Difference between revisions of "User:Jon Awbrey/ATLAS"

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o-----------o---------o------------o------------o------------o-----------o
 
o-----------o---------o------------o------------o------------o-----------o
 
</pre>
 
</pre>
 +
 +
<font face="courier new">
 +
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ Table 51.  Computation of an Analytic Series in Symbolic Terms
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| ''u'' || ''v''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| ''J''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| E''J''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| D''J''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| d''J''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
 +
| d<sup>2</sup>''J''
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" 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="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0
 +
|-
 +
| 0
 +
|-
 +
| 0
 +
|-
 +
| 1
 +
|}
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|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| &nbsp;d''u''&nbsp;&nbsp;d''v''&nbsp;
 +
|-
 +
| &nbsp;d''u''&nbsp;(d''v'')
 +
|-
 +
| (d''u'')&nbsp;d''v''&nbsp;
 +
|-
 +
| (d''u'')(d''v'')
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| &nbsp;&nbsp;d''u''&nbsp;&nbsp;d''v''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;&nbsp;d''u''&nbsp;(d''v'')&nbsp;
 +
|-
 +
| &nbsp;(d''u'')&nbsp;d''v''&nbsp;&nbsp;
 +
|-
 +
| ((d''u'')(d''v''))
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| ()
 +
|-
 +
| d''u''
 +
|-
 +
| d''v''
 +
|-
 +
| (d''u'', d''v'')
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="8" 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''
 +
|}
 +
|}
 +
</font><br>
  
 
===Figure 52.  Decomposition of the Enlarged Conjunction EJ = (J, DJ)===
 
===Figure 52.  Decomposition of the Enlarged Conjunction EJ = (J, DJ)===

Revision as of 19:20, 14 June 2007

Differential Logic and Dynamic Systems

Table 1. Syntax & Semantics of a Calculus for Propositional Logic

Table 1.  Syntax & Semantics of a Calculus for Propositional Logic
o-------------------o-------------------o-------------------o
|    Expression     |  Interpretation   |  Other Notations  |
o-------------------o-------------------o-------------------o
|  " "              | True.             |  1                |
o-------------------o-------------------o-------------------o
|  ()               | False.            |  0                |
o-------------------o-------------------o-------------------o
|  A                | A.                |  A                |
o-------------------o-------------------o-------------------o
|  (A)              | Not A.            |  A'               |
|                   |                   |  ~A               |
o-------------------o-------------------o-------------------o
|  A B C            | A and B and C.    |  A & B & C        |
o-------------------o-------------------o-------------------o
|  ((A)(B)(C))      | A or B or C.      |  A v B v C        |
o-------------------o-------------------o-------------------o
|  (A (B))          | A implies B.      |  A => B           |
|                   | If A then B.      |                   |
o-------------------o-------------------o-------------------o
|  (A, B)           | A not equal to B. |  A =/= B          |
|                   | A exclusive-or B. |  A  +  B          |
o-------------------o-------------------o-------------------o
|  ((A, B))         | A is equal to B.  |  A  =  B          |
|                   | A if & only if B. |  A <=> B          |
o-------------------o-------------------o-------------------o
|  (A, B, C)        | Just one of       |  A'B C  v         |
|                   | A, B, C           |  A B'C  v         |
|                   | is false.         |  A B C'           |
o-------------------o-------------------o-------------------o
|  ((A),(B),(C))    | Just one of       |  A B'C' v         |
|                   | A, B, C           |  A'B C' v         |
|                   | is true.          |  A'B'C            |
|                   |                   |                   |
|                   | Partition all     |                   |
|                   | into A, B, C.     |                   |
o-------------------o-------------------o-------------------o
|  ((A, B), C)      | Oddly many of     |  A + B + C        |
|  (A, (B, C))      | A, B, C           |                   |
|                   | are true.         |  A B C  v         |
|                   |                   |  A B'C' v         |
|                   |                   |  A'B C' v         |
|                   |                   |  A'B'C            |
o-------------------o-------------------o-------------------o
|  (Q, (A),(B),(C)) | Partition  Q      |  Q'A'B'C' v       |
|                   | into A, B, C.     |  Q A B'C' v       |
|                   |                   |  Q A'B C' v       |
|                   | Genus Q comprises |  Q A'B'C          |
|                   | species A, B, C.  |                   |
o-------------------o-------------------o-------------------o

Table 1. Syntax and Semantics of a Calculus for Propositional Logic
Expression Interpretation Other Notations
" " True. 1
( ) False. 0
A A. A
(A) Not A.  A’
~A
¬A
A B C A and B and C. A ∧ B ∧ C
((A)(B)(C)) A or B or C. A ∨ B ∨ C
(A (B)) A implies B.
If A then B. 		A ⇒ B
(A, B) A not equal to B.
A exclusive-or B. 		A ≠ B
A + B
((A, B)) A is equal to B.
A if & only if B. 		A = B
A ⇔ B
(A, B, C) Just one of
A, B, C
is false.

A’B C ∨
A B’C ∨
A B C’

((A),(B),(C)) Just one of
A, B, C
is true.

Partition all
into A, B, C.

A B’C’ ∨
A’B C’ ∨
A’B’C

((A, B), C)
 
(A, (B, C)) 		Oddly many of
A, B, C
are true.

A + B + C
 
A B C  ∨
A B’C’ ∨
A’B C’ ∨
A’B’C

(Q, (A),(B),(C)) Partition Q
into A, B, C.

Genus Q comprises
species A, B, C.

Q’A’B’C’ ∨
Q A B’C’ ∨
Q A’B C’ ∨
Q A’B’C


Table 2. Fundamental Notations for Propositional Calculus

Table 2.  Fundamental Notations for Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol  | Notation          | Description       | Type              |
o---------o-------------------o-------------------o-------------------o
| !A!     | {a_1, ..., a_n}   | Alphabet          | [n]  =  #n#       |
o---------o-------------------o-------------------o-------------------o
|  A_i    | {(a_i), a_i}      | Dimension i       |  B                |
o---------o-------------------o-------------------o-------------------o
|  A      | <|!A!|>           | Set of cells,     |  B^n              |
|         | <|a_i, ..., a_n|> | coordinate tuples,|                   |
|         | {<a_i, ..., a_n>} | interpretations,  |                   |
|         | A_1 x ... x A_n   | points, or vectors|                   |
|         | Prod_i A_i        | in the universe   |                   |
o---------o-------------------o-------------------o-------------------o
|  A*     | (hom : A -> B)    | Linear functions  | (B^n)*  =  B^n    |
o---------o-------------------o-------------------o-------------------o
|  A^     | (A -> B)          | Boolean functions |  B^n -> B         |
o---------o-------------------o-------------------o-------------------o
|  A%     | [!A!]             | Universe of Disc. | (B^n, (B^n -> B)) |
|         | (A, A^)           | based on features | (B^n +-> B)       |
|         | (A +-> B)         | {a_1, ..., a_n}   | [B^n]             |
|         | (A, (A -> B))     |                   |                   |
|         | [a_1, ..., a_n]   |                   |                   |
o---------o-------------------o-------------------o-------------------o

Table 2. Fundamental Notations for Propositional Calculus
Symbol Notation Description Type
A {a1, …, an} Alphabet [n] = n
Ai {(ai), ai} Dimension i B
A

〈A〉
〈a1, …, an〉
{‹a1, …, an›}
A1 × … × An
∏i Ai

Set of cells,
coordinate tuples,
points, or vectors
in the universe
of discourse

Bn
A* (hom : A → B) Linear functions (Bn)* = Bn
A^ (A → B) Boolean functions Bn → B
A•

[A]
(A, A^)
(A +→ B)
(A, (A → B))
[a1, …, an]

Universe of discourse
based on the features
{a1, …, an}

(Bn, (Bn → B))
(Bn +→ B)
[Bn]


Table 3. Analogy of Real and Boolean Types

Table 3.  Analogy of Real and Boolean Types
o-------------------------o-------------------------o-------------------------o
|      Real Domain R      |           <->           |    Boolean Domain B     |
o-------------------------o-------------------------o-------------------------o
|           R^n           |       Basic Space       |           B^n           |
o-------------------------o-------------------------o-------------------------o
|        R^n -> R         |     Function Space      |        B^n -> B         |
o-------------------------o-------------------------o-------------------------o
|     (R^n -> R) -> R     |     Tangent Vector      |     (B^n -> B) -> B     |
o-------------------------o-------------------------o-------------------------o
| R^n -> ((R^n -> R) -> R)|      Vector Field       | B^n -> ((B^n -> B) -> B)|
o-------------------------o-------------------------o-------------------------o
| (R^n x (R^n -> R)) -> R |          ditto          | (B^n x (B^n -> B)) -> B |
o-------------------------o-------------------------o-------------------------o
| ((R^n -> R) x R^n) -> R |          ditto          | ((B^n -> B) x B^n) -> B |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^n -> R)|       Derivation        | (B^n -> B) -> (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
|        R^n -> R^m       |  Basic Transformation   |        B^n -> B^m       |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)|
o-------------------------o-------------------------o-------------------------o
|           ...           |           ...           |           ...           |
o-------------------------o-------------------------o-------------------------o

Table 3. Analogy of Real and Boolean Types
Real Domain R ←→ Boolean Domain B
Rn Basic Space Bn
Rn → R Function Space Bn → B
(Rn→R) → R Tangent Vector (Bn→B) → B
Rn → ((Rn→R)→R) Vector Field Bn → ((Bn→B)→B)
(Rn × (Rn→ R)) → R ditto (Bn × (Bn→ B)) → B
((Rn→R) × Rn) → R ditto ((Bn→B) × Bn) → B
(Rn→R) → (Rn→R) Derivation (Bn→B) → (Bn→B)
Rn → Rm Basic Transformation Bn → Bm
(Rn→R) → (Rm→R) Function Transformation (Bn→B) → (Bm→B)
... ... ...


Table 4. An Equivalence Based on the Propositions as Types Analogy

Table 4.  An Equivalence Based on the Propositions as Types Analogy
o-------------------------o------------------------o--------------------------o
|         Pattern         |      Construction      |        Instance          |
o-------------------------o------------------------o--------------------------o
|      X -> (Y -> Z)      |      Vector Field      | K^n -> ((K^n -> K) -> K) |
o-------------------------o------------------------o--------------------------o
|     (X x Y)  -> Z       |                        | (K^n x (K^n -> K)) -> K  |
o-------------------------o------------------------o--------------------------o
|     (Y x X)  -> Z       |                        | ((K^n -> K) x K^n) -> K  |
o-------------------------o------------------------o--------------------------o
|      Y -> (X -> Z)      |       Derivation       | (K^n -> K) -> (K^n -> K) |
o-------------------------o------------------------o--------------------------o

Table 4. An Equivalence Based on the Propositions as Types Analogy
Pattern Construction Instance
X → (Y → Z) Vector Field Kn → ((Kn → K) → K)
(X × Y) → Z   (Kn × (Kn → K)) → K
(Y × X) → Z   ((Kn → K) × Kn) → K
Y → (X → Z) Derivation (Kn → K) → (Kn → K)


Table 5. A Bridge Over Troubled Waters

Table 5.  A Bridge Over Troubled Waters
o-------------------------o-------------------------o-------------------------o
|      Linear Space       |      Liminal Space      |      Logical Space      |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| !X!                     | !`X`!                   | !A!                     |
|                         |                         |                         |
| {x_1, ..., x_n}         | {`x`_1, ..., `x`_n}     | {a_1, ..., a_n}         |
|                         |                         |                         |
| cardinality n           | cardinality n           | cardinality n           |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X_i                     | `X`_i                   | A_i                     |
|                         |                         |                         |
| <|x_i|>                 | {(`x`_i), `x`_i}        | {(a_i), a_i}            |
|                         |                         |                         |
| isomorphic to K         | isomorphic to B         | isomorphic to B         |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X                       | `X`                     | A                       |
|                         |                         |                         |
| <|!X!|>                 | <|!`X`!|>               | <|!A!|>                 |
|                         |                         |                         |
| <|x_1, ..., x_n|>       | <|`x`_1, ..., `x`_n|>   | <|a_1, ..., a_n|>       |
|                         |                         |                         |
| {<x_1, ..., x_n>}       | {<`x`_1, ..., `x`_n>}   | {<a_1, ..., a_n>}       |
|                         |                         |                         |
| X_1 x ... x X_n         | `X`_1 x ... x `X`_n     | A_1 x ... x A_n         |
|                         |                         |                         |
| Prod_i X_i              | Prod_i `X`_i            | Prod_i A_i              |
|                         |                         |                         |
| isomorphic to K^n       | isomorphic to B^n       | isomorphic to B^n       |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X*                      | `X`*                    | A*                      |
|                         |                         |                         |
| (hom : X -> K)          | (hom : `X` -> B)        | (hom : A -> B)          |
|                         |                         |                         |
| isomorphic to K^n       | isomorphic to B^n       | isomorphic to B^n       |
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X^                      | `X`^                    | A^                      |
|                         |                         |                         |
| (X -> K)                | (`X` -> B)              | (A -> B)                |
|                         |                         |                         |
| isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
|                         |                         |                         |
| X%                      | `X`%                    | A%                      |
|                         |                         |                         |
| [!X!]                   | [!`X`!]                 | [!A!]                   |
|                         |                         |                         |
| [x_1, ..., x_n]         | [`x`_1, ..., `x`_n]     | [a_1, ..., a_n]         |
|                         |                         |                         |
| (X, X^)                 | (`X`, `X`^)             | (A, A^)                 |
|                         |                         |                         |
| (X +-> K)               | (`X` +-> B)             | (A +-> B)               |
|                         |                         |                         |
| (X, (X -> K))           | (`X`, (`X` -> B))       | (A, (A -> B))           |
|                         |                         |                         |
| isomorphic to:          | isomorphic to:          | isomorphic to:          |
|                         |                         |                         |
| (K^n, (K^n -> K))       | (B^n, (B^n -> B))       | (B^n, (B^n -> K))       |
|                         |                         |                         |
| (K^n +-> K)             | (B^n +-> B)             | (B^n +-> B)             |
|                         |                         |                         |
| [K^n]                   | [B^n]                   | [B^n]                   |
o-------------------------o-------------------------o-------------------------o

Table 5. A Bridge Over Troubled Waters
Linear Space Liminal Space Logical Space

X
{x1, …, xn}
cardinality n

X
{x1, …, xn}
cardinality n

A
{a1, …, an}
cardinality n

Xi
〈xi〉
isomorphic to K

Xi
{(xi), xi}
isomorphic to B

Ai
{(ai), ai}
isomorphic to B

X
〈X〉
〈x1, …, xn〉
{‹x1, …, xn›}
X1 × … × Xn
∏i Xi
isomorphic to Kn

X
〈X〉
〈x1, …, xn〉
{‹x1, …, xn›}
X1 × … × Xn
∏i Xi
isomorphic to Bn

A
〈A〉
〈a1, …, an〉
{‹a1, …, an›}
A1 × … × An
∏i Ai
isomorphic to Bn

X*
(hom : X → K)
isomorphic to Kn

X*
(hom : X → B)
isomorphic to Bn

A*
(hom : A → B)
isomorphic to Bn

X^
(X → K)
isomorphic to:
(Kn → K)

X^
(X → B)
isomorphic to:
(Bn → B)

A^
(A → B)
isomorphic to:
(Bn → B)

X•
[X]
[x1, …, xn]
(X, X^)
(X +→ K)
(X, (X → K))
isomorphic to:
(Kn, (Kn → K))
(Kn +→ K)
[Kn]

X•
[X]
[x1, …, xn]
(X, X^)
(X +→ B)
(X, (X → B))
isomorphic to:
(Bn, (Bn → B))
(Bn +→ B)
[Bn]

A•
[A]
[a1, …, an]
(A, A^)
(A +→ B)
(A, (A → B))
isomorphic to:
(Bn, (Bn → B))
(Bn +→ B)
[Bn]


Table 6. Propositional Forms on One Variable

Table 6.  Propositional Forms on One Variable
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x :   1 0   |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     |  f_00   |   0 0   |   ( )    | false            |    0     |
|         |         |         |          |                  |          |
| f_1     |  f_01   |   0 1   |   (x)    | not x            |   ~x     |
|         |         |         |          |                  |          |
| f_2     |  f_10   |   1 0   |    x     | x                |    x     |
|         |         |         |          |                  |          |
| f_3     |  f_11   |   1 1   |  (( ))   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
Table 6. Propositional Forms on One Variable
L1
Decimal 		L2
Binary 		L3
Vector 		L4
Cactus 		L5
English 		L6
Ordinary
  x : 1 0      
f0 f00 0 0 ( ) false 0
f1 f01 0 1 (x) not x ~x
f2 f10 1 0 x x x
f3 f11 1 1 (( )) true 1


