | | | |
+ o-----------------------------------------------------o
−| | 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
−| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
−| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
−| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
−| ////////o///////\ | |\\\\/ o \\\\\|
−| | | | | | | | |
+| //////////\///////\ | |\\\/ /\\ \\\\|
−| | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 |
+| o///////o///o///////o | |\\o o\\\o o\\|
−| |// u //|///|// v //| | |\\| u |\\\| v |\\|
−| | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 |
+| o///////o///o///////o | |\\o o\\\o o\\|
−| | | | | | | | |
+| \///////\////////// | |\\\\ \\/ /\\\|
−o--------o-------o-------o--------o-------o-------o-------o-------o
+| \///////o//////// | |\\\\\ 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-----o------------o----------o----------o----------o----------o----------o
+ \ | | /
− \ | | /
−| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |
+ \ | | /
−| | | | | | | | |
+ \ | | /
−| 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |
+ \ | | /
−| | | | | | | | |
+ \ | | /
−| 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |
+ o-------\----|---------------------------|----/-------o
− | X \ | | / |
−| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |
+ | \| |/ |
−| | | | | | | | |
+ | o-----------o o-----------o |
− | //////////////\ /\\\\\\\\\\\\\\ |
−</pre>
+ | ////////////////o\\\\\\\\\\\\\\\\ |
− | /////////////////X\\\\\\\\\\\\\\\\\ |
−===Formula Display 18===
+ | /////////////////XXX\\\\\\\\\\\\\\\\\ |
− | o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
−<pre>
+ | |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
−o-------------------------------------------------------------------------o
+ | |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
−| |
+ | |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
−| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
+ | o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
−| |
+ | \///////////////\XXX/\\\\\\\\\\\\\\\/ |
−| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
+ | \///////////////\X/\\\\\\\\\\\\\\\/ |
−| |
+ | \///////////////o\\\\\\\\\\\\\\\/ |
−o-------------------------------------------------------------------------o
+ | \////////////// \\\\\\\\\\\\\\/ |
−</pre>
+ | o-----------o o-----------o |
− | |
−<br><font face="courier new">
+ | |
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+ o-----------------------------------------------------o
−|
+Figure 61. Propositional Transformation
−{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+</pre>
−|
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]</p>
+<p><center><font size="+1">'''Figure 61. Propositional Transformation'''</font></center></p>
+===Figure 62. Propositional Transformation (Short Form)===
+<pre>
+o-------------------------o o-------------------------o
+| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
+| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
+| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
+| ////////o///////\ | |\\\\/ o \\\\\|
+| //////////\///////\ | |\\\/ /\\ \\\\|
+| o///////o///o///////o | |\\o o\\\o o\\|
+| |// u //|///|// v //| | |\\| u |\\\| v |\\|
+| o///////o///o///////o | |\\o o\\\o o\\|
+| \///////\////////// | |\\\\ \\/ /\\\|
+| \///////o//////// | |\\\\\ o /\\\\|
+| \////// \////// | |\\\\\\ /\\ /\\\\\|
+| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
+| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
+o-------------------------o o-------------------------o
+ \ / \ /
+ \ / \ /
+ \ / \ /
+ \ f / \ g /
+ \ / \ /
+ \ / \ /
+ \ / \ /
+ \ / \ /
+ \ / \ /
+o---------\-----/---------------------\-----/---------o
+| X \ / \ / |
+| \ / \ / |
+| o-----------o o-----------o |
+| //////////////\ /\\\\\\\\\\\\\\ |
+| ////////////////o\\\\\\\\\\\\\\\\ |
+| /////////////////X\\\\\\\\\\\\\\\\\ |
+| /////////////////XXX\\\\\\\\\\\\\\\\\ |
+| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
+| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
+| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
+| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
+| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
+| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
+| \///////////////\X/\\\\\\\\\\\\\\\/ |
+| \///////////////o\\\\\\\\\\\\\\\/ |
+| \////////////// \\\\\\\\\\\\\\/ |
+| o-----------o o-----------o |
+| |
+| |
+o-----------------------------------------------------o
+Figure 62. Propositional Transformation (Short Form)
+</pre>
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]</p>