Table 7. Propositional Forms on Two Variables

Table 7.  Propositional Forms on Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x : 1 1 0 0 |          |                  |          |
|         |       y : 1 0 1 0 |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
|         |         |         |          |                  |          |
| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
|         |         |         |          |                  |          |
| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
|         |         |         |          |                  |          |
| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
|         |         |         |          |                  |          |
| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
|         |         |         |          |                  |          |
| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
|         |         |         |          |                  |          |
| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
|         |         |         |          |                  |          |
| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
|         |         |         |          |                  |          |
| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
|         |         |         |          |                  |          |
| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
|         |         |         |          |                  |          |
| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
|         |         |         |          |                  |          |
| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
|         |         |         |          |                  |          |
| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
|         |         |         |          |                  |          |
| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
|         |         |         |          |                  |          |
| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
|         |         |         |          |                  |          |
| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
Table 7. Propositional Forms on Two Variables
L1
Decimal 		L2
Binary 		L3
Vector 		L4
Cactus 		L5
English 		L6
Ordinary
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x → y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1


Table 8. Notation for the Differential Extension of Propositional Calculus

Table 8.  Notation for the Differential Extension of Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol  | Notation          | Description       | Type              |
o---------o-------------------o-------------------o-------------------o
| d!A!    | {da_1, ..., da_n} | Alphabet of       | [n]  =  #n#       |
|         |                   | differential      |                   |
|         |                   | features          |                   |
o---------o-------------------o-------------------o-------------------o
| dA_i    | {(da_i), da_i}    | Differential      |  D                |
|         |                   | dimension i       |                   |
o---------o-------------------o-------------------o-------------------o
| dA      | <|d!A!|>          | Tangent space     |  D^n              |
|         | <|da_i,...,da_n|> | at a point:       |                   |
|         | {<da_i,...,da_n>} | Set of changes,   |                   |
|         | dA_1 x ... x dA_n | motions, steps,   |                   |
|         | Prod_i dA_i       | tangent vectors   |                   |
|         |                   | at a point        |                   |
o---------o-------------------o-------------------o-------------------o
| dA*     | (hom : dA -> B)   | Linear functions  | (D^n)*  ~=~  D^n  |
|         |                   | on dA             |                   |
o---------o-------------------o-------------------o-------------------o
| dA^     | (dA -> B)         | Boolean functions |  D^n -> B         |
|         |                   | on dA             |                   |
o---------o-------------------o-------------------o-------------------o
| dA%     | [d!A!]            | Tangent universe  | (D^n, (D^n -> B)) |
|         | (dA, dA^)         | at a point of A%, | (D^n +-> B)       |
|         | (dA +-> B)        | based on the      | [D^n]             |
|         | (dA, (dA -> B))   | tangent features  |                   |
|         | [da_1, ..., da_n] | {da_1, ..., da_n} |                   |
o---------o-------------------o-------------------o-------------------o

Table 8. Notation for the Differential Extension of Propositional Calculus
Symbol Notation Description Type
dA {da1, …, dan}

Alphabet of
differential
features

[n] = n
dAi {(dai), dai}

Differential
dimension i

D
dA

〈dA〉
〈da1, …, dan〉
{‹da1, …, dan›}
dA1 × … × dAn
∏i dAi

Tangent space
at a point:
Set of changes,
motions, steps,
tangent vectors
at a point

Dn
dA* (hom : dA → B)

Linear functions
on dA

(Dn)* = Dn
dA^ (dA → B)

Boolean functions
on dA

Dn → B
dA•

[dA]
(dA, dA^)
(dA +→ B)
(dA, (dA → B))
[da1, …, dan]

Tangent universe
at a point of A•,
based on the
tangent features
{da1, …, dan}

(Dn, (Dn → B))
(Dn +→ B)
[Dn]


Table 9. Higher Order Differential Features

Table 9.  Higher Order Differential Features
o----------------------------------------o----------------------------------------o
|                                        |                                        |
| !A!   = d^0.!A! = {a_1, ..., a_n}      | E^0.!A!  = d^0.!A!                     |
|                                        |                                        |
| d!A!  = d^1.!A! = {da_1, ..., da_n}    | E^1.!A!  = d^0.!A! |_| d^1.!A!         |
|                                        |                                        |
|         d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A!  = d^0.!A! |_| ... |_| d^k.!A! |
|                                        |                                        |
| d*!A! = {d^0.!A!, ..., d^k.!A!, ...}   | E^oo.!A! = |_| d*!A!                       |
|                                        |                                        |
o----------------------------------------o----------------------------------------o

Table 9. Higher Order Differential Features

A = d0A = {a1, …, an}

dA = d1A = {da1, …, dan}

dkA = {dka1, …, dkan}

d*A = {d0A, …, dkA, …}

E0A = d0A

E1A = d0A ∪ d1A

EkA = d0A ∪ … ∪ dkA

E∞A = ∪ d*A


Table 9. Higher Order Differential Features
A = d0A = {a1, …, an}
dA = d1A = {da1, …, dan}
    dkA = {dka1, …, dkan}
d*A = {d0A, …, dkA, …}
E0A = d0A
E1A = d0A ∪ d1A
EkA = d0A ∪ … ∪ dkA
E∞A = ∪ d*A


Table 10. A Realm of Intentional Features

Table 10.  A Realm of Intentional Features
o---------------------------------------o----------------------------------------o
|                                       |                                        |
| p^0.!A!  =  !A!  =  {a_1, ..., a_n}   | Q^0.!A!  =  !A!                        |
|                                       |                                        |
| p^1.!A!  =  !A!' =  {a_1', ..., a_n'} | Q^1.!A!  =  !A! |_| !A!'               |
|                                       |                                        |
| p^2.!A!  =  !A!" =  {a_1", ..., a_n"} | Q^2.!A!  =  !A! |_| !A!' |_| !A!"      |
|                                       |                                        |
| ...         ...     ...               | ...         ...                        |
|                                       |                                        |
| p^k.!A!  =  {p^k.a_1, ..., p^k.a_n}   | Q^k.!A!  =  !A! |_| ... |_| p^k.!A!    |
|                                       |                                        |
o---------------------------------------o----------------------------------------o

Table 10. A Realm of Intentional Features
p0A = A = {a1 , …, an }
p1A = A′ = {a1′, …, an′}
p2A = A″ = {a1″, …, an″}
...       ...
pkA =     {pka1, …, pkan}
Q0A = A
Q1A = A ∪ A′
Q2A = A ∪ A′ ∪ A″
...   ...
QkA = A ∪ A′ ∪ … ∪ pkA


Formula Display 1

o-------------------------------------------------o
|                                                 |
|      From  (A) & (dA)  infer  (A)  next.        |
|                                                 |
|      From  (A) &  dA   infer   A   next.        |
|                                                 |
|      From   A  & (dA)  infer   A   next.        |
|                                                 |
|      From   A  &  dA   infer  (A)  next.        |
|                                                 |
o-------------------------------------------------o


  From (A) and (dA) infer (A) next.  
  From (A) and dA infer A next.  
  From A and (dA) infer A next.  
  From A and dA infer (A) next.  


Table 11. A Pair of Commodious Trajectories

Table 11.  A Pair of Commodious Trajectories
o---------o-------------------o-------------------o
| Time    | Trajectory 1      | Trajectory 2      |
o---------o-------------------o-------------------o
|         |                   |                   |
| 0       |  A   dA  (d^2.A)  | (A) (dA)  d^2.A   |
|         |                   |                   |
| 1       | (A)  dA   d^2.A   | (A)  dA   d^2.A   |
|         |                   |                   |
| 2       |  A  (dA) (d^2.A)  |  A  (dA) (d^2.A)  |
|         |                   |                   |
| 3       |  A  (dA) (d^2.A)  |  A  (dA) (d^2.A)  |
|         |                   |                   |
| 4       |  "    "    "      |  "    "    "      |
|         |                   |                   |
o---------o-------------------o-------------------o

Table 11. A Pair of Commodious Trajectories
Time Trajectory 1 Trajectory 2
0
1
2
3
4
A dA (d2A)
(A) dA d2A
A (dA) (d2A)
A (dA) (d2A)
" " "
(A) (dA) d2A
(A) dA d2A
A (dA) (d2A)
A (dA) (d2A)
" " "


Figure 12. The Anchor

o-------------------------------------------------o
| E^2.X                                           |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /        A        \               |
|              /                   \              |
|             /         ->-         \             |
|            o         /   \         o            |
|            |         \   /         |            |
|            |          -o-          |            |
|            |           ^           |            |
|        o---o---------o | o---------o---o        |
|       /     \         \|/         /     \       |
|      /       \    o    |         /       \      |
|     /         \   |   /|\       /         \     |
|    /           \  |  / | \     /           \    |
|   o             o-|-o--|--o---o             o   |
|   |               | |  |  |                 |   |
|   |               ---->o<----o              |   |
|   |                 |     |                 |   |
|   o       dA        o     o      d^2.A      o   |
|    \                 \   /                 /    |
|     \                 \ /                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 12.  The Anchor

Figure 13. The Tiller

o-------------------------------------------------o
|                                                 |
|                                   ->-           |
|                                  /   \          |
|                                  \   /          |
|                 o-------------o   -o-           |
|                /               \  ^             |
|               /       dA        \/         A    |
|              /                  /\              |
|             /                  /  \             |
|            o    o             /    o            |
|            |     \           /     |            |
|            |      \         /      |            |
o------------|-------\-------/-------|------------o
|            |        \     /        |            |
|            |         \   /         |            |
|            o          v /          o            |
|             \          o          /             |
|              \         ^         /              |
|               \        |        /        d^2.A  |
|                \       |       /                |
|                 o------|------o                 |
|                        |                        |
|                        |                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o
Figure 13.  The Tiller

Table 14. Differential Propositions

Table 14.  Differential Propositions
o-------o--------o---------o-----------o-------------------o----------o
|       |      A : 1 1 0 0 |           |                   |          |
|       |     dA : 1 0 1 0 |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
| f_0   | g_0    | 0 0 0 0 |    ()     | False             |    0     |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_1    | 0 0 0 1 |  (A)(dA)  | Neither A nor dA  | ~A & ~dA |
|       |        |         |           |                   |          |
|       | g_2    | 0 0 1 0 |  (A) dA   | Not A but dA      | ~A &  dA |
|       |        |         |           |                   |          |
|       | g_4    | 0 1 0 0 |   A (dA)  | A but not dA      |  A & ~dA |
|       |        |         |           |                   |          |
|       | g_8    | 1 0 0 0 |   A  dA   | A and dA          |  A &  dA |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
| f_1   | g_3    | 0 0 1 1 |  (A)      | Not A             | ~A       |
|       |        |         |           |                   |          |
| f_2   | g_12   | 1 1 0 0 |   A       | A                 |  A       |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_6    | 0 1 1 0 |  (A, dA)  | A not equal to dA |  A + dA  |
|       |        |         |           |                   |          |
|       | g_9    | 1 0 0 1 | ((A, dA)) | A equal to dA     |  A = dA  |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_5    | 0 1 0 1 |     (dA)  | Not dA            |      ~dA |
|       |        |         |           |                   |          |
|       | g_10   | 1 0 1 0 |      dA   | dA                |       dA |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
|       | g_7    | 0 1 1 1 |  (A  dA)  | Not both A and dA | ~A v ~dA |
|       |        |         |           |                   |          |
|       | g_11   | 1 0 1 1 |  (A (dA)) | Not A without dA  |  A => dA |
|       |        |         |           |                   |          |
|       | g_13   | 1 1 0 1 | ((A) dA)  | Not dA without A  |  A <= dA |
|       |        |         |           |                   |          |
|       | g_14   | 1 1 1 0 | ((A)(dA)) | A or dA           |  A v  dA |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
|       |        |         |           |                   |          |
| f_3   | g_15   | 1 1 1 1 |   (())    | True              |    1     |
|       |        |         |           |                   |          |
o-------o--------o---------o-----------o-------------------o----------o
Table 14. Differential Propositions
  A : 1 1 0 0      
  dA : 1 0 1 0      
f0 g0 0 0 0 0 ( ) False 0
  g1 0 0 0 1 (A)(dA) Neither A nor dA ¬A ∧ ¬dA
  g2 0 0 1 0 (A) dA Not A but dA ¬A ∧ dA
  g4 0 1 0 0 A (dA) A but not dA A ∧ ¬dA
  g8 1 0 0 0 A dA A and dA A ∧ dA
f1 g3 0 0 1 1 (A) Not A ¬A
f2 g12 1 1 0 0 A A A
  g6 0 1 1 0 (A, dA) A not equal to dA A ≠ dA
  g9 1 0 0 1 ((A, dA)) A equal to dA A = dA
  g5 0 1 0 1 (dA) Not dA ¬dA
  g10 1 0 1 0 dA dA dA
  g7 0 1 1 1 (A dA) Not both A and dA ¬A ∨ ¬dA
  g11 1 0 1 1 (A (dA)) Not A without dA A → dA
  g13 1 1 0 1 ((A) dA) Not dA without A A ← dA
  g14 1 1 1 0 ((A)(dA)) A or dA A ∨ dA
f3 g15 1 1 1 1 (( )) True 1


Table 14. Differential Propositions
  A : 1 1 0 0      
  dA : 1 0 1 0      
f0 g0 0 0 0 0 ( ) False 0

 
 
 
 

g1
g2
g4
g8

0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0

(A)(dA)
(A) dA
A (dA)
A dA

Neither A nor dA
Not A but dA
A but not dA
A and dA

¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA

f1
f2

g3
g12

0 0 1 1
1 1 0 0

(A)
A

Not A
A

¬A
A

 
 

g6
g9

0 1 1 0
1 0 0 1

(A, dA)
((A, dA))

A not equal to dA
A equal to dA

A ≠ dA
A = dA

 
 

g5
g10

0 1 0 1
1 0 1 0

(dA)
dA

Not dA
dA

¬dA
dA

 
 
 
 

g7
g11
g13
g14

0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0

(A dA)
(A (dA))
((A) dA)
((A)(dA))

Not both A and dA
Not A without dA
Not dA without A
A or dA

¬A ∨ ¬dA
A → dA
A ← dA
A ∨ dA

f3 g15 1 1 1 1 (( )) True 1


Table 15. Tacit Extension of [A] to [A, dA]

Table 15.  Tacit Extension of [A] to [A, dA]
o---------------------------------------------------------------------o
|                                                                     |
|    0    =      0  . ((dA), dA)        =              0              |
|                                                                     |
|   (A)   =     (A) . ((dA), dA)        =      (A)(dA) + (A) dA       |
|                                                                     |
|    A    =      A  . ((dA), dA)        =       A (dA) +  A  dA       |
|                                                                     |
|    1    =      1  . ((dA), dA)        =              1              |
|                                                                     |
o---------------------------------------------------------------------o

Table 15. Tacit Extension of [A] to [A, dA]
  0 = 0 · ((dA), dA) = 0  
  (A) = (A) · ((dA), dA) = (A)(dA) + (A) dA   
  A = A · ((dA), dA) =  A (dA) +  A  dA   
  1 = 1 · ((dA), dA) = 1


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

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                       / \                       |
|                      /   \                      |
|                     /     \                     |
|                    /       \                    |
|                   o         o                   |
|                  / \       / \                  |
|                 /   \     /   \                 |
|                /     \   /     \                |
|               /       \ /       \               |
|              o         o         o              |
|             / \       / \       / \             |
|            /   \     /   \     /   \            |
|           /     \   /     \   /     \           |
|          /       \ /       \ /       \          |
|         o    5    o    7    o         o         |
|        / \  ^|   / \  ^|   / \       / \        |
|       /   \/ |  /   \/ |  /   \     /   \       |
|      /    /\ | /    /\ | /     \   /     \      |
|     /    /  \|/    /  \|/       \ /       \     |
|    o    4<---|----/----|----3    o         o    |
|    |\       /|\  /    /|\  ^    / \       /|    |
|    | \     / | \/    / | \/    /   \     / |    |
|    |  \   /  | /\   /  | /\   /     \   /  |    |
|    |   \ /   v/  \ /   |/  \ /       \ /   |    |
|    |    o    6    o    |    o         o    |    |
|    |    |\       / \  /|   / \       /|    |    |
|    |    | \     /   \/ |  /   \     / |    |    |
|    |    |  \   /    /\ | /     \   /  |    |    |
|    | d^0.A  \ /    /  \|/       \ /  d^1.A |    |
|    o----+----o    2<---|----1    o----+----o    |
|         |     \       /|\  ^    /     |         |
|         |      \     / | \/    /      |         |
|         |       \   /  | /\   /       |         |
|         | d^2.A  \ /   v/  \ /  d^3.A |         |
|         o---------o    0    o---------o         |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        o                        |
|                                                 |
o-------------------------------------------------o
Figure 16-a.  A Couple of Fourth Gear Orbits:  1

Figure 16-b. A Couple of Fourth Gear Orbits: 2

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                       / \                       |
|                      /   \                      |
|                     /     \                     |
|                    /       \                    |
|                   o    0    o                   |
|                  / \       / \                  |
|                 /   \     /   \                 |
|                /     \   /     \                |
|               /       \ /       \               |
|              o    5    o    2    o              |
|             / \       / \       / \             |
|            /   \     /   \     /   \            |
|           /     \   /     \   /     \           |
|          /       \ /       \ /       \          |
|         o         o         o    6    o         |
|        / \       / \       / \       / \        |
|       /   \     /   \     /   \     /   \       |
|      /     \   /     \   /     \   /     \      |
|     /       \ /       \ /       \ /       \     |
|    o         o    7    o         o    4    o    |
|    |\       / \       / \       / \       /|    |
|    | \     /   \     /   \     /   \     / |    |
|    |  \   /     \   /     \   /     \   /  |    |
|    |   \ /       \ /       \ /       \ /   |    |
|    |    o         o    3    o    1    o    |    |
|    |    |\       / \       / \       /|    |    |
|    |    | \     /   \     /   \     / |    |    |
|    |    |  \   /     \   /     \   /  |    |    |
|    | d^0.A  \ /       \ /       \ /  d^1.A |    |
|    o----+----o         o         o----+----o    |
|         |     \       / \       /     |         |
|         |      \     /   \     /      |         |
|         |       \   /     \   /       |         |
|         | d^2.A  \ /       \ /  d^3.A |         |
|         o---------o         o---------o         |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        o                        |
|                                                 |
o-------------------------------------------------o
Figure 16-b.  A Couple of Fourth Gear Orbits:  2

Formula Display 2

o-------------------------------------------------------------------------------o
|                                                                               |
|  r(q)    =   Sum_k d_k . 2^(-k)          =   Sum_k d^k.A(q) . 2^(-k)          |
|                                                                               |
|  =                                                                            |
|                                                                               |
|  s(q)/t  =  (Sum_k d_k . 2^(m-k)) / 2^m  =  (Sum_k d^k.A(q) . 2^(m-k)) / 2^m  |
|                                                                               |
o-------------------------------------------------------------------------------o


r(q) = ∑k dk · 2-k = ∑k dkA(q) · 2-k
=
s(q)/t = (∑k dk · 2(m-k)) / 2m = (∑k dkA(q) · 2(m-k)) / 2m



\(r(q)\!\) \(=\) \(\sum_k d_k \cdot 2^{-k}\) \(=\) \(\sum_k \mbox{d}^k A(q) \cdot 2^{-k}\)
\(=\)
\(\frac{s(q)}{t}\) \(=\) \(\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}\) \(=\) \(\frac{\sum_k \mbox{d}^k A(q) \cdot 2^{(m-k)}}{2^m}\)


Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1

Table 17-a.  A Couple of Orbits in Fourth Gear:  Orbit 1
o---------o---------o---------o---------o---------o---------o---------o
| Time    | State   |    A    |   dA    |         |         |         |
|  p_i    |  q_j    |  d^0.A  |  d^1.A  |  d^2.A  |  d^3.A  |  d^4.A  |
o---------o---------o---------o---------o---------o---------o---------o
|         |         |                                                 |
|  p_0    |  q_01   |    0.        0         0         0         1    |
|         |         |                                                 |
|  p_1    |  q_03   |    0.        0         0         1         1    |
|         |         |                                                 |
|  p_2    |  q_05   |    0.        0         1         0         1    |
|         |         |                                                 |
|  p_3    |  q_15   |    0.        1         1         1         1    |
|         |         |                                                 |
|  p_4    |  q_17   |    1.        0         0         0         1    |
|         |         |                                                 |
|  p_5    |  q_19   |    1.        0         0         1         1    |
|         |         |                                                 |
|  p_6    |  q_21   |    1.        0         1         0         1    |
|         |         |                                                 |
|  p_7    |  q_31   |    1.        1         1         1         1    |
|         |         |                                                 |
o---------o---------o---------o---------o---------o---------o---------o
Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1
Time State A dA      
pi qj d0A d1A d2A d3A d4A
p0
p1
p2
p3
p4
p5
p6
p7
q01
q03
q05
q15
q17
q19
q21
q31
0. 0 0 0 1
0. 0 0 1 1
0. 0 1 0 1
0. 1 1 1 1
1. 0 0 0 1
1. 0 0 1 1
1. 0 1 0 1
1. 1 1 1 1


Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2

Table 17-b.  A Couple of Orbits in Fourth Gear:  Orbit 2
o---------o---------o---------o---------o---------o---------o---------o
| Time    | State   |    A    |   dA    |         |         |         |
|  p_i    |  q_j    |  d^0.A  |  d^1.A  |  d^2.A  |  d^3.A  |  d^4.A  |
o---------o---------o---------o---------o---------o---------o---------o
|         |         |                                                 |
|  p_0    |  q_25   |    1.        1         0         0         1    |
|         |         |                                                 |
|  p_1    |  q_11   |    0.        1         0         1         1    |
|         |         |                                                 |
|  p_2    |  q_29   |    1.        1         1         0         1    |
|         |         |                                                 |
|  p_3    |  q_07   |    0.        0         1         1         1    |
|         |         |                                                 |
|  p_4    |  q_09   |    0.        1         0         0         1    |
|         |         |                                                 |
|  p_5    |  q_27   |    1.        1         0         1         1    |
|         |         |                                                 |
|  p_6    |  q_13   |    0.        1         1         0         1    |
|         |         |                                                 |
|  p_7    |  q_23   |    1.        0         1         1         1    |
|         |         |                                                 |
o---------o---------o---------o---------o---------o---------o---------o
Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2
Time State A dA      
pi qj d0A d1A d2A d3A d4A
p0
p1
p2
p3
p4
p5
p6
p7
q25
q11
q29
q07
q09
q27
q13
q23
1. 1 0 0 1
0. 1 0 1 1
1. 1 1 0 1
0. 0 1 1 1
0. 1 0 0 1
1. 1 0 1 1
0. 1 1 0 1
1. 0 1 1 1


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

o-----------------------------------------------------------o
|                                                           |
|              o                             o              |
|             / \                           / \             |
|            /   \                         /   \            |
|           /     \                       /     \           |
|          /       \                     /       \          |
|         /         o                   o   1 1   o         |
|        /         / \                 / \       / \        |
|       /         /   \               /   \     /   \       |
|      /    1    /     \             /     \   /     \      |
|     /         /       \    !e!    /       \ /       \     |
|    o         /         o  ---->  o   1 0   o   0 1   o    |
|    |\       /         /          |\       / \       /|    |
|    | \     /    0    /           | \     /   \     / |    |
|    |  \   /         /            |  \   /     \   /  |    |
|    |x_1\ /         /             |x_1\ /       \ /x_2|    |
|    o----o         /              o----o   0 0   o----o    |
|          \       /                     \       /          |
|           \     /                       \     /           |
|            \   /                         \   /            |
|             \ /                           \ /             |
|              o                             o              |
|                                                           |
o-----------------------------------------------------------o
Figure 18-a.  Extension from 1 to 2 Dimensions:  Areal

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

o-----------------------------o         o-------------------o
|                             |         |                   |
|                             |         |     o-------o     |
|         o---------o         |         |    /         \    |
|        /           \        |         |   o           o   |
|       /      o------------------------|   |    dx     |   |
|      /               \      |         |   o           o   |
|     /                 \     |         |    \         /    |
|    o                   o    |         |     o-------o     |
|    |                   |    |         |                   |
|    |                   |    |         o-------------------o
|    |         x         |    |
|    |                   |    |         o-------------------o
|    |                   |    |         |                   |
|    o                   o    |         |     o-------o     |
|     \                 /     |         |    /         \    |
|      \               /      |         |   o           o   |
|       \             /    o------------|   |    dx     |   |
|        \           /        |         |   o           o   |
|         o---------o         |         |    \         /    |
|                             |         |     o-------o     |
|                             |         |                   |
o-----------------------------o         o-------------------o
Figure 18-b.  Extension from 1 to 2 Dimensions:  Bundle

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

o-----------------------------------------------------------o
|                                                           |
|                                                           |
|               o-----------------o                         |
|              /         o         \                        |
|             /    (dx) / \         \ dx                    |
|            /         v   o--------------------->o         |
|           /           \ /           \                     |
|          /             o             \                    |
|         o                             o                   |
|         |                             |                   |
|         |                             |                   |
|         |              x              |        (x)        |
|         |                             |                   |
|         |                             |                   |
|         o                             o                   |
|          \                           /          o         |
|           \                         /          / \        |
|            \           o<---------------------o   v       |
|             \                     / dx         \ / (dx)   |
|              \                   /              o         |
|               o-----------------o                         |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 18-c.  Extension from 1 to 2 Dimensions:  Compact

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

o-----------------------------------------------------------o
|                                                           |
|                                                           |
|                            dx                             |
|           .--.   .---------->----------.   .--.           |
|           |   \ /                       \ /   |           |
|     (dx)  ^    @  x                 (x)  @    v  (dx)     |
|           |   / \                       / \   |           |
|           *--*   *----------<----------*   *--*           |
|                             dx                            |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 18-d.  Extension from 1 to 2 Dimensions:  Digraph

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

o-------------------------------------------------------------------------------o
|                                                                               |
|                   o                                       o                   |
|                  / \                                     / \                  |
|                 /   \                                   /   \                 |
|                /     \                                 /     \                |
|               /       \                               o 1100  o               |
|              /         \                             / \     / \              |
|             /           \                           /   \   /   \             |
|            /             \           !e!           /     \ /     \            |
|           o      1 1      o         ---->         o 1101  o 1110  o           |
|          / \             / \                     / \     / \     / \          |
|         /   \           /   \                   /   \   /   \   /   \         |
|        /     \         /     \                 /     \ /     \ /     \        |
|       /       \       /       \               o 1001  o 1111  o 0110  o       |
|      /         \     /         \             / \     / \     / \     / \      |
|     /           \   /           \           /   \   /   \   /   \   /   \     |
|    /             \ /             \         /     \ /     \ /     \ /     \    |
|   o      1 0      o      0 1      o       o 1000  o 1011  o 0111  o 0100  o   |
|   |\             / \             /|       |\     / \     / \     / \     /|   |
|   | \           /   \           / |       | \   /   \   /   \   /   \   / |   |
|   |  \         /     \         /  |       |  \ /     \ /     \ /     \ /  |   |
|   |   \       /       \       /   |       |   o 1010  o 0011  o 0101  o   |   |
|   |    \     /         \     /    |       |   |\     / \     / \     /|   |   |
|   |     \   /           \   /     |       |   | \   /   \   /   \   / |   |   |
|   | x_1  \ /             \ /  x_2 |       |x_1|  \ /     \ /     \ /  |x_2|   |
|   o-------o      0 0      o-------o       o---+---o 0010  o 0001  o---+---o   |
|            \             /                    |    \     / \     /    |       |
|             \           /                     |     \   /   \   /     |       |
|              \         /                      | x_3  \ /     \ /  x_4 |       |
|               \       /                       o-------o 0000  o-------o       |
|                \     /                                 \     /                |
|                 \   /                                   \   /                 |
|                  \ /                                     \ /                  |
|                   o                                       o                   |
|                                                                               |
o-------------------------------------------------------------------------------o
Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal

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

                                                  o-----------------------------o
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         / \         \   |
                                                  |  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                 /|  o         o   o         o  |
                                                / |   \         \ /         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     o-----------------------------o
                                           /
o-----------------------------------------/---o   o-----------------------------o
|                                        /    |   |      o-----o   o-----o      |
|                                       @     |   |     /       \ /       \     |
|          o---------o   o---------o          |   |    /         o         \    |
|         /           \ /           \         |   |   /         / \         \   |
|        /             o             \        |   |  o         o   o         o  |
|       /             / \     @-------\-----------@  |   du    |   |    dv   |  |
|      /             / @ \             \      |   |  o         o   o         o  |
|     /             /   \ \             \     |   |   \         \ /         /   |
|    /             /     \ \             \    |   |    \         o         /    |
|   o             o       \ o             o   |   |     \       / \       /     |
|   |             |        \|             |   |   |      o-----o   o-----o      |
|   |             |         |             |   |   o-----------------------------o
|   |      u      |         |\     v      |   |
|   |             |         | \           |   |   o-----------------------------o
|   |             |         |  \          |   |   |      o-----o   o-----o      |
|   o             o         o   \         o   |   |     /       \ /       \     |
|    \             \       /     \       /    |   |    /         o         \    |
|     \             \     /       \     /     |   |   /         / \         \   |
|      \             \   /         \   /      |   |  o         o   o         o  |
|       \       @-----\-/-----------\-------------@  |   du    |   |    dv   |  |
|        \             o             /        |   |  o         o   o         o  |
|         \           / \           / \       |   |   \         \ /         /   |
|          o---------o   o---------o   \      |   |    \         o         /    |
|                                       \     |   |     \       / \       /     |
|                                        \    |   |      o-----o   o-----o      |
o-----------------------------------------\---o   o-----------------------------o
                                           \
                                            \     o-----------------------------o
                                             \    |      o-----o   o-----o      |
                                              \   |     /       \ /       \     |
                                               \  |    /         o         \    |
                                                \ |   /         / \         \   |
                                                 \|  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                  |  o         o   o         o  |
                                                  |   \         \ /         /   |
                                                  |    \         o         /    |
                                                  |     \       / \       /     |
                                                  |      o-----o   o-----o      |
                                                  o-----------------------------o
Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle

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

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (du).(dv) o                       o    |
|    |                       |   -->--   |                       |    |
|    |                       |   \   /   |                       |    |
|    |              dv .(du) |    \ /    | du .(dv)              |    |
|    |      u      <---------------@--------------->      v      |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  V                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact

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

o-----------------------------------------------------------o
|                                                           |
|                           .->-.                           |
|                           |   |                           |
|                           *   *                           |
|                            \ /                            |
|                       .-->--@--<--.                       |
|                      /     / \     \                      |
|                     /     /   \     \                     |
|                    /     .     .     \                    |
|                   /      |     |      \                   |
|                  /       |     |       \                  |
|                 /        |     |        \                 |
|                .         |     |         .                |
|                |         |     |         |                |
|                v         |     |         v                |
|           .--. | .---------->----------. | .--.           |
|           |   \|/        |     |        \|/   |           |
|           ^    @         ^     v         @    v           |
|           |   /|\        |     |        /|\   |           |
|           *--* | *----------<----------* | *--*           |
|                ^         |     |         ^                |
|                |         |     |         |                |
|                *         |     |         *                |
|                 \        |     |        /                 |
|                  \       |     |       /                  |
|                   \      |     |      /                   |
|                    \     .     .     /                    |
|                     \     \   /     /                     |
|                      \     \ /     /                      |
|                       *-->--@--<--*                       |
|                            / \                            |
|                           .   .                           |
|                           |   |                           |
|                           *-<-*                           |
|                                                           |
o-----------------------------------------------------------o
Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph

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

o-------------------------------o     o-------------------------------o
|                               |     |                               |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|      /       \ /       \      |     |      /       \ /       \      |
|     /         o         \     |     |     /         o         \     |
|    /         /`\         \    |     |    /         /`\         \    |
|   o         o```o         o   |     |   o         o```o         o   |
|   |    u    |```|    v    |   |     |   |    u    |```|    v    |   |
|   o         o```o         o   |     |   o         o```o         o   |
|    \         \`/         /    |     |    \         \`/         /    |
|     \         o         /     |     |     \         o         /     |
|      \       / \       /      |     |      \       / \       /      |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
                                       \                             /
                                         \                         /
                                           \                     /
               u v                           \        J        /
                                               \             /
                                                 \         /
                                                   \     /
                                                     \ /
                                                      o
Figure 20-i.  Thematization of Conjunction (Stage 1)

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

o-------------------------------o     o-------------------------------o
|                               |     |                               |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|      /       \ /       \      |     |      /       \ /       \      |
|     /         o         \     |     |     /         o         \     |
|    /         /`\         \    |     |    /         /`\         \    |
|   o         o```o         o   |     |   o         o```o         o   |
|   |    u    |```|    v    |   |     |   |    u    |```|    v    |   |
|   o         o```o         o   |     |   o         o```o         o   |
|    \         \`/         /    |     |    \         \`/         /    |
|     \         o         /     |     |     \         o         /     |
|      \       / \       /      |     |      \       / \       /      |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
 \                             /       \                             /
   \                         /           \                         /
     \                     /               \          J          /
       \                 /                   \                 /
         \             /                       \             /
o----------\---------/----------o     o----------\---------/----------o
|            \     /            |     |            \     /            |
|              \ /              |     |              \ /              |
|         o-----@-----o         |     |         o-----@-----o         |
|        /`````````````\        |     |        /`````````````\        |
|       /```````````````\       |     |       /```````````````\       |
|      /`````````````````\      |     |      /`````````````````\      |
|     o```````````````````o     |     |     o```````````````````o     |
|     |```````````````````|     |     |     |```````````````````|     |
|     |```````` J ````````|     |     |     |```````` x ````````|     |
|     |```````````````````|     |     |     |```````````````````|     |
|     o```````````````````o     |     |     o```````````````````o     |
|      \`````````````````/      |     |      \`````````````````/      |
|       \```````````````/       |     |       \```````````````/       |
|        \`````````````/        |     |        \`````````````/        |
|         o-----------o         |     |         o-----------o         |
|                               |     |                               |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
             J = u v                             x = J<u, v>