+<p><center><font size="+1">'''Figure 62. Propositional Transformation (Short Form)'''</font></center></p>
+===Figure 63. Transformation of Positions===
+<pre>
+ o-----------------------------------------------------o
+ |`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
+ |` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
+ |` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
+ |` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
+ |` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
+ |` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
+ |` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
+ |` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
+ |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
+ |` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
+ |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
+ |` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
+ |` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
+ |` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
+ |` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
+ |` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
+ |` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
+ |` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
+ |` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
+ o-----------\----|---------|---------|----------------o
+ " " \ | | | " "
+ " " \ | | | " "
+ " " \ | | | " "
+ " " \| | | " "
+o-------------------------o \ | | o-------------------------o
+| U | |\ | | |`U```````````````````````|
+| o---o o---o | | \ | | |``````o---o```o---o``````|
+| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
+| /'''''''o'''''''\ | | \ | | |````/ o \````|
+| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
+| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
+| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
+| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
+| \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|
+| \'''''''o'''''''/ | | \ | |````\ o /````|
+| \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|
+| o---o o---o | | | \ | |``````o---o```o---o``````|
+| | | | \ * |`````````````````````````|
+o-------------------------o | | \ / o-------------------------o
+ \ | | | \ / | /
+ \ ((u)(v)) | | | \/ | ((u, v)) /
+ \ | | | /\ | /
+ \ | | | / \ | /
+ \ | | | / \ | /
+ \ | | | / * | /
+ \ | | | / | | /
+ \ | | |/ | | /
+ \ | | / | | /
+ \ | | /| | | /
+ o-------\----|---|-------/-|---------|---|----/-------o
+ | X \ | | / | | | / |
+ | \| | / | | |/ |
+ | o---|----/--o | o-------|---o |
+ | /' ' | ' / ' '\|/` ` ` ` | ` `\ |
+ | / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |
+ | /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |
+ | / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |
+ | @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |
+ | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
+ | |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |
+ | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
+ | o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |
+ | \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |
+ | \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |
+ | \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |
+ | \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |
+ | o-----------o o-----------o |
+ | |
+ | |
+ o-----------------------------------------------------o
+Figure 63. Transformation of Positions
+</pre>
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]</p>
+<p><center><font size="+1">'''Figure 63. Transformation of Positions'''</font></center></p>
+===Table 64. Transformation of Positions===
+<pre>
+Table 64. Transformation of Positions
+o-----o----------o----------o-------o-------o--------o--------o-------------o
+| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |
+o-----o----------o----------o-------o-------o--------o--------o-------------o
+| | | | | | | | ^ |
+| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |
+| | | | | | | | |
+| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |
+| | | | | | | | = |
+| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |
+| | | | | | | | |
+| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |
+| | | | | | | | | |
+o-----o----------o----------o-------o-------o--------o--------o-------------o
+| | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] |
+o-----o----------o----------o-------o-------o--------o--------o-------------o
+</pre>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ '''Table 64. Transformation of Positions'''
+|- style="background:paleturquoise"
+| ''u'' ''v''
+| ''x''
+| ''y''
+| ''x'' ''y''
+| ''x'' (''y'')
+| (''x'') ''y''
+| (''x'')(''y'')