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

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

o-------------------------------o     o-------------------------------o
|                               |     |```````````````````````````````|
|                               |     |````````````o-----o````````````|
|                               |     |```````````/       \```````````|
|                               |     |``````````/         \``````````|
|                               |     |`````````/           \`````````|
|                               |     |````````/             \````````|
|               J               |     |```````o       x       o```````|
|                               |     |```````|               |```````|
|                               |     |```````|               |```````|
|                               |     |```````|               |```````|
|       o-----o   o-----o       |     |```````o-----o   o-----o```````|
|      /       \ /       \      |     |``````/`\     \ /     /`\``````|
|     /         o         \     |     |`````/```\     o     /```\`````|
|    /         /`\         \    |     |````/`````\   /`\   /`````\````|
|   /         /```\         \   |     |```/```````\ /```\ /```````\```|
|  o         o`````o         o  |     |``o`````````o-----o`````````o``|
|  |    u    |`````|    v    |  |     |``|`````````|     |`````````|``|
o--o---------o-----o---------o--o     |``|``` u ```|     |``` v ```|``|
|``|`````````|     |`````````|``|     |``|`````````|     |`````````|``|
|``o`````````o     o`````````o``|     |``o`````````o     o`````````o``|
|```\`````````\   /`````````/```|     |```\`````````\   /`````````/```|
|````\`````````\ /`````````/````|     |````\`````````\ /`````````/````|
|`````\`````````o`````````/`````|     |`````\`````````o`````````/`````|
|``````\```````/`\```````/``````|     |``````\```````/`\```````/``````|
|```````o-----o```o-----o```````|     |```````o-----o```o-----o```````|
|```````````````````````````````|     |```````````````````````````````|
o-------------------------------o     o-------------------------------o
                                       \                             /
                                         \                         /
          J   =   u v                      \                     /
                                             \       !j!       /
                                               \             /
         !j!  =   (( x , u v ))                  \         /
                                                   \     /
                                                     \ /
                                                      @
Figure 20-iii.  Thematization of Conjunction (Stage 3)

Figure 21. Thematization of Disjunction and Equality

                f                                     g
o-------------------------------o     o-------------------------------o
|                               |     |```````````````````````````````|
|       o-----o   o-----o       |     |```````o-----o```o-----o```````|
|      /```````\ /```````\      |     |``````/       \`/       \``````|
|     /`````````o`````````\     |     |`````/         o         \`````|
|    /`````````/`\`````````\    |     |````/         /`\         \````|
|   /`````````/```\`````````\   |     |```/         /```\         \```|
|  o`````````o`````o```````` o  |     |``o         o`````o         o``|
|  |`````````|`````|`````````|  |     |``|         |`````|         |``|
|  |``` u ```|`````|``` v ```|  |     |``|    u    |`````|    v    |``|
|  |`````````|`````|`````````|  |     |``|         |`````|         |``|
|  o`````````o`````o`````````o  |     |``o         o`````o         o``|
|   \`````````\```/`````````/   |     |```\         \```/         /```|
|    \`````````\`/`````````/    |     |````\         \`/         /````|
|     \`````````o`````````/     |     |`````\         o         /`````|
|      \```````/ \```````/      |     |``````\       /`\       /``````|
|       o-----o   o-----o       |     |```````o-----o```o-----o```````|
|                               |     |```````````````````````````````|
o-------------------------------o     o-------------------------------o
            ((u)(v))                              ((u , v))

                |                                     |
                |                                     |
              theta                                 theta
                |                                     |
                |                                     |
                v                                     v

               !f!                                   !g!
o-------------------------------o     o-------------------------------o
|```````````````````````````````|     |                               |
|````````````o-----o````````````|     |            o-----o            |
|```````````/       \```````````|     |           /```````\           |
|``````````/         \``````````|     |          /`````````\          |
|`````````/           \`````````|     |         /```````````\         |
|````````/             \````````|     |        /`````````````\        |
|```````o       f       o```````|     |       o`````` g ``````o       |
|```````|               |```````|     |       |```````````````|       |
|```````|               |```````|     |       |```````````````|       |
|```````|               |```````|     |       |```````````````|       |
|```````o-----o   o-----o```````|     |       o-----o```o-----o       |
|``````/ \`````\ /`````/ \``````|     |      /`\     \`/     /`\      |
|`````/   \`````o`````/   \`````|     |     /```\     o     /```\     |
|````/     \```/`\```/     \````|     |    /`````\   /`\   /`````\    |
|```/       \`/```\`/       \```|     |   /```````\ /```\ /```````\   |
|``o         o-----o         o``|     |  o`````````o-----o`````````o  |
|``|         |     |         |``|     |  |`````````|     |`````````|  |
|``|    u    |     |    v    |``|     |  |``` u ```|     |``` v ```|  |
|``|         |     |         |``|     |  |`````````|     |`````````|  |
|``o         o     o         o``|     |  o`````````o     o`````````o  |
|```\         \   /         /```|     |   \`````````\   /`````````/   |
|````\         \ /         /````|     |    \`````````\ /`````````/    |
|`````\         o         /`````|     |     \`````````o`````````/     |
|``````\       /`\       /``````|     |      \```````/ \```````/      |
|```````o-----o```o-----o```````|     |       o-----o   o-----o       |
|```````````````````````````````|     |                               |
o-------------------------------o     o-------------------------------o
        ((f , ((u)(v)) ))                    ((g , ((u , v)) ))

Figure 21.  Thematization of Disjunction and Equality

Table 22. Disjunction f and Equality g

Table 22.  Disjunction f and Equality g
o-------------------o-------------------o
|    u         v    |    f         g    |
o-------------------o-------------------o
|                   |                   |
|    0         0    |    0         1    |
|                   |                   |
|    0         1    |    1         0    |
|                   |                   |
|    1         0    |    1         0    |
|                   |                   |
|    1         1    |    1         1    |
|                   |                   |
o-------------------o-------------------o

Table 22. Disjunction f and Equality g
u v
f g
0 0
0 1
1 0
1 1
0 1
1 0
1 0
1 1


Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)

Tables 23-i and 23-ii.  Thematics of Disjunction and Equality (1)
o-----------------o-----------o         o-----------------o-----------o
|  u     v     f  |  x    !f! |         |  u     v     g  |  y    !g! |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0    --> |  0     1  |         |  0     0    --> |  1     1  |
|                 |           |         |                 |           |
|  0     1    --> |  1     1  |         |  0     1    --> |  0     1  |
|                 |           |         |                 |           |
|  1     0    --> |  1     1  |         |  1     0    --> |  0     1  |
|                 |           |         |                 |           |
|  1     1    --> |  1     1  |         |  1     1    --> |  1     1  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0        |  1     0  |         |  0     0        |  0     0  |
|                 |           |         |                 |           |
|  0     1        |  0     0  |         |  0     1        |  1     0  |
|                 |           |         |                 |           |
|  1     0        |  0     0  |         |  1     0        |  1     0  |
|                 |           |         |                 |           |
|  1     1        |  0     0  |         |  1     1        |  0     0  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)
Table 23-i. Disjunction f
u v f
x φ
0 0 →
0 1 →
1 0 →
1 1 →
0 1
1 1
1 1
1 1
0 0   
0 1   
1 0   
1 1   
1 0
0 0
0 0
0 0
Table 23-ii. Equality g
u v g
y γ
0 0 →
0 1 →
1 0 →
1 1 →
1 1
0 1
0 1
1 1
0 0   
0 1   
1 0   
1 1   
0 0
1 0
1 0
0 0


Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)

Tables 24-i and 24-ii.  Thematics of Disjunction and Equality (2)
o-----------------------o-----o         o-----------------------o-----o
|  u     v     f     x  | !f! |         |  u     v     g     y  | !g! |
o-----------------------o-----o         o-----------------------o-----o
|                       |     |         |                       |     |
|  0     0    -->    0  |  1  |         |  0     0           0  |  0  |
|                       |     |         |                       |     |
|  0     0           1  |  0  |         |  0     0    -->    1  |  1  |
|                       |     |         |                       |     |
|  0     1           0  |  0  |         |  0     1    -->    0  |  1  |
|                       |     |         |                       |     |
|  0     1    -->    1  |  1  |         |  0     1           1  |  0  |
|                       |     |         |                       |     |
o-----------------------o-----o         o-----------------------o-----o
|                       |     |         |                       |     |
|  1     0           0  |  0  |         |  1     0    -->    0  |  1  |
|                       |     |         |                       |     |
|  1     0    -->    1  |  1  |         |  1     0           1  |  0  |
|                       |     |         |                       |     |
|  1     1           0  |  0  |         |  1     1           0  |  0  |
|                       |     |         |                       |     |
|  1     1    -->    1  |  1  |         |  1     1    -->    1  |  1  |
|                       |     |         |                       |     |
o-----------------------o-----o         o-----------------------o-----o
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)
Table 24-i. Disjunction f
u v f x
φ
0 0 → 0
0 0    1
0 1    0
0 1 → 1
1
0
0
1
1 0    0
1 0 → 1
1 1    0
1 1 → 1
0
1
0
1
Table 24-ii. Equality g
u v g y
γ
0 0    0
0 0 → 1
0 1 → 0
0 1    1
0
1
1
0
1 0 → 0
1 0    1
1 1    0
1 1 → 1
1
0
0
1


Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)

Tables 25-i and 25-ii.  Thematics of Disjunction and Equality (3)
o-----------------------o-----o         o-----------------------o-----o
|  u     v     f     x  | !f! |         |  u     v     g     y  | !g! |
o-----------------------o-----o         o-----------------------o-----o
|                       |     |         |                       |     |
|  0     0    -->    0  |  1  |         |  0     0           0  |  0  |
|                       |     |         |                       |     |
|  0     1           0  |  0  |         |  0     1    -->    0  |  1  |
|                       |     |         |                       |     |
|  1     0           0  |  0  |         |  1     0    -->    0  |  1  |
|                       |     |         |                       |     |
|  1     1           0  |  0  |         |  1     1           0  |  0  |
|                       |     |         |                       |     |
o-----------------------o-----o         o-----------------------o-----o
|                       |     |         |                       |     |
|  0     0           1  |  0  |         |  0     0    -->    1  |  1  |
|                       |     |         |                       |     |
|  0     1    -->    1  |  1  |         |  0     1           1  |  0  |
|                       |     |         |                       |     |
|  1     0    -->    1  |  1  |         |  1     0           1  |  0  |
|                       |     |         |                       |     |
|  1     1    -->    1  |  1  |         |  1     1    -->    1  |  1  |
|                       |     |         |                       |     |
o-----------------------o-----o         o-----------------------o-----o
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)
Table 25-i. Disjunction f
u v f x
φ
0 0 → 0
0 1    0
1 0    0
1 1    0
1
0
0
0
0 0    1
0 1 → 1
1 0 → 1
1 1 → 1
0
1
1
1
Table 25-ii. Equality g
u v g y
γ
0 0    0
0 1 → 0
1 0 → 0
1 1    0
0
1
1
0
0 0 → 1
0 1    1
1 0    1
1 1 → 1
1
0
0
1


Tables 26-i and 26-ii. Tacit Extension and Thematization

Tables 26-i and 26-ii.  Tacit Extension and Thematization
o-----------------o-----------o         o-----------------o-----------o
|  u     v     x  | !e!f  !f! |         |  u     v     y  | !e!g  !g! |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  0     0     0  |  0     1  |         |  0     0     0  |  1     0  |
|                 |           |         |                 |           |
|  0     0     1  |  0     0  |         |  0     0     1  |  1     1  |
|                 |           |         |                 |           |
|  0     1     0  |  1     0  |         |  0     1     0  |  0     1  |
|                 |           |         |                 |           |
|  0     1     1  |  1     1  |         |  0     1     1  |  0     0  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
|                 |           |         |                 |           |
|  1     0     0  |  1     0  |         |  1     0     0  |  0     1  |
|                 |           |         |                 |           |
|  1     0     1  |  1     1  |         |  1     0     1  |  0     0  |
|                 |           |         |                 |           |
|  1     1     0  |  1     0  |         |  1     1     0  |  1     0  |
|                 |           |         |                 |           |
|  1     1     1  |  1     1  |         |  1     1     1  |  1     1  |
|                 |           |         |                 |           |
o-----------------o-----------o         o-----------------o-----------o
Tables 26-i and 26-ii. Tacit Extension and Thematization
Table 26-i. Disjunction f
u v x
εf θf
0 0 0
0 0 1
0 1 0
0 1 1
0 1
0 0
1 0
1 1
1 0 0
1 0 1
1 1 0
1 1 1
1 0
1 1
1 0
1 1
Table 26-ii. Equality g
u v y
εg θg
0 0 0
0 0 1
0 1 0
0 1 1
1 0
1 1
0 1
0 0
1 0 0
1 0 1
1 1 0
1 1 1
0 1
0 0
1 0
1 1


Table 27. Thematization of Bivariate Propositions

Table 27.  Thematization of Bivariate Propositions
o---------o---------o----------o--------------------o--------------------o
|       u : 1 1 0 0 |    f     |     theta (f)      |     theta (f)      |
|       v : 1 0 1 0 |          |                    |                    |
o---------o---------o----------o--------------------o--------------------o
|         |         |          |                    |                    |
| f_0     | 0 0 0 0 |    ()    | (( f ,    ()    )) | f              + 1 |
|         |         |          |                    |                    |
| f_1     | 0 0 0 1 |  (u)(v)  | (( f ,  (u)(v)  )) | f + u + v + uv     |
|         |         |          |                    |                    |
| f_2     | 0 0 1 0 |  (u) v   | (( f ,  (u) v   )) | f     + v + uv + 1 |
|         |         |          |                    |                    |
| f_3     | 0 0 1 1 |  (u)     | (( f ,  (u)     )) | f + u              |
|         |         |          |                    |                    |
| f_4     | 0 1 0 0 |   u (v)  | (( f ,   u (v)  )) | f + u     + uv + 1 |
|         |         |          |                    |                    |
| f_5     | 0 1 0 1 |     (v)  | (( f ,     (v)  )) | f     + v          |
|         |         |          |                    |                    |
| f_6     | 0 1 1 0 |  (u, v)  | (( f ,  (u, v)  )) | f + u + v      + 1 |
|         |         |          |                    |                    |
| f_7     | 0 1 1 1 |  (u  v)  | (( f ,  (u  v)  )) | f         + uv     |
|         |         |          |                    |                    |
o---------o---------o----------o--------------------o--------------------o
|         |         |          |                    |                    |
| f_8     | 1 0 0 0 |   u  v   | (( f ,   u  v   )) | f         + uv + 1 |
|         |         |          |                    |                    |
| f_9     | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v          |
|         |         |          |                    |                    |
| f_10    | 1 0 1 0 |      v   | (( f ,      v   )) | f     + v      + 1 |
|         |         |          |                    |                    |
| f_11    | 1 0 1 1 |  (u (v)) | (( f ,  (u (v)) )) | f + u     + uv     |
|         |         |          |                    |                    |
| f_12    | 1 1 0 0 |   u      | (( f ,   u      )) | f + u          + 1 |
|         |         |          |                    |                    |
| f_13    | 1 1 0 1 | ((u) v)  | (( f , ((u) v)  )) | f     + v + uv     |
|         |         |          |                    |                    |
| f_14    | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 |
|         |         |          |                    |                    |
| f_15    | 1 1 1 1 |   (())   | (( f ,   (())   )) | f                  |
|         |         |          |                    |                    |
o---------o---------o----------o--------------------o--------------------o


Table 27. Thematization of Bivariate Propositions
u :
v :
1100
1010
f
 
θf
 
θf
 
f0
f1
f2
f3
f4
f5
f6
f7
0000
0001
0010
0011
0100
0101
0110
0111
()
 (u)(v) 
 (u) v  
 (u)    
  u (v) 
    (v) 
 (u, v) 
 (u  v) 
(( f ,    ()    ))
(( f ,  (u)(v)  ))
(( f ,  (u) v   ))
(( f ,  (u)     ))
(( f ,   u (v)  ))
(( f ,     (v)  ))
(( f ,  (u, v)  ))
(( f ,  (u  v)  ))
 f + 1
 f + u + v + uv
 f + v + uv + 1
 f + u
 f + u + uv + 1
 f + v
 f + u + v + 1
 f + uv
f8
f9
f10
f11
f12
f13
f14
f15
1000
1001
1010
1011
1100
1101
1110
1111
  u  v  
((u, v))
     v  
 (u (v))
  u     
((u) v) 
((u)(v))
(())
(( f ,   u  v   ))
(( f , ((u, v)) ))
(( f ,      v   ))
(( f ,  (u (v)) ))
(( f ,   u      ))
(( f , ((u) v)  ))
(( f , ((u)(v)) ))
(( f ,   (())   ))
 f + uv + 1
 f + u + v
 f + v + 1
 f + u + uv
 f + u + 1
 f + v + uv
 f + u + v + uv + 1
 f


Table 28. Propositions on Two Variables

Table 28.  Propositions on Two Variables
o-------o-----o----------------------------------------------------------------o
| u   v |     | f   f   f   f   f   f   f   f   f   f   f   f   f   f   f   f  |
|       |     | 00  01  02  03  04  05  06  07  08  09  10  11  12  13  14  15 |
o-------o-----o----------------------------------------------------------------o
|       |     |                                                                |
| 0   0 |     | 0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1  |
|       |     |                                                                |
| 0   1 |     | 0   0   1   1   0   0   1   1   0   0   1   1   0   0   1   1  |
|       |     |                                                                |
| 1   0 |     | 0   0   0   0   1   1   1   1   0   0   0   0   1   1   1   1  |
|       |     |                                                                |
| 1   1 |     | 0   0   0   0   0   0   0   0   1   1   1   1   1   1   1   1  |
|       |     |                                                                |
o-------o-----o----------------------------------------------------------------o


Table 28. Propositions on Two Variables
u v   f0 f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15
0 0   0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1   0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 0   0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1   0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1



Table 28. Propositions on Two Variables
u v   f0 f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15
0 0   0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1   0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 0   0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1   0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1



Table 28. Propositions on Two Variables
u v   f00 f01 f02 f03 f04 f05 f06 f07 f08 f09 f10 f11 f12 f13 f14 f15
0 0   0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1   0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 0   0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1   0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Table 29. Thematic Extensions of Bivariate Propositions

Table 29.  Thematic Extensions of Bivariate Propositions
o-------o-----o----------------------------------------------------------------o
| u   v | f^¢ |!f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! |
|       |     | 00  01  02  03  04  05  06  07  08  09  10  11  12  13  14  15 |
o-------o-----o----------------------------------------------------------------o
|       |     |                                                                |
| 0  0  |  0  | 1   0   1   0   1   0   1   0   1   0   1   0   1   0   1   0  |
|       |     |                                                                |
| 0  0  |  1  | 0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1  |
|       |     |                                                                |
| 0  1  |  0  | 1   1   0   0   1   1   0   0   1   1   0   0   1   1   0   0  |
|       |     |                                                                |
| 0  1  |  1  | 0   0   1   1   0   0   1   1   0   0   1   1   0   0   1   1  |
|       |     |                                                                |
| 1  0  |  0  | 1   1   1   1   0   0   0   0   1   1   1   1   0   0   0   0  |
|       |     |                                                                |
| 1  0  |  1  | 0   0   0   0   1   1   1   1   0   0   0   0   1   1   1   1  |
|       |     |                                                                |
| 1  1  |  0  | 1   1   1   1   1   1   1   1   0   0   0   0   0   0   0   0  |
|       |     |                                                                |
| 1  1  |  1  | 0   0   0   0   0   0   0   0   1   1   1   1   1   1   1   1  |
|       |     |                                                                |
o-------o-----o----------------------------------------------------------------o


Table 29. Thematic Extensions of Bivariate Propositions
u v f¢ φ00 φ01 φ02 φ03 φ04 φ05 φ06 φ07 φ08 φ09 φ10 φ11 φ12 φ13 φ14 φ15
0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0
1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Figure 30. Generic Frame of a Logical Transformation

             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
| U                       | | U                       | | U                       |
|      o---o   o---o      | |      o---o   o---o      | |      o---o   o---o      |
|     /     \ /     \     | |     /     \ /     \     | |     /     \ /     \     |
|    /       o       \    | |    /       o       \    | |    /       o       \    |
|   /       / \       \   | |   /       / \       \   | |   /       / \       \   |
|  o       o   o       o  | |  o       o   o       o  | |  o       o   o       o  |
|  |   u   |   |   v   |  | |  |   u   |   |   v   |  | |  |   u   |   |   v   |  |
|  o       o   o       o  | |  o       o   o       o  | |  o       o   o       o  |
|   \       \ /       /   | |   \       \ /       /   | |   \       \ /       /   |
|    \       o       /    | |    \       o       /    | |    \       o       /    |
|     \     / \     /     | |     \     / \     /     | |     \     / \     /     |
|      o---o   o---o      | |      o---o   o---o      | |      o---o   o---o      |
|                         | |                         | |                         |
o-------------------------o o-------------------------o o-------------------------o
 \                        |  \                       /  |                        /
  \                       |   \                     /   |                       /
   \                      |    \                   /    |                      /
    \                     |     \                 /     |                     /
     \       g            |      \       f       /      |            h       /
      \                   |       \             /       |                   /
       \                  |        \           /        |                  /
        \                 |         \         /         |                 /
         \                |          \       /          |                /
          \    o----------|-----------\-----/-----------|----------o    /
           \   | X        |            \   /            |          |   /
            \  |          |             \ /             |          |  /
             \ |          |        o-----o-----o        |          | /
              \|          |       /             \       |          |/
               \          |      /               \      |          /
               |\         |     /                 \     |         /|
               | \        |    /                   \    |        / |
               |  \       |   /                     \   |       /  |
               |   \      |  o           x           o  |      /   |
               |    \     |  |                       |  |     /    |
               |     \    |  |                       |  |    /     |
               |      \   |  |                       |  |   /      |
               |       \  |  |                       |  |  /       |
               |        \ |  |                       |  | /        |
               |         \|  |                       |  |/         |
               |          o--o--------o     o--------o--o          |
               |         /    \        \   /        /    \         |
               |        /      \        \ /        /      \        |
               |       /        \        o        /        \       |
               |      /          \      / \      /          \      |
               |     /            \    /   \    /            \     |
               |    o              o--o-----o--o              o    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    |        y        |     |        z        |    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    o                 o     o                 o    |
               |     \                 \   /                 /     |
               |      \                 \ /                 /      |
               |       \                 o                 /       |
               |        \               / \               /        |
               |         \             /   \             /         |
               |          o-----------o     o-----------o          |
               |                                                   |
               |                                                   |
               o---------------------------------------------------o
                \                                                 /
                  \                                             /
                    \                                         /
                      \                                     /
                        \                                 /
                          \            p , q            /
                            \                         /
                              \                     /
                                \                 /
                                  \             /
                                    \         /
                                      \     /
                                        \ /
                                         o

Figure 30.  Generic Frame of a Logical Transformation

Formula Display 3

o-------------------------------------------------o
|                                                 |
|         x              =           f<u, v>      |
|                                                 |
|         y              =           g<u, v>      |
|                                                 |
|         z              =           h<u, v>      |
|                                                 |
o-------------------------------------------------o


  x = f‹u, v›  
  y = g‹u, v›  
  z = h‹u, v›  


Figure 31. Operator Diagram (1)

o---------------------------------------o
|                                       |
|                                       |
|      U%           F           X%      |
|         o------------------>o         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|     !W! |                   | !W!     |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         v                   v         |
|         o------------------>o         |
|   !W!U%         !W!F          !W!X%   |
|                                       |
|                                       |
o---------------------------------------o
Figure 31.  Operator Diagram (1)

Figure 32. Operator Diagram (2)

o---------------------------------------o
|                                       |
|                                       |
|      U%          !W!          !W!U%   |
|         o------------------>o         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|      F  |                   | !W!F    |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         v                   v         |
|         o------------------>o         |
|      X%          !W!          !W!X%   |
|                                       |
|                                       |
o---------------------------------------o
Figure 32.  Operator Diagram (2)

Figure 33-i. Analytic Diagram (1)

U%          $E$      $E$U%        $E$U%        $E$U%
   o------------------>o============o============o
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
F  |                   | $E$F   =   | $d$^0.F  + | $r$^0.F
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
   v                   v            v            v
   o------------------>o============o============o
X%          $E$      $E$X%        $E$X%        $E$X%

Figure 33-i.  Analytic Diagram (1)

Figure 33-ii. Analytic Diagram (2)

U%          $E$      $E$U%        $E$U%        $E$U%        $E$U%
   o------------------>o============o============o============o
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
F  |                   | $E$F   =   | $d$^0.F  + | $d$^1.F  + | $r$^1.F
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
   v                   v            v            v            v
   o------------------>o============o============o============o
X%          $E$      $E$X%        $E$X%        $E$X%        $E$X%

Figure 33-ii.  Analytic Diagram (2)

Formula Display 4

o--------------------------------------------------------------------------------------o
|                                                                                      |
|  x_1  =  !e!F_1 <u_1, ..., u_n, du_1, ..., du_n>  =  F_1 <u_1, ..., u_n>             |
|                                                                                      |
|  ...                                                                                 |
|                                                                                      |
|  x_k  =  !e!F_k <u_1, ..., u_n, du_1, ..., du_n>  =  F_k <u_1, ..., u_n>             |
|                                                                                      |
|                                                                                      |
| dx_1  =  EF_1 <u_1, ..., u_n, du_1, ..., du_n>  =  F_1 <u_1 + du_1, ..., u_n + du_n> |
|                                                                                      |
|  ...                                                                                 |
|                                                                                      |
| dx_k  =  EF_k <u_1, ..., u_n, du_1, ..., du_n>  =  F_k <u_1 + du_1, ..., u_n + du_n> |
|                                                                                      |
o--------------------------------------------------------------------------------------o


x1 = \(\epsilon\)F1‹u1, …, un, du1, …, dun› = F1‹u1, …, un›
...
xk = \(\epsilon\)Fk‹u1, …, un, du1, …, dun› = Fk‹u1, …, un›
dx1 = EF1‹u1, …, un, du1, …, dun› = F1‹u1 + du1, …, un + dun›
...
dxk = EFk‹u1, …, un, du1, …, dun› = Fk‹u1 + du1, …, un + dun›


Formula Display 5

o--------------------------------------------------------------------------------o
|                                                                                |
|  x_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
|  x_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
|                                                                                |
| dx_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
| dx_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
o--------------------------------------------------------------------------------o


x1 = \(\epsilon\)F1‹u1, …, un, du1, …, dun› = F1‹u1, …, un›
...
xk = \(\epsilon\)Fk‹u1, …, un, du1, …, dun› = Fk‹u1, …, un›
dx1 = \(\epsilon\)F1‹u1, …, un, du1, …, dun› = F1‹u1, …, un›
...
dxk = \(\epsilon\)Fk‹u1, …, un, du1, …, dun› = Fk‹u1, …, un›


Formula Display 6

o--------------------------------------------------------------------------------o
|                                                                                |
| dx_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
| dx_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
o--------------------------------------------------------------------------------o


dx1 = \(\epsilon\)F1‹u1, …, un, du1, …, dun› = F1‹u1, …, un›
...
dxk = \(\epsilon\)Fk‹u1, …, un, du1, …, dun› = Fk‹u1, …, un›


Formula Display 7

o-------------------------------------------------o
|                                                 |
| $D$   =   $E$ - $e$                             |
|                                                 |
|       =   $r$^0                                 |
|                                                 |
|       =   $d$^1  +  $r$^1                       |
|                                                 |
|       =   $d$^1  +  ...  +  $d$^m  +  $r$^m     |
|                                                 |
|       =   Sum_(i = 1 ... m) $d$^i  +  $r$^m     |
|                                                 |
o-------------------------------------------------o


D = E – e
  = r0
  = d1 + r1
  = d1 + … + dm + rm
  = ∑(i = 1 … m) di + rm


Figure 34. Tangent Functor Diagram

U%          $T$      $T$U%        $T$U%
   o------------------>o============o
   |                   |            |
   |                   |            |
   |                   |            |
   |                   |            |
F  |                   | $T$F   =   | <!e!, d> F
   |                   |            |
   |                   |            |
   |                   |            |
   v                   v            v
   o------------------>o============o
X%          $T$      $T$X%        $T$X%

Figure 34.  Tangent Functor Diagram

Figure 35. Conjunction as Transformation

o---------------------------------------o
|                                       |
|                                       |
|       o---------o   o---------o       |
|      /           \ /           \      |
|     /             o             \     |
|    /             /`\             \    |
|   /             /```\             \   |
|  o             o`````o             o  |
|  |             |`````|             |  |
|  |      u      |`````|      v      |  |
|  |             |`````|             |  |
|  o             o`````o             o  |
|   \             \```/             /   |
|    \             \`/             /    |
|     \             o             /     |
|      \           / \           /      |
|       o---------o   o---------o       |
|                                       |
|                                       |
o---------------------------------------o
 \                                     /
   \                                 /
     \                             /
       \            J            /
         \                     /
           \                 /
             \             /
o--------------\---------/--------------o
|                \     /                |
|                  \ /                  |
|            o------@------o            |
|           /```````````````\           |
|          /`````````````````\          |
|         /```````````````````\         |
|        /`````````````````````\        |
|       o```````````````````````o       |
|       |```````````````````````|       |
|       |`````````` x ``````````|       |
|       |```````````````````````|       |
|       o```````````````````````o       |
|        \`````````````````````/        |
|         \```````````````````/         |
|          \`````````````````/          |
|           \```````````````/           |
|            o-------------o            |
|                                       |
|                                       |
o---------------------------------------o
Figure 35.  Conjunction as Transformation

Table 36. Computation of !e!J

Table 36.  Computation of !e!J
o---------------------------------------------------------------------o
|                                                                     |
| !e!J  =  J<u, v>                                                    |
|                                                                     |
|       =  u v                                                        |
|                                                                     |
|       =  u v (du)(dv)  +  u v (du) dv  +  u v du (dv)  +  u v du dv |
|                                                                     |
o---------------------------------------------------------------------o
|                                                                     |
| !e!J  =  u v (du)(dv)  +                                            |
|          u v (du) dv   +                                            |
|          u v  du (dv)  +                                            |
|          u v  du  dv                                                |
|                                                                     |
o---------------------------------------------------------------------o

Table 36. Computation of \(\epsilon\)J
\(\epsilon\)J = J‹u, v›
  = u v
  = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv
\(\epsilon\)J = u v (du)(dv) +  
    u v (du) dv +  
    u v  du (dv) +  
    u v  du  dv    


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

o---------------------------------------o
|                                       |
|                   o                   |
|                  /%\                  |
|                 /%%%\                 |
|                /%%%%%\                |
|               o%%%%%%%o               |
|              /%\%%%%%/%\              |
|             /%%%\%%%/%%%\             |
|            /%%%%%\%/%%%%%\            |
|           o%%%%%%%o%%%%%%%o           |
|          / \%%%%%/%\%%%%%/ \          |
|         /   \%%%/%%%\%%%/   \         |
|        /     \%/%%%%%\%/     \        |
|       o       o%%%%%%%o       o       |
|      / \     / \%%%%%/ \     / \      |
|     /   \   /   \%%%/   \   /   \     |
|    /     \ /     \%/     \ /     \    |
|   o       o       o       o       o   |
|   |\     / \     / \     / \     /|   |
|   | \   /   \   /   \   /   \   / |   |
|   |  \ /     \ /     \ /     \ /  |   |
|   |   o       o       o       o   |   |
|   |   |\     / \     / \     /|   |   |
|   |   | \   /   \   /   \   / |   |   |
|   | u |  \ /     \ /     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 37-a.  Tacit Extension of J (Areal)

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

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         / \         \   |
                                                  |  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                 /|  o         o   o         o  |
                                                / |   \         \ /         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /       \ /       \     |
|          o---------o   o---------o          |   |    /         o         \    |
|         /           \ /           \         |   |   /         / \         \   |
|        /             o             \        |   |  o         o   o         o  |
|       /             /`\      @------\-----------@  |   du    |   |    dv   |  |
|      /             /```\             \      |   |  o         o   o         o  |
|     /             /`````\             \     |   |   \         \ /         /   |
|    /             /```````\             \    |   |    \         o         /    |
|   o             o`````````o             o   |   |     \       / \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /       \     |
|    \             \```````/   \         /    |   |    /         o         \    |
|     \             \`````/     \       /     |   |   /         / \         \   |
|      \             \```/       \     /      |   |  o         o   o         o  |
|       \      @------\-/---------\---------------@  |   du    |   |    dv   |  |
|        \             o           \ /        |   |  o         o   o         o  |
|         \           / \           /         |   |   \         \ /         /   |
|          o---------o   o---------o \        |   |    \         o         /    |
|                                     \       |   |     \       / \       /     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |`````````````````````````````|
                                             \    |````` o-----o```o-----o``````|
                                              \   |`````/```````\`/```````\`````|
                                               \  |````/`````````o`````````\````|
                                                \ |```/`````````/`\`````````\```|
                                                 \|``o`````````o```o`````````o``|
                                                  @``|```du````|```|````dv```|``|
                                                  |``o`````````o```o`````````o``|
                                                  |```\`````````\`/`````````/```|
                                                  |````\`````````o`````````/````|
                                                  |`````\```````/`\```````/`````|
                                                  |``````o-----o```o-----o``````|
                                                  |`````````````````````````````|
                                                  o-----------------------------o
Figure 37-b.  Tacit Extension of J (Bundle)

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

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (du).(dv) o                       o    |
|    |                       |   -->--   |                       |    |
|    |                       |   \   /   |                       |    |
|    |              dv .(du) |    \ /    | du .(dv)              |    |
|    |      u      <---------------@--------------->      v      |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  V                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 37-c.  Tacit Extension of J (Compact)

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

o-----------------------------------------------------------o
|                                                           |
|                         (du).(dv)                         |
|                          --->---                          |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                           u @ v                           |
|                            /|\                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|               (du) dv /     |     \ du (dv)               |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  v          |          v                  |
|                 @           |           @                 |
|               u (v)         |         (u) v               |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du . dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             v                             |
|                             @                             |
|                                                           |
|                          (u).(v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 37-d.  Tacit Extension of J (Digraph)

Table 38. Computation of EJ (Method 1)