+| ''X''<sup> •</sup> = [''x'', ''y'' ]
+|-
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 0
+|-
+| 0 1
+|-
+| 1 0
+|-
+| 1 1
+|}
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0
+|-
+| 1
+|-
+| 1
+|-
+| 1
+|}
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1
+|-
+| 0
+|-
+| 0
+|-
+| 1
+|}
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0
+|-
+| 0
+|-
+| 0
+|-
+| 1
+|}
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0
+|-
+| 1
+|-
+| 1
+|-
+| 0
+|}
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1
+|-
+| 0
+|-
+| 0
+|-
+| 0
+|}
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0
+|-
+| 0
+|-
+| 0
+|-
+| 0
+|}
+| width="12%" |
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| ↑
+|-
+| ''F''
+|-
+| ‹''f'', ''g'' ›
+|-
+| ↑
+|}
+|-
+|
+| ((''u'')(''v''))
+| ((''u'', ''v''))
+| ''u'' ''v''
+| (''u'', ''v'')
+| (''u'')(''v'')
+| ( )
+| ''U''<sup> •</sup> = [''u'', ''v'' ]
+|}
+<br>
+===Table 65. Induced Transformation on Propositions===
+<pre>
+Table 65. Induced Transformation on Propositions
+o------------o---------------------------------o------------o
+| X% | <--- F = <f , g> <--- | U% |
+o------------o----------o-----------o----------o------------o
+| | u = | 1 1 0 0 | = u | |
+| | v = | 1 0 1 0 | = v | |
+| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
+| | x = | 1 1 1 0 | = f<u,v> | |
+| | y = | 1 0 0 1 | = g<u,v> | |
+o------------o----------o-----------o----------o------------o
+| | | | | |
+| f_0 | () | 0 0 0 0 | () | f_0 |
+| | | | | |
+| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |
+| | | | | |
+| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |
+| | | | | |
+| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |
+| | | | | |
+| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |
+| | | | | |
+| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |
+| | | | | |
+| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |
+| | | | | |
+| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |
+| | | | | |
+o------------o----------o-----------o----------o------------o
+| | | | | |
+| f_8 | x y | 1 0 0 0 | u v | f_8 |
+| | | | | |
+| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |
+| | | | | |
+| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |
+| | | | | |
+| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |
+| | | | | |
+| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |
+| | | | | |
+| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |
+| | | | | |
+| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |
+| | | | | |
+| f_15 | (()) | 1 1 1 1 | (()) | f_15 |
+| | | | | |
+o------------o----------o-----------o----------o------------o
+</pre>
+<br><font face="courier new">
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ Table 65. Induced Transformation on Propositions
+|- style="background:paleturquoise"
+| ''X''<sup> •</sup>
+| colspan="3" |
+{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"
+| ←
+| ''F'' = ‹''f'' , ''g''›
+| ←
+|}
+| ''U''<sup> •</sup>
+|- style="background:paleturquoise"
+| rowspan="2" | ''f''<sub>''i''</sub>‹''x'', ''y''›
+|
+{| align="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''<sub>''j''</sub>‹''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''<sub>0</sub>
+|-
+| ''f''<sub>1</sub>
+|-
+| ''f''<sub>2</sub>
+|-
+| ''f''<sub>3</sub>
+|-
+| ''f''<sub>4</sub>
+|-
+| ''f''<sub>5</sub>
+|-
+| ''f''<sub>6</sub>
+|-
+| ''f''<sub>7</sub>
+|}
+|
+{| 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''<sub>0</sub>
+|-
+| ''f''<sub>0</sub>
+|-
+| ''f''<sub>1</sub>
+|-
+| ''f''<sub>1</sub>
+|-
+| ''f''<sub>6</sub>
+|-
+| ''f''<sub>6</sub>
+|-
+| ''f''<sub>7</sub>
+|-
+| ''f''<sub>7</sub>
+|}
+|-
+|
+{| cellpadding="2" style="background:lightcyan"
+| ''f''<sub>8</sub>
+|-
+| ''f''<sub>9</sub>
+|-
+| ''f''<sub>10</sub>
+|-
+| ''f''<sub>11</sub>
+|-
+| ''f''<sub>12</sub>
+|-
+| ''f''<sub>13</sub>
+|-
+| ''f''<sub>14</sub>
+|-
+| ''f''<sub>15</sub>
+|}
+|
+{| 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''<sub>8</sub>
+|-
+| ''f''<sub>8</sub>
+|-
+| ''f''<sub>9</sub>
+|-
+| ''f''<sub>9</sub>
+|-
+| ''f''<sub>14</sub>
+|-
+| ''f''<sub>14</sub>
+|-
+| ''f''<sub>15</sub>
+|-
+| ''f''<sub>15</sub>
+|}
+|}
+</font><br>
+===Formula Display 14===
+<pre>
+o-------------------------------------------------o
+| |
+| EG_i = G_i <u + du, v + dv> |
+| |
+o-------------------------------------------------o
+</pre>
+<br><font face="courier new">
+{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
+|
+{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+| width="8%" | E''G''<sub>''i''</sub>
+| width="4%" | =
+| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
+|}
+|}
+</font><br>