Table 38.  Computation of EJ (Method 1)
o-------------------------------------------------------------------------------o
|                                                                               |
| EJ  =  J<u + du, v + dv>                                                      |
|                                                                               |
|     =  (u, du)(v, dv)                                                         |
|                                                                               |
|     =   u  v  J<1 + du, 1 + dv>  +                                            |
|                                                                               |
|         u (v) J<1 + du, 0 + dv>  +                                            |
|                                                                               |
|        (u) v  J<0 + du, 1 + dv>  +                                            |
|                                                                               |
|        (u)(v) J<0 + du, 0 + dv>                                               |
|                                                                               |
|     =   u  v  J<(du), (dv)>  +                                                |
|                                                                               |
|         u (v) J<(du),  dv >  +                                                |
|                                                                               |
|        (u) v  J< du , (dv)>  +                                                |
|                                                                               |
|        (u)(v) J< du ,  dv >                                                   |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| EJ  =   u  v (du)(dv)                                                         |
|                        +   u (v)(du) dv                                       |
|                                           +  (u) v  du (dv)                   |
|                                                              +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 38. Computation of EJ (Method 1)
EJ = J‹u + du, v + dv›  
 
  = (u, du)(v, dv)  
 
  =  u  v  J‹1 + du, 1 + dv› +
     u (v) J‹1 + du, 0 + dv› +
    (u) v  J‹0 + du, 1 + dv› +
    (u)(v) J‹0 + du, 0 + dv›  
 
  =  u  v  J‹(du), (dv)› +
     u (v) J‹(du),  dv › +
    (u) v  J‹ du , (dv)› +
    (u)(v) J‹ du ,  dv ›  
EJ = u v (du)(dv)      
    + u (v) (du) dv    
      + (u) v du (dv)  
        + (u)(v) du dv


Table 39. Computation of EJ (Method 2)

Table 39.  Computation of EJ (Method 2)
o-------------------------------------------------------------------------------o
|                                                                               |
| EJ  =  <u + du> <v + dv>                                                      |
|                                                                               |
|     =       u v        +       u dv       +       v du       +      du dv     |
|                                                                               |
| EJ  =   u  v (du)(dv)  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 39. Computation of EJ (Method 2)
EJ = ‹u + du› \(\cdot\) ‹v + dv›    
 
  = u v + u dv + v du + du dv    
 
EJ = u v (du)(dv) + u (v) (du) dv + (u) v du (dv) + (u)(v) du dv


Figure 40-a. Enlargement of J (Areal)

o---------------------------------------o
|                                       |
|                   o                   |
|                  /%\                  |
|                 /%%%\                 |
|                /%%%%%\                |
|               o%%%%%%%o               |
|              / \%%%%%/ \              |
|             /   \%%%/   \             |
|            /     \%/     \            |
|           o       o       o           |
|          /%\     / \     /%\          |
|         /%%%\   /   \   /%%%\         |
|        /%%%%%\ /     \ /%%%%%\        |
|       o%%%%%%%o       o%%%%%%%o       |
|      / \%%%%%/ \     / \%%%%%/ \      |
|     /   \%%%/   \   /   \%%%/   \     |
|    /     \%/     \ /     \%/     \    |
|   o       o       o       o       o   |
|   |\     / \     /%\     / \     /|   |
|   | \   /   \   /%%%\   /   \   / |   |
|   |  \ /     \ /%%%%%\ /     \ /  |   |
|   |   o       o%%%%%%%o       o   |   |
|   |   |\     / \%%%%%/ \     /|   |   |
|   |   | \   /   \%%%/   \   / |   |   |
|   | u |  \ /     \%/     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 40-a.  Enlargement of J (Areal)

Figure 40-b. Enlargement of J (Bundle)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         /%\         \   |
                                                  |  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                 /|  o         o%%%o         o  |
                                                / |   \         \%/         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /%%%%%%%\ /       \     |
|          o---------o   o---------o          |   |    /%%%%%%%%%o         \    |
|         /           \ /           \         |   |   /%%%%%%%%%/ \         \   |
|        /             o             \        |   |  o%%%%%%%%%o   o         o  |
|       /             /`\      @------\-----------@  |%% du %%%|   |    dv   |  |
|      /             /```\             \      |   |  o%%%%%%%%%o   o         o  |
|     /             /`````\             \     |   |   \%%%%%%%%%\ /         /   |
|    /             /```````\             \    |   |    \%%%%%%%%%o         /    |
|   o             o`````````o             o   |   |     \%%%%%%%/ \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /%%%%%%%\     |
|    \             \```````/   \         /    |   |    /         o%%%%%%%%%\    |
|     \             \`````/     \       /     |   |   /         / \%%%%%%%%%\   |
|      \             \```/       \     /      |   |  o         o   o%%%%%%%%%o  |
|       \      @------\-/---------\---------------@  |   du    |   |%%% dv %%|  |
|        \             o           \ /        |   |  o         o   o%%%%%%%%%o  |
|         \           / \           /         |   |   \         \ /%%%%%%%%%/   |
|          o---------o   o---------o \        |   |    \         o%%%%%%%%%/    |
|                                     \       |   |     \       / \%%%%%%%/     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
                                             \    |%%%%%%o-----o%%%o-----o%%%%%%|
                                              \   |%%%%%/       \%/       \%%%%%|
                                               \  |%%%%/         o         \%%%%|
                                                \ |%%%/         / \         \%%%|
                                                 \|%%o         o   o         o%%|
                                                  @%%|   du    |   |    dv   |%%|
                                                  |%%o         o   o         o%%|
                                                  |%%%\         \ /         /%%%|
                                                  |%%%%\         o         /%%%%|
                                                  |%%%%%\       /%\       /%%%%%|
                                                  |%%%%%%o-----o%%%o-----o%%%%%%|
                                                  |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
                                                  o-----------------------------o
Figure 40-b.  Enlargement of J (Bundle)

Figure 40-c. Enlargement of J (Compact)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (du).(dv) o                       o    |
|    |                       |   -->--   |                       |    |
|    |                       |   \   /   |                       |    |
|    |              dv .(du) |    \ /    | du .(dv)              |    |
|    |     u     o---------------->@<----------------o     v     |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 40-c.  Enlargement of J (Compact)

Figure 40-d. Enlargement of J (Digraph)

o-----------------------------------------------------------o
|                                                           |
|                         (du).(dv)                         |
|                          --->---                          |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                           u @ v                           |
|                            ^^^                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|               (du) dv /     |     \ du (dv)               |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  /          |          \                  |
|                 @           |           @                 |
|               u (v)         |         (u) v               |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du . dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             @                             |
|                                                           |
|                          (u).(v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 40-d.  Enlargement of J (Digraph)

Table 41. Computation of DJ (Method 1)

Table 41.  Computation of DJ (Method 1)
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  EJ                 +  !e!J                                             |
|                                                                               |
|     =  J<u + du, v + dv>  +  J<u, v>                                          |
|                                                                               |
|     =  (u, du)(v, dv)     +  u v                                              |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =        0                                                                |
|                                                                               |
|     +   u  v (du) dv   +   u (v)(du) dv                                       |
|                                                                               |
|     +   u  v  du (dv)                     +  (u) v  du (dv)                   |
|                                                                               |
|     +   u  v  du  dv                                         +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  u v ((du)(dv))  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 41. Computation of DJ (Method 1)
DJ = EJ + \(\epsilon\)J
  = J‹u + du, v + dv› + J‹u, v›
  = (u, du)(v, dv) + u v
DJ = 0      
  + u v (du) dv + u (v)(du) dv    
  + u v  du (dv)   + (u) v du (dv)  
  + u v  du  dv     + (u)(v) du dv
DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv


Table 42. Computation of DJ (Method 2)

Table 42.  Computation of DJ (Method 2)
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  !e!J            +  EJ                                                  |
|                                                                               |
|     =  J<u, v>         +  J<u + du, v + dv>                                   |
|                                                                               |
|     =  u v             +  (u, du)(v, dv)                                      |
|                                                                               |
|     =  0               +  u dv            +  v du            +  du dv         |
|                                                                               |
|     =  0               +  u (du) dv       +  v du (dv)       + ((u, v)) du dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 42. Computation of DJ (Method 2)
DJ = \(\epsilon\)J + EJ
  = J‹u, v› + J‹u + du, v + dv›
  = u v + (u, du)(v, dv)
  = 0 + u dv + v du + du dv
DJ = 0 + u (du) dv + v du (dv) + ((u, v)) du dv


Table 43. Computation of DJ (Method 3)

Table 43.  Computation of DJ (Method 3)
o-------------------------------------------------------------------------------o
|                                                                               |
|  DJ  =  !e!J           +   EJ                                                 |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| !e!J =  u  v (du)(dv)  +   u  v (du) dv   +   u  v  du (dv)  +   u  v  du  dv |
|                                                                               |
|  EJ  =  u  v (du)(dv)  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du  dv |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
|  DJ  =   0 . (du)(dv)  +    u . (du) dv   +     v . du (dv)  + ((u, v)) du dv |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 43. Computation of DJ (Method 3)
DJ = \(\epsilon\)J + EJ  
\(\epsilon\)J = u v (du)(dv) + u  v (du) dv +  u  v du (dv) +  u  v  du dv
EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv
DJ = 0 \(\cdot\) (du)(dv) + u \(\cdot\) (du) dv + v \(\cdot\) du (dv) + ((u, v)) du dv


Formula Display 8

o-------------------------------------------------------------------------------o
|                                                                               |
| !e!J  =  {Dispositions from  J  to  J }  +  {Dispositions from  J  to (J)}    |
|                                                                               |
|  EJ   =  {Dispositions from  J  to  J }  +  {Dispositions from (J) to  J }    |
|                                                                               |
|  DJ   =  (!e!J, EJ)                                                           |
|                                                                               |
|  DJ   =  {Dispositions from  J  to (J)}  +  {Dispositions from (J) to  J }    |
|                                                                               |
o-------------------------------------------------------------------------------o

\(\epsilon\)J = {Dispositions from J to J } + {Dispositions from J to (J) }
 
EJ = {Dispositions from J to J } + {Dispositions from (J) to J }
 
DJ = (\(\epsilon\)J, EJ)  
 
DJ = {Dispositions from J to (J) } + {Dispositions from (J) to J }


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

o---------------------------------------o
|                                       |
|                   o                   |
|                  / \                  |
|                 /   \                 |
|                /     \                |
|               o       o               |
|              /%\     /%\              |
|             /%%%\   /%%%\             |
|            /%%%%%\ /%%%%%\            |
|           o%%%%%%%o%%%%%%%o           |
|          /%\%%%%%/%\%%%%%/%\          |
|         /%%%\%%%/%%%\%%%/%%%\         |
|        /%%%%%\%/%%%%%\%/%%%%%\        |
|       o%%%%%%%o%%%%%%%o%%%%%%%o       |
|      / \%%%%%/ \%%%%%/ \%%%%%/ \      |
|     /   \%%%/   \%%%/   \%%%/   \     |
|    /     \%/     \%/     \%/     \    |
|   o       o       o       o       o   |
|   |\     / \     /%\     / \     /|   |
|   | \   /   \   /%%%\   /   \   / |   |
|   |  \ /     \ /%%%%%\ /     \ /  |   |
|   |   o       o%%%%%%%o       o   |   |
|   |   |\     / \%%%%%/ \     /|   |   |
|   |   | \   /   \%%%/   \   / |   |   |
|   | u |  \ /     \%/     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 44-a.  Difference Map of J (Areal)

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

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         /%\         \   |
                                                  |  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                 /|  o         o%%%o         o  |
                                                / |   \         \%/         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /%%%%%%%\ /       \     |
|          o---------o   o---------o          |   |    /%%%%%%%%%o         \    |
|         /           \ /           \         |   |   /%%%%%%%%%/ \         \   |
|        /             o             \        |   |  o%%%%%%%%%o   o         o  |
|       /             /`\      @------\-----------@  |%% du %%%|   |    dv   |  |
|      /             /```\             \      |   |  o%%%%%%%%%o   o         o  |
|     /             /`````\             \     |   |   \%%%%%%%%%\ /         /   |
|    /             /```````\             \    |   |    \%%%%%%%%%o         /    |
|   o             o`````````o             o   |   |     \%%%%%%%/ \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /%%%%%%%\     |
|    \             \```````/   \         /    |   |    /         o%%%%%%%%%\    |
|     \             \`````/     \       /     |   |   /         / \%%%%%%%%%\   |
|      \             \```/       \     /      |   |  o         o   o%%%%%%%%%o  |
|       \      @------\-/---------\---------------@  |   du    |   |%%% dv %%|  |
|        \             o           \ /        |   |  o         o   o%%%%%%%%%o  |
|         \           / \           /         |   |   \         \ /%%%%%%%%%/   |
|          o---------o   o---------o \        |   |    \         o%%%%%%%%%/    |
|                                     \       |   |     \       / \%%%%%%%/     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |                             |
                                             \    |      o-----o   o-----o      |
                                              \   |     /%%%%%%%\ /%%%%%%%\     |
                                               \  |    /%%%%%%%%%o%%%%%%%%%\    |
                                                \ |   /%%%%%%%%%/%\%%%%%%%%%\   |
                                                 \|  o%%%%%%%%%o%%%o%%%%%%%%%o  |
                                                  @  |%% du %%%|%%%|%%% dv %%|  |
                                                  |  o%%%%%%%%%o%%%o%%%%%%%%%o  |
                                                  |   \%%%%%%%%%\%/%%%%%%%%%/   |
                                                  |    \%%%%%%%%%o%%%%%%%%%/    |
                                                  |     \%%%%%%%/ \%%%%%%%/     |
                                                  |      o-----o   o-----o      |
                                                  |                             |
                                                  o-----------------------------o
Figure 44-b.  Difference Map of J (Bundle)

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

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |              dv .(du) |           | du .(dv)              |    |
|    |     u     @<--------------->@<--------------->@     v     |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  v                                  |
|                                  @                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 44-c.  Difference Map of J (Compact)

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

o-----------------------------------------------------------o
|                                                           |
|                            u v                            |
|                                                           |
|                             @                             |
|                            ^^^                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|               (du) dv /     |     \ du (dv)               |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  v          |          v                  |
|                 @           |           @                 |
|               u (v)         |         (u) v               |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du | dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             v                             |
|                             @                             |
|                                                           |
|                          (u) (v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 44-d.  Difference Map of J (Digraph)

Table 45. Computation of dJ

Table 45.  Computation of dJ
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  u v ((du)(dv))  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du dv  |
|                                                                               |
| =>                                                                            |
|                                                                               |
| dJ  =  u v  (du, dv)   +   u (v) dv       +  (u) v  du       +  (u)(v) . 0    |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 45. Computation of dJ
DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv
⇒
dJ = u v (du, dv) + u (v) dv + (u) v du + (u)(v) \(\cdot\) 0


Figure 46-a. Differential of J (Areal)

o---------------------------------------o
|                                       |
|                   o                   |
|                  / \                  |
|                 /   \                 |
|                /     \                |
|               o       o               |
|              /%\     /%\              |
|             /%%%\   /%%%\             |
|            /%%%%%\ /%%%%%\            |
|           o%%%%%%%o%%%%%%%o           |
|          /%\%%%%%/ \%%%%%/%\          |
|         /%%%\%%%/   \%%%/%%%\         |
|        /%%%%%\%/     \%/%%%%%\        |
|       o%%%%%%%o       o%%%%%%%o       |
|      / \%%%%%/%\     /%\%%%%%/ \      |
|     /   \%%%/%%%\   /%%%\%%%/   \     |
|    /     \%/%%%%%\ /%%%%%\%/     \    |
|   o       o%%%%%%%o%%%%%%%o       o   |
|   |\     / \%%%%%/ \%%%%%/ \     /|   |
|   | \   /   \%%%/   \%%%/   \   / |   |
|   |  \ /     \%/     \%/     \ /  |   |
|   |   o       o       o       o   |   |
|   |   |\     / \     / \     /|   |   |
|   |   | \   /   \   /   \   / |   |   |
|   | u |  \ /     \ /     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 46-a.  Differential of J (Areal)

Figure 46-b. Differential of J (Bundle)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         / \         \   |
                                                  |  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                 /|  o         o   o         o  |
                                                / |   \         \ /         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /%%%%%%%\ /       \     |
|          o---------o   o---------o          |   |    /%%%%%%%%%o         \    |
|         /           \ /           \         |   |   /%%%%%%%%%/%\         \   |
|        /             o             \        |   |  o%%%%%%%%%o%%%o         o  |
|       /             /`\      @------\-----------@  |%% du %%%|%%%|    dv   |  |
|      /             /```\             \      |   |  o%%%%%%%%%o%%%o         o  |
|     /             /`````\             \     |   |   \%%%%%%%%%\%/         /   |
|    /             /```````\             \    |   |    \%%%%%%%%%o         /    |
|   o             o`````````o             o   |   |     \%%%%%%%/ \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /%%%%%%%\     |
|    \             \```````/   \         /    |   |    /         o%%%%%%%%%\    |
|     \             \`````/     \       /     |   |   /         /%\%%%%%%%%%\   |
|      \             \```/       \     /      |   |  o         o%%%o%%%%%%%%%o  |
|       \      @------\-/---------\---------------@  |   du    |%%%|%%% dv %%|  |
|        \             o           \ /        |   |  o         o%%%o%%%%%%%%%o  |
|         \           / \           /         |   |   \         \%/%%%%%%%%%/   |
|          o---------o   o---------o \        |   |    \         o%%%%%%%%%/    |
|                                     \       |   |     \       / \%%%%%%%/     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |                             |
                                             \    |      o-----o   o-----o      |
                                              \   |     /%%%%%%%\ /%%%%%%%\     |
                                               \  |    /%%%%%%%%%o%%%%%%%%%\    |
                                                \ |   /%%%%%%%%%/ \%%%%%%%%%\   |
                                                 \|  o%%%%%%%%%o   o%%%%%%%%%o  |
                                                  @  |%% du %%%|   |%%% dv %%|  |
                                                  |  o%%%%%%%%%o   o%%%%%%%%%o  |
                                                  |   \%%%%%%%%%\ /%%%%%%%%%/   |
                                                  |    \%%%%%%%%%o%%%%%%%%%/    |
                                                  |     \%%%%%%%/ \%%%%%%%/     |
                                                  |      o-----o   o-----o      |
                                                  |                             |
                                                  o-----------------------------o
Figure 46-b.  Differential of J (Bundle)

Figure 46-c. Differential of J (Compact)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /   @   \                       \      |
|     /                       /   ^ ^   \                       \     |
|    o                       o   /   \   o                       o    |
|    |                       |  /     \  |                       |    |
|    |                       | /       \ |                       |    |
|    |                       |/         \|                       |    |
|    |         u         (du)/ dv     du \(dv)         v         |    |
|    |                      /|           |\                      |    |
|    |                     / |           | \                     |    |
|    |                    /  |           |  \                    |    |
|    o                   /   o           o   \                   o    |
|     \                 /     \         /     \                 /     |
|      \               v       \ du dv /       v               /      |
|       \             @<----------------------->@             /       |
|        \                       \   /                       /        |
|         \                       \ /                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 46-c.  Differential of J (Compact)

Figure 46-d. Differential of J (Digraph)

o-----------------------------------------------------------o
|                                                           |
|                            u v                            |
|                             @                             |
|                            ^ ^                            |
|                           /   \                           |
|                          /     \                          |
|                         /       \                         |
|                        /         \                        |
|               (du) dv /           \ du (dv)               |
|                      /             \                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  v                     v                  |
|           u (v) @<--------------------->@ (u) v           |
|                           du dv                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                             @                             |
|                          (u) (v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 46-d.  Differential of J (Digraph)

Table 47. Computation of rJ

Table 47.  Computation of rJ
o-------------------------------------------------------------------------------o
|                                                                               |
| rJ  =        DJ        +        dJ                                            |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| DJ  =  u v ((du)(dv))  +   u (v)(du) dv   +  (u) v  du (dv)  +  (u)(v) du dv  |
|                                                                               |
| dJ  =  u v  (du, dv)   +   u (v) dv       +  (u) v  du       +  (u)(v) . 0    |
|                                                                               |
o-------------------------------------------------------------------------------o
|                                                                               |
| rJ  =  u v   du  dv    +   u (v) du  dv   +  (u) v  du  dv   +  (u)(v) du dv  |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 47. Computation of rJ
rJ = DJ + dJ  
DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv
dJ = u v  (du, dv) + u (v) dv + (u) v du + (u)(v) \(\cdot\) 0
rJ = u v   du dv + u (v) du dv + (u) v du dv + (u)(v) du dv


Figure 48-a. Remainder of J (Areal)

o---------------------------------------o
|                                       |
|                   o                   |
|                  / \                  |
|                 /   \                 |
|                /     \                |
|               o       o               |
|              / \     / \              |
|             /   \   /   \             |
|            /     \ /     \            |
|           o       o       o           |
|          / \     /%\     / \          |
|         /   \   /%%%\   /   \         |
|        /     \ /%%%%%\ /     \        |
|       o       o%%%%%%%o       o       |
|      / \     /%\%%%%%/%\     / \      |
|     /   \   /%%%\%%%/%%%\   /   \     |
|    /     \ /%%%%%\%/%%%%%\ /     \    |
|   o       o%%%%%%%o%%%%%%%o       o   |
|   |\     / \%%%%%/%\%%%%%/ \     /|   |
|   | \   /   \%%%/%%%\%%%/   \   / |   |
|   |  \ /     \%/%%%%%\%/     \ /  |   |
|   |   o       o%%%%%%%o       o   |   |
|   |   |\     / \%%%%%/ \     /|   |   |
|   |   | \   /   \%%%/   \   / |   |   |
|   | u |  \ /     \%/     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 48-a.  Remainder of J (Areal)

Figure 48-b. Remainder of J (Bundle)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         /%\         \   |
                                                  |  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                 /|  o         o%%%o         o  |
                                                / |   \         \%/         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /       \ /       \     |
|          o---------o   o---------o          |   |    /         o         \    |
|         /           \ /           \         |   |   /         /%\         \   |
|        /             o             \        |   |  o         o%%%o         o  |
|       /             /`\      @------\-----------@  |   du    |%%%|    dv   |  |
|      /             /```\             \      |   |  o         o%%%o         o  |
|     /             /`````\             \     |   |   \         \%/         /   |
|    /             /```````\             \    |   |    \         o         /    |
|   o             o`````````o             o   |   |     \       / \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /       \     |
|    \             \```````/   \         /    |   |    /         o         \    |
|     \             \`````/     \       /     |   |   /         /%\         \   |
|      \             \```/       \     /      |   |  o         o%%%o         o  |
|       \      @------\-/---------\---------------@  |   du    |%%%|    dv   |  |
|        \             o           \ /        |   |  o         o%%%o         o  |
|         \           / \           /         |   |   \         \%/         /   |
|          o---------o   o---------o \        |   |    \         o         /    |
|                                     \       |   |     \       / \       /     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |                             |
                                             \    |      o-----o   o-----o      |
                                              \   |     /       \ /       \     |
                                               \  |    /         o         \    |
                                                \ |   /         /%\         \   |
                                                 \|  o         o%%%o         o  |
                                                  @  |   du    |%%%|    dv   |  |
                                                  |  o         o%%%o         o  |
                                                  |   \         \%/         /   |
                                                  |    \         o         /    |
                                                  |     \       / \       /     |
                                                  |      o-----o   o-----o      |
                                                  |                             |
                                                  o-----------------------------o
Figure 48-b.  Remainder of J (Bundle)

Figure 48-c. Remainder of J (Compact)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |   du dv   |                       |    |
|    |       u       @<------------------------->@       v       |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    o                       o     @     o                       o    |
|     \                       \    ^    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                    du | dv                    /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                                  |                                  |
|                                  v                                  |
|                                  @                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 48-c.  Remainder of J (Compact)

Figure 48-d. Remainder of J (Digraph)

o-----------------------------------------------------------o
|                                                           |
|                            u v                            |
|                             @                             |
|                             ^                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du | dv                          |
|           u (v) @<----------|---------->@ (u) v           |
|                          du | dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             v                             |
|                             @                             |
|                          (u) (v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 48-d.  Remainder of J (Digraph)

Table 49. Computation Summary for J

Table 49.  Computation Summary for J
o-------------------------------------------------------------------------------o
|                                                                               |
| !e!J  =  uv .     1       + u(v) .    0    + (u)v .   0     + (u)(v) .   0    |
|                                                                               |
|   EJ  =  uv .  (du)(dv)   + u(v) . (du)dv  + (u)v . du(dv)  + (u)(v) . du dv  |
|                                                                               |
|   DJ  =  uv . ((du)(dv))  + u(v) . (du)dv  + (u)v . du(dv)  + (u)(v) . du dv  |
|                                                                               |
|   dJ  =  uv .  (du, dv)   + u(v) .     dv  + (u)v . du      + (u)(v) .   0    |
|                                                                               |
|   rJ  =  uv .   du  dv    + u(v) .  du dv  + (u)v . du dv   + (u)(v) . du dv  |
|                                                                               |
o-------------------------------------------------------------------------------o

Table 49. Computation Summary for J
\(\epsilon\)J = uv \(\cdot\) 1 + u(v) \(\cdot\) 0 + (u)v \(\cdot\) 0 + (u)(v) \(\cdot\) 0
EJ = uv \(\cdot\) (du)(dv) + u(v) \(\cdot\) (du)dv + (u)v \(\cdot\) du(dv) + (u)(v) \(\cdot\) du dv
DJ = uv \(\cdot\) ((du)(dv)) + u(v) \(\cdot\) (du)dv + (u)v \(\cdot\) du(dv) + (u)(v) \(\cdot\) du dv
dJ = uv \(\cdot\) (du, dv) + u(v) \(\cdot\) dv + (u)v \(\cdot\) du + (u)(v) \(\cdot\) 0
rJ = uv \(\cdot\) du dv + u(v) \(\cdot\) du dv + (u)v \(\cdot\) du dv + (u)(v) \(\cdot\) du dv


Table 50. Computation of an Analytic Series in Terms of Coordinates

Table 50.  Computation of an Analytic Series in Terms of Coordinates
o-----------o-------------o-------------oo-------------o---------o-------------o
|  u     v  |  du     dv  |  u'     v'  || !e!J    EJ  |   DJ    |  dJ   d^2.J |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  0     0  |  0      0   |  0      0   ||  0      0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  0      1   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  1      0   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  1      1   ||         1   |    1    |  0      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  0     1  |  0      0   |  0      1   ||  0      0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  0      0   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  1      1   ||         1   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  1      0   ||         0   |    0    |  1      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  1     0  |  0      0   |  1      0   ||  0      0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  1      1   ||         1   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  0      0   ||         0   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  0      1   ||         0   |    0    |  1      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o
|           |             |             ||             |         |             |
|  1     1  |  0      0   |  1      1   ||  1      1   |    0    |  0      0   |
|           |             |             ||             |         |             |
|           |  0      1   |  1      0   ||         0   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      0   |  0      1   ||         0   |    1    |  1      0   |
|           |             |             ||             |         |             |
|           |  1      1   |  0      0   ||         0   |    1    |  0      1   |
|           |             |             ||             |         |             |
o-----------o-------------o-------------oo-------------o---------o-------------o

Formula Display 9

o-------------------------------------------------o
|                                                 |
|         u'   =   u + du   =   (u, du)           |
|                                                 |
|         v'   =   v + du   =   (v, dv)           |
|                                                 |
o-------------------------------------------------o


  u’ = u + du = (u, du)  
  v’ = v + du = (v, dv)  


Formula Display 10

o--------------------------------------------------------------o
|                                                              |
|   EJ<u, v, du, dv>   =   J<u + du, v + dv>   =   J<u', v'>   |
|                                                              |
o--------------------------------------------------------------o


EJ‹u, v, du, dv› = J‹u + du, v + dv› = J‹u’, v’›


Table 51. Computation of an Analytic Series in Symbolic Terms

Table 51.  Computation of an Analytic Series in Symbolic Terms
o-----------o---------o------------o------------o------------o-----------o
|  u     v  |    J    |     EJ     |     DJ     |     dJ     |   d^2.J   |
o-----------o---------o------------o------------o------------o-----------o
|           |         |            |            |            |           |
|  0     0  |    0    |   du  dv   |   du  dv   |     ()     |   du dv   |
|           |         |            |            |            |           |
|  0     1  |    0    |   du (dv)  |   du (dv)  |     du     |   du dv   |
|           |         |            |            |            |           |
|  1     0  |    0    |  (du) dv   |  (du) dv   |     dv     |   du dv   |
|           |         |            |            |            |           |
|  1     1  |    1    |  (du)(dv)  | ((du)(dv)) |  (du, dv)  |   du dv   |
|           |         |            |            |            |           |
o-----------o---------o------------o------------o------------o-----------o

Table 51. Computation of an Analytic Series in Symbolic Terms
u v
J
EJ
DJ
dJ
d2J
0 0
0 1
1 0
1 1
0
0
0
1
 du  dv 
 du (dv)
(du) dv 
(du)(dv)
  du  dv  
  du (dv) 
 (du) dv  
((du)(dv))
()
du
dv
(du, dv)
du dv
du dv
du dv
du dv


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

            o                           o                           o
           /%\                         /%\                         / \
          /%%%\                       /%%%\                       /   \
         o%%%%%o                     o%%%%%o                     o     o
        / \%%%/ \                   /%\%%%/%\                   /%\   /%\
       /   \%/   \                 /%%%\%/%%%\                 /%%%\ /%%%\
      o     o     o               o%%%%%o%%%%%o               o%%%%%o%%%%%o
     /%\   / \   /%\             / \%%%/%\%%%/ \             /%\%%%/%\%%%/%\
    /%%%\ /   \ /%%%\           /   \%/%%%\%/   \           /%%%\%/%%%\%/%%%\
   o%%%%%o     o%%%%%o         o     o%%%%%o     o         o%%%%%o%%%%%o%%%%%o
  / \%%%/ \   / \%%%/ \       / \   / \%%%/ \   / \       / \%%%/ \%%%/ \%%%/ \
 /   \%/   \ /   \%/   \     /   \ /   \%/   \ /   \     /   \%/   \%/   \%/   \
o     o     o     o     o   o     o     o     o     o   o     o     o     o     o
|\   / \   /%\   / \   /|   |\   / \   / \   / \   /|   |\   / \   /%\   / \   /|
| \ /   \ /%%%\ /   \ / |   | \ /   \ /   \ /   \ / |   | \ /   \ /%%%\ /   \ / |
|  o     o%%%%%o     o  |   |  o     o     o     o  |   |  o     o%%%%%o     o  |
|  |\   / \%%%/ \   /|  |   |  |\   / \   / \   /|  |   |  |\   / \%%%/ \   /|  |
|u | \ /   \%/   \ / | v|   |u | \ /   \ /   \ / | v|   |u | \ /   \%/   \ / | v|
o--+--o     o     o--+--o   o--+--o     o     o--+--o   o--+--o     o     o--+--o
   |   \   / \   /   |         |   \   / \   /   |         |   \   / \   /   |
   | du \ /   \ / dv |         | du \ /   \ / dv |         | du \ /   \ / dv |
   o-----o     o-----o         o-----o     o-----o         o-----o     o-----o
          \   /                       \   /                       \   /
           \ /                         \ /                         \ /
            o                           o                           o

           EJ             =             J             +            DJ

o-----------------------o   o-----------------------o   o-----------------------o
|                       |   |                       |   |                       |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|     /    \ /    \     |   |     /    \ /    \     |   |     /    \ /    \     |
|    /      o      \    |   |    /      o      \    |   |    /      o      \    |
|   /  u   / \   v  \   |   |   /  u   / \   v  \   |   |   /  u   / \   v  \   |
|  o      /->-\      o  |   |  o      /->-\      o  |   |  o      /   \      o  |
|  |     o \ / o     |  |   |  |     o \ / o     |  |   |  |     o     o     |  |
|  |  @--|->@<-|--@  |  |   |  |  @<-|--@--|->@  |  |   |  |  @<-|->@<-|->@  |  |
|  |     o  ^  o     |  |   |  |     o  |  o     |  |   |  |     o  ^  o     |  |
|  o      \ | /      o  |   |  o      \ | /      o  |   |  o      \ | /      o  |
|   \      \|/      /   |   |   \      \|/      /   |   |   \      \|/      /   |
|    \      |      /    |   |    \      |      /    |   |    \      |      /    |
|     \    /|\    /     |   |     \    /|\    /     |   |     \    /|\    /     |
|      o--o | o--o      |   |      o--o v o--o      |   |      o--o v o--o      |
|           @           |   |           @           |   |           @           |
o-----------------------o   o-----------------------o   o-----------------------o
Figure 52.  Decomposition of the Enlarged Conjunction EJ = (J, DJ)

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

            o                           o                           o
           / \                         / \                         / \
          /   \                       /   \                       /   \
         o     o                     o     o                     o     o
        /%\   /%\                   /%\   /%\                   / \   / \
       /%%%\ /%%%\                 /%%%\%/%%%\                 /   \ /   \
      o%%%%%o%%%%%o               o%%%%%o%%%%%o               o     o     o
     /%\%%%/%\%%%/%\             /%\%%%/ \%%%/%\             / \   /%\   / \
    /%%%\%/%%%\%/%%%\           /%%%\%/   \%/%%%\           /   \ /%%%\ /   \
   o%%%%%o%%%%%o%%%%%o         o%%%%%o     o%%%%%o         o     o%%%%%o     o
  / \%%%/ \%%%/ \%%%/ \       / \%%%/%\   /%\%%%/ \       / \   /%\%%%/%\   / \
 /   \%/   \%/   \%/   \     /   \%/%%%\ /%%%\%/   \     /   \ /%%%\%/%%%\ /   \
o     o     o     o     o   o     o%%%%%o%%%%%o     o   o     o%%%%%o%%%%%o     o
|\   / \   /%\   / \   /|   |\   / \%%%/ \%%%/ \   /|   |\   / \%%%/%\%%%/ \   /|
| \ /   \ /%%%\ /   \ / |   | \ /   \%/   \%/   \ / |   | \ /   \%/%%%\%/   \ / |
|  o     o%%%%%o     o  |   |  o     o     o     o  |   |  o     o%%%%%o     o  |
|  |\   / \%%%/ \   /|  |   |  |\   / \   / \   /|  |   |  |\   / \%%%/ \   /|  |
|u | \ /   \%/   \ / | v|   |u | \ /   \ /   \ / | v|   |u | \ /   \%/   \ / | v|
o--+--o     o     o--+--o   o--+--o     o     o--+--o   o--+--o     o     o--+--o
   |   \   / \   /   |         |   \   / \   /   |         |   \   / \   /   |
   | du \ /   \ / dv |         | du \ /   \ / dv |         | du \ /   \ / dv |
   o-----o     o-----o         o-----o     o-----o         o-----o     o-----o
          \   /                       \   /                       \   /
           \ /                         \ /                         \ /
            o                           o                           o