+===Formula Display 15===
+<pre>
+o-------------------------------------------------o
+| |
+| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |
+| |
+| = G_i <u, v> + G_i <u + du, v + dv> |
+| |
+o-------------------------------------------------o
+</pre>
+<br><font face="courier new">
+{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
+|
+{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+| width="8%" | D''G''<sub>''i''</sub>
+| width="4%" | =
+| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
+| width="4%" | +
+| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
+|-
+| width="8%" |
+| width="4%" | =
+| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
+| width="4%" | +
+| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
+|}
+|}
+</font><br>
+===Formula Display 16===
+<pre>
+o-------------------------------------------------o
+| |
+| Ef = ((u + du)(v + dv)) |
+| |
+| Eg = ((u + du, v + dv)) |
+| |
+o-------------------------------------------------o
+</pre>
+<br><font face="courier new">
+{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
+|
+{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+| 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''))
+|}
+|}
+</font><br>
+===Formula Display 17===
+<pre>
+o-------------------------------------------------o
+| |
+| Df = ((u)(v)) + ((u + du)(v + dv)) |
+| |
+| Dg = ((u, v)) + ((u + du, v + dv)) |
+| |
+o-------------------------------------------------o
+</pre>
+<br><font face="courier new">
+{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
+|
+{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
+| width="8%" | D''f''
+| width="4%" | =
+| width="20%" | ((''u'')(''v''))
+| width="4%" | +
+| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
+|-
+| width="8%" | D''g''
+| width="4%" | =
+| width="20%" | ((''u'', ''v''))
+| width="4%" | +
+| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
+|}
+|}
+</font><br>
+===Table 66-i. Computation Summary for f‹u, v› = ((u)(v))===
+<pre>
+Table 66-i. Computation Summary for f<u, v> = ((u)(v))
+o--------------------------------------------------------------------------------o
+| |
+| !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
+| |
+| Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) |
+| |
+| Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) |
+| |
+| df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) |
+| |
+| rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv |
+| |
+o--------------------------------------------------------------------------------o
+</pre>
+<font face="courier new">
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))
+|
+{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| <math>\epsilon</math>''f''
+| = || ''uv'' || <math>\cdot</math> || 1
+| + || ''u''(''v'') || <math>\cdot</math> || 1
+| + || (''u'')''v'' || <math>\cdot</math> || 1
+| + || (''u'')(''v'') || <math>\cdot</math> || 0
+|-
+| E''f''
+| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')
+| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))
+| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')
+| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
+|-
+| D''f''
+| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
+| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')
+| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
+| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
+|-
+| d''f''
+| = || ''uv'' || <math>\cdot</math> || 0
+| + || ''u''(''v'') || <math>\cdot</math> || d''u''
+| + || (''u'')''v'' || <math>\cdot</math> || d''v''
+| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+|-
+| r''f''
+| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
+| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''
+| + || (''u'')''v'' || <math>\cdot</math> || d''u'' d''v''
+| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
+|}
+|}
+</font><br>
+===Table 66-ii. Computation Summary for g‹u, v› = ((u, v))===
+<pre>
+Table 66-ii. Computation Summary for g<u, v> = ((u, v))
+o--------------------------------------------------------------------------------o
+| |
+| !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |
+| |
+| Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) |
+| |
+| Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
+| |
+| dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
+| |
+| rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 |
+| |
+o--------------------------------------------------------------------------------o
+</pre>
+<font face="courier new">
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))
+|
+{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| <math>\epsilon</math>''g''
+| = || ''uv'' || <math>\cdot</math> || 1
+| + || ''u''(''v'') || <math>\cdot</math> || 0
+| + || (''u'')''v'' || <math>\cdot</math> || 0
+| + || (''u'')(''v'') || <math>\cdot</math> || 1
+|-
+| E''g''
+| = || ''uv'' || <math>\cdot</math> || ((d''u'', d''v''))
+| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
+| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
+|-
+| D''g''
+| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
+| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
+| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+|-
+| d''g''
+| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
+| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
+| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+|-
+| r''g''
+| = || ''uv'' || <math>\cdot</math> || 0
+| + || ''u''(''v'') || <math>\cdot</math> || 0
+| + || (''u'')''v'' || <math>\cdot</math> || 0
+| + || (''u'')(''v'') || <math>\cdot</math> || 0
+|}
+|}
+</font><br>
+===Table 67. Computation of an Analytic Series in Terms of Coordinates===
+<pre>
+Table 67. Computation of an Analytic Series in Terms of Coordinates
+o--------o-------o-------o--------o-------o-------o-------o-------o
+| u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |
+o--------o-------o-------o--------o-------o-------o-------o-------o
+| | | | | | | | |
+| 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 |
+| | | | | | | | |
+| | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
+| | | | | | | | |
+| | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
+| | | | | | | | |
+| | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 |
+| | | | | | | | |
+o--------o-------o-------o--------o-------o-------o-------o-------o
+| | | | | | | | |
+| 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
+| | | | | | | | |
+| | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
+| | | | | | | | |
+| | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
+| | | | | | | | |
+| | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 |
+| | | | | | | | |
+o--------o-------o-------o--------o-------o-------o-------o-------o
+| | | | | | | | |
+| 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
+| | | | | | | | |
+| | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
+| | | | | | | | |
+| | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
+| | | | | | | | |
+| | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 |
+| | | | | | | | |
+o--------o-------o-------o--------o-------o-------o-------o-------o
+| | | | | | | | |
+| 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 |
+| | | | | | | | |
+| | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 |
+| | | | | | | | |
+| | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 |
+| | | | | | | | |
+| | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 |
+| | | | | | | | |
+o--------o-------o-------o--------o-------o-------o-------o-------o
+</pre>
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ Table 67. Computation of an Analytic Series in Terms of Coordinates
+|
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| ''u''
+| ''v''
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| d''u''
+| d''v''
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| ''u''<font face="courier new">’</font>
+| ''v''<font face="courier new">’</font>
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 1 || 0
+|-
+| 1 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 1 || 0
+|-
+| 1 || 1
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 1 || 0
+|-
+| 1 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 1
+|-
+| 0 || 0
+|-
+| 1 || 1
+|-
+| 1 || 0
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 1 || 0
+|-
+| 1 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 0
+|-
+| 1 || 1
+|-
+| 0 || 0
+|-
+| 0 || 1
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 1
+|-
+| 1 || 0
+|-
+| 0 || 1
+|-
+| 0 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 1 || 0
+|-
+| 1 || 1
+|}
+|}
+|
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| <math>\epsilon</math>''f''
+| <math>\epsilon</math>''g''
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| E''f''
+| E''g''
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| D''f''
+| D''g''
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| d''f''
+| d''g''
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
+| d<sup>2</sup>''f''
+| d<sup>2</sup>''g''
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 1
+|-
+| 1 || 0
+|-
+| 1 || 0
+|-
+| 1 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 1 || 1
+|-
+| 1 || 1
+|-
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 1 || 1
+|-
+| 1 || 1
+|-
+| 0 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 1 || 0
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 0
+|-
+| 0 || 1
+|-
+| 1 || 1
+|-
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 1 || 1
+|-
+| 0 || 1
+|-
+| 0 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 1 || 1
+|-
+| 0 || 1
+|-
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 1 || 0
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 0
+|-
+| 1 || 1
+|-
+| 0 || 1
+|-
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 1 || 1
+|-
+| 0 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 1 || 1
+|-
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 1 || 0
+|}
+|-
+| valign="top" |
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 1 || 1
+|-
+| 1 || 0
+|-
+| 1 || 0
+|-
+| 0 || 1
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 0 || 1
+|-
+| 1 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 1
+|-
+| 0 || 1
+|-
+| 0 || 0
+|}
+|
+{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 0 || 0
+|-
+| 1 || 0
+|}
+|}
+|}
+<br>
+===Table 68. Computation of an Analytic Series in Symbolic Terms===
+<pre>
+Table 68. Computation of an Analytic Series in Symbolic Terms
+o-----o-----o------------o----------o----------o----------o----------o----------o
+| u v | f g | Df | Dg | df | dg | rf | rg |
+o-----o-----o------------o----------o----------o----------o----------o----------o
+| | | | | | | | |
+| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |
+| | | | | | | | |
+| 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |
+| | | | | | | | |
+| 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |
+| | | | | | | | |
+| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |
+| | | | | | | | |
+o-----o-----o------------o----------o----------o----------o----------o----------o
+</pre>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ '''Table 68. Computation of an Analytic Series in Symbolic Terms'''
+|- style="background:paleturquoise"
+| ''u'' ''v''
+| ''f'' ''g''
+| D''f''
+| D''g''
+| d''f''
+| d''g''
+| d<sup>2</sup>''f''
+| d<sup>2</sup>''g''
+|-
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 0
+|-
+| 0 1
+|-
+| 1 0
+|-
+| 1 1
+|}
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| 0 1
+|-
+| 1 0
+|-
+| 1 0
+|-
+| 1 1
+|}
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| ((d''u'')(d''v''))
+|-
+| (d''u'') d''v''
+|-
+| d''u'' (d''v'')
+|-
+| d''u'' d''v''
+|}
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| (d''u'', d''v'')
+|-
+| (d''u'', d''v'')
+|-
+| (d''u'', d''v'')
+|-
+| (d''u'', d''v'')
+|}
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| (d''u'', d''v'')
+|-
+| d''v''
+|-
+| d''u''
+|-
+| ( )
+|}
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| (d''u'', d''v'')
+|-
+| (d''u'', d''v'')
+|-
+| (d''u'', d''v'')
+|-
+| (d''u'', d''v'')
+|}
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| d''u'' d''v''
+|-
+| d''u'' d''v''
+|-
+| d''u'' d''v''
+|-
+| d''u'' d''v''
+|}
+|
+{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+| ( )
+|-
+| ( )
+|-
+| ( )
+|-
+| ( )
+|}
+|}
+<br>
+===Formula Display 18===
+<pre>
+o-------------------------------------------------------------------------o
+| |
+| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
+| |
+| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
+| |
+o-------------------------------------------------------------------------o
+</pre>
+<br><font face="courier new">
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|
+{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+|
|-
|-
| D''f''
| D''f''
Line 8,891:
Line 10,449:
</font><br>
</font><br>
−===Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›===
+===Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›===
<pre>
<pre>
Line 8,957:
Line 10,515:
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]</p>
+<p><center><font size="+1">'''Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›'''</font></center></p>
===Formula Display 19===
===Formula Display 19===
Line 8,995:
Line 10,557:
</font><br>
</font><br>
−===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===
+===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===
<pre>
<pre>
Line 9,080:
Line 10,642:
Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
</pre>
</pre>
+
+<br>
+<p>[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]</p>
+<p><center><font size="+1">'''Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font></center></p>
===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
−<pre>
<pre>
Line 9,263:
Line 10,827:
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
</pre>
</pre>
+
+[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|<font size="3">'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]