           DJ             =            dJ             +            ddJ

o-----------------------o   o-----------------------o   o-----------------------o
|                       |   |                       |   |                       |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|     /    \ /    \     |   |     /    \ /    \     |   |     /    \ /    \     |
|    /      o      \    |   |    /      o      \    |   |    /      o      \    |
|   /  u   / \   v  \   |   |   /  u   / \   v  \   |   |   /  u   / \   v  \   |
|  o      /   \      o  |   |  o      /   \      o  |   |  o      /   \      o  |
|  |     o     o     |  |   |  |     o     o     |  |   |  |     o     o     |  |
|  |  @<-|->@<-|->@  |  |   |  |  @<-|->@<-|->@  |  |   |  |  @<-|-----|->@  |  |
|  |     o  ^  o     |  |   |  |   ^ o     o ^   |  |   |  |     o  @  o     |  |
|  o      \ | /      o  |   |  o    \ \   / /    o  |   |  o      \ ^ /      o  |
|   \      \|/      /   |   |   \    --\-/--    /   |   |   \      \|/      /   |
|    \      |      /    |   |    \      o      /    |   |    \      |      /    |
|     \    /|\    /     |   |     \    / \    /     |   |     \    /|\    /     |
|      o--o v o--o      |   |      o--o   o--o      |   |      o--o v o--o      |
|           @           |   |           @           |   |           @           |
o-----------------------o   o-----------------------o   o-----------------------o
Figure 53.  Decomposition of the Differed Conjunction DJ = (dJ, ddJ)

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

Table 54.  Cast of Characters:  Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation                | Description      | Type                       |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| U%   | = [u, v]                | Source Universe  | [B^2]                      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| X%   | = [x]                   | Target Universe  | [B^1]                      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EU%  | = [u, v, du, dv]        | Extended         | [B^2 x D^2]                |
|      |                         | Source Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EX%  | = [x, dx]               | Extended         | [B^1 x D^1]                |
|      |                         | Target Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| J    | J : U -> B              | Proposition      | (B^2 -> B) c [B^2]         |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| J    | J : U% -> X%            | Transformation,  | [B^2] -> [B^1]             |
|      |                         | or Mapping       |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| W    | W :                     | Operator         |                            |
|      | U% -> EU%,              |                  | [B^2] -> [B^2 x D^2],      |
|      | X% -> EX%,              |                  | [B^1] -> [B^1 x D^1],      |
|      | (U%->X%)->(EU%->EX%),   |                  | ([B^2] -> [B^1])           |
|      | for each W among:       |                  | ->                         |
|      | e!, !h!, E, D, d        |                  | ([B^2 x D^2]->[B^1 x D^1]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| !e!  |                         | Tacit Extension Operator   !e!                |
| !h!  |                         | Trope Extension Operator   !h!                |
|  E   |                         | Enlargement Operator        E                 |
|  D   |                         | Difference Operator         D                 |
|  d   |                         | Differential Operator       d                 |
|      |                         |                                               |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| $W$  | $W$ :                   | Operator         |                            |
|      | U% -> $T$U% = EU%,      |                  | [B^2] -> [B^2 x D^2],      |
|      | X% -> $T$X% = EX%,      |                  | [B^1] -> [B^1 x D^1],      |
|      | (U%->X%)->($T$U%->$T$X%)|                  | ([B^2] -> [B^1])           |
|      | for each $W$ among:     |                  | ->                         |
|      | $e$, $E$, $D$, $T$      |                  | ([B^2 x D^2]->[B^1 x D^1]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| $e$  |                         | Radius Operator            $e$ = <!e!, !h!>   |
| $E$  |                         | Secant Operator            $E$ = <!e!,  E >   |
| $D$  |                         | Chord Operator             $D$ = <!e!,  D >   |
| $T$  |                         | Tangent Functor            $T$ = <!e!,  d >   |
|      |                         |                                               |
o------o-------------------------o-----------------------------------------------o

Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes

Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
|              | Operator             | Proposition        | Map                  |
o--------------o----------------------o--------------------o----------------------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--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Trope        | !h! :                | !h!J :             | !h!J :               |
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------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--------------o----------------------o--------------------o----------------------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--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Differential | d :                  | dJ :               | dJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Remainder    | r :                  | rJ :               | rJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Radius       | $e$ = <!e!, !h!> :   |                    | $e$J :               |
| Operator     | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------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--------------o----------------------o--------------------o----------------------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--------------o----------------------o--------------------o----------------------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--------------o----------------------o--------------------o----------------------o

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

                              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  |  =
              =   |  |\   / \   / \   /|  |   =
             =    |u | \ /   \ /   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       /\\
          ////\      | du \ /   \ / dv |      /\\\\
         o/////o     o-----o     o-----o     o\\\\\o
        //\/////\           \   /           /\\\\\/\\
       ////\/////\           \ /           /\\\\\/\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         / \\\\/\\\\\/ \
    /   \/////\//   \       =   =       /   \\/\\\\\/   \
   o     o/////o     o     =     =     o     o\\\\\o     o
  / \   / \//// \   / \   =       =   / \   / \\\\/ \   / \
 /   \ /   \//   \ /   \ =         = /   \ /   \\/   \ /   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   / \   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /   \ /   \ / |
|  o     o     o     o  |           |  o     o     o     o  |
|  |\   / \   / \   /|  |           |  |\   / \   / \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \ /   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\          !h!J
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       /
      x = uv               \     /             dx = uv
                            \   /
                             \ /
                              o

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

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

                              o
                             /X\
                            /XXX\
                           oXXXXXo
                          //\XXX//\
                         ////\X////\
                        o/////o/////o
                       /\\/////\////\\
                      /\\\\/////\//\\\\
                     o\\\\\o/////o\\\\\o
                    / \\\\/ \//// \\\\/ \
                   /   \\/   \//   \\/   \
                  o     o     o     o     o
                 =|\   / \   /\\   / \   /|=
                = | \ /   \ /\\\\ /   \ / | =
               =  |  o     o\\\\\o     o  |  =
              =   |  |\   / \\\\/ \   /|  |   =
             =    |u | \ /   \\/   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       /\\
          ////\      | du \ /   \ / dv |      /\\\\
         o/////o     o-----o     o-----o     o\\\\\o
        //\/////\           \   /           / \\\\/ \
       ////\/////\           \ /           /   \\/   \
      o/////o/////o           o           o     o     o
     / \/////\//// \         = =         /\\   / \   /\\
    /   \/////\//   \       =   =       /\\\\ /   \ /\\\\
   o     o/////o     o     =     =     o\\\\\o     o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/ \   / \\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/   \ /   \\/   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   /\\   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /\\\\ /   \ / |
|  o     o     o     o  |           |  o     o\\\\\o     o  |
|  |\   / \   / \   /|  |           |  |\   / \\\\/ \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \\/   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           EJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |. 
                  o-----o           o-----o 
                         \         /
                          \       / dx = (u, du)(v, dv)
      x = uv               \     /
                            \   /   dx = uv + u dv + v du + du dv
                             \ /
                              o

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

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

                              o
                             //\
                            ////\
                           o/////o
                          /X\////X\
                         /XXX\//XXX\
                        oXXXXXoXXXXXo
                       /\\XXX/X\XXX/\\
                      /\\\\X/XXX\X/\\\\
                     o\\\\\oXXXXXo\\\\\o
                    / \\\\/ \XXX/ \\\\/ \
                   /   \\/   \X/   \\/   \
                  o     o     o     o     o
                 =|\   / \   /\\   / \   /|=
                = | \ /   \ /\\\\ /   \ / | =
               =  |  o     o\\\\\o     o  |  =
              =   |  |\   / \\\\/ \   /|  |   =
             =    |u | \ /   \\/   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       / \
          ////\      | du \ /   \ / dv |      /   \
         o/////o     o-----o     o-----o     o     o
        //\/////\           \   /           /\\   /\\
       ////\/////\           \ /           /\\\\ /\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         /\\\\\/\\\\\/\\
    /   \/////\//   \       =   =       /\\\\\/\\\\\/\\\\
   o     o/////o     o     =     =     o\\\\\o\\\\\o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/ \\\\/ \\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/   \\/   \\/   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   /\\   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /\\\\ /   \ / |
|  o     o     o     o  |           |  o     o\\\\\o     o  |
|  |\   / \   / \   /|  |           |  |\   / \\\\/ \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \\/   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           DJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       / dx = (u, du)(v, dv) - uv
      x = uv               \     /
                            \   /   dx = u dv + v du + du dv
                             \ /
                              o

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

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

                              o
                             //\
                            ////\
                           o/////o
                          /X\////X\
                         /XXX\//XXX\
                        oXXXXXoXXXXXo
                       /\\XXX//\XXX/\\
                      /\\\\X////\X/\\\\
                     o\\\\\o/////o\\\\\o
                    / \\\\/\\////\\\\\/ \
                   /   \\/\\\\//\\\\\/   \
                  o     o\\\\\o\\\\\o     o
                 =|\   / \\\\/ \\\\/ \   /|=
                = | \ /   \\/   \\/   \ / | =
               =  |  o     o     o     o  |  =
              =   |  |\   / \   / \   /|  |   =
             =    |u | \ /   \ /   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       / \
          ////\      | du \ /   \ / dv |      /   \
         o/////o     o-----o     o-----o     o     o
        //\/////\           \   /           /\\   /\\
       ////\/////\           \ /           /\\\\ /\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         /\\\\\/ \\\\/\\
    /   \/////\//   \       =   =       /\\\\\/   \\/\\\\
   o     o/////o     o     =     =     o\\\\\o     o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/\\   /\\\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/\\\\ /\\\\\/   \
o     o     o     o     o           o     o\\\\\o\\\\\o     o
|\   / \   / \   / \   /|           |\   / \\\\/ \\\\/ \   /|
| \ /   \ /   \ /   \ / |           | \ /   \\/   \\/   \ / |
|  o     o     o     o  |           |  o     o     o     o  |
|  |\   / \   / \   /|  |           |  |\   / \   / \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \ /   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           dJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       /
      x = uv               \     /  dx = u dv + v du
                            \   /
                             \ /
                              o

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

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

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  / \  dv  \   |
|  o      /   \      o  |
|  |     o     o     |  |
|  |     |     |     |  |
|  |     o     o     |  |
|  o      \   /      o  |
|   \      \ /      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /    \ /    \     |      \
|    /      o      \    |       \
|   /  du  / \  dv  \   |        \
|  o      /   \      o  |         \
|  |     o     o     |  @          \
|  |     |     |     |  |\          \
|  |     o     o     |  | \          \
|  o      \   /      o  |  \          \
|   \      \ /      /   |   \          \
|    \      o      /    |    \          \
|     \    / \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |```````````````````````|
|                       | \ |          \          @ |   |```````````````````````|
|                       |  \|           \           |   |```````````````````````|
|      o--o   o--o      |   \      o--o  \o--o      |   |``````o--o```o--o``````|
|     /    \ /    \     |   |\    /    \ /\   \     |   |`````/````\`/````\`````|
|    /      o      \    |   | \  /      o  @   \    |   |````/``````o``````\````|
|   /  du  / \  dv  \   |   |  \/  du  /`\  dv  \   |   |```/``du``/`\``dv``\```|
|  o      /   \      o  |   |  o\     /```\      o  |   |``o``````/```\``````o``|
|  |     o     o     |  |   |  | \   o`````o     |  |   |``|`````o`````o`````|``|
|  |     |     |     |  |   |  |  @  |``@--|-----|------@``|`````|`````|`````|``|
|  |     o     o     |  |   |  |     o`````o     |  |   |``|`````o`````o`````|``|
|  o      \   /      o  |   |  o      \```/      o  |   |``o``````\```/``````o``|
|   \      \ /      /   |   |   \      \`/      /   |   |```\``````\`/``````/```|
|    \      o      /    |   |    \      o      /    |   |````\``````o``````/````|
|     \    / \    /     |   |     \    / \    /     |   |`````\````/`\````/`````|
|      o--o   o--o      |   |      o--o   o--o      |   |``````o--o```o--o``````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \       !h!J        /        \        J        /        \       !h!J        /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o

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

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

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  /`\  dv  \   |
|  o      /```\      o  |
|  |     o`````o     |  |
|  |     |`````|     |  |
|  |     o`````o     |  |
|  o      \```/      o  |
|   \      \`/      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /````\ /    \     |      \
|    /``````o      \    |       \
|   /``du``/ \  dv  \   |        \
|  o``````/   \      o  |         \
|  |`````o     o     |  @          \
|  |`````|     |     |  |\          \
|  |`````o     o     |  | \          \
|  o``````\   /      o  |  \          \
|   \``````\ /      /   |   \          \
|    \``````o      /    |    \          \
|     \````/ \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |```````````````````````|
|                       | \ |          \          @ |   |```````````````````````|
|                       |  \|           \           |   |```````````````````````|
|      o--o   o--o      |   \      o--o  \o--o      |   |``````o--o```o--o``````|
|     /    \ /````\     |   |\    /    \ /\   \     |   |`````/    \`/    \`````|
|    /      o``````\    |   | \  /      o  @   \    |   |````/      o      \````|
|   /  du  / \``dv``\   |   |  \/  du  /`\  dv  \   |   |```/  du  / \  dv  \```|
|  o      /   \``````o  |   |  o\     /```\      o  |   |``o      /   \      o``|
|  |     o     o`````|  |   |  | \   o`````o     |  |   |``|     o     o     |``|
|  |     |     |`````|  |   |  |  @  |``@--|-----|------@``|     |     |     |``|
|  |     o     o`````|  |   |  |     o`````o     |  |   |``|     o     o     |``|
|  o      \   /``````o  |   |  o      \```/      o  |   |``o      \   /      o``|
|   \      \ /``````/   |   |   \      \`/      /   |   |```\      \ /      /```|
|    \      o``````/    |   |    \      o      /    |   |````\      o      /````|
|     \    / \````/     |   |     \    / \    /     |   |`````\    /`\    /`````|
|      o--o   o--o      |   |      o--o   o--o      |   |``````o--o```o--o``````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
|                       |   |                       |   |```````````````````````|
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \        EJ         /        \        J        /        \        EJ         /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o

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

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

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  /`\  dv  \   |
|  o      /```\      o  |
|  |     o`````o     |  |
|  |     |`````|     |  |
|  |     o`````o     |  |
|  o      \```/      o  |
|   \      \`/      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /````\ /    \     |      \
|    /``````o      \    |       \
|   /``du``/ \  dv  \   |        \
|  o``````/   \      o  |         \
|  |`````o     o     |  @          \
|  |`````|     |     |  |\          \
|  |`````o     o     |  | \          \
|  o``````\   /      o  |  \          \
|   \``````\ /      /   |   \          \
|    \``````o      /    |    \          \
|     \````/ \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |                       |
|                       | \ |          \          @ |   |                       |
|                       |  \|           \           |   |                       |
|      o--o   o--o      |   \      o--o  \o--o      |   |      o--o   o--o      |
|     /    \ /````\     |   |\    /    \ /\   \     |   |     /````\ /````\     |
|    /      o``````\    |   | \  /      o  @   \    |   |    /``````o``````\    |
|   /  du  / \``dv``\   |   |  \/  du  /`\  dv  \   |   |   /``du``/`\``dv``\   |
|  o      /   \``````o  |   |  o\     /```\      o  |   |  o``````/```\``````o  |
|  |     o     o`````|  |   |  | \   o`````o     |  |   |  |`````o`````o`````|  |
|  |     |     |`````|  |   |  |  @  |``@--|-----|------@  |`````|`````|`````|  |
|  |     o     o`````|  |   |  |     o`````o     |  |   |  |`````o`````o`````|  |
|  o      \   /``````o  |   |  o      \```/      o  |   |  o``````\```/``````o  |
|   \      \ /``````/   |   |   \      \`/      /   |   |   \``````\`/``````/   |
|    \      o``````/    |   |    \      o      /    |   |    \``````o``````/    |
|     \    / \````/     |   |     \    / \    /     |   |     \````/ \````/     |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \        DJ         /        \        J        /        \        DJ         /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o

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

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

o-----------------------o
|                       |
|                       |
|                       |
|      o--o   o--o      |
|     /    \ /    \     |
|    /      o      \    |
|   /  du  / \  dv  \   |
|  o      /   \      o  |
|  |     o     o     |  |
|  |     |     |     |  |
|  |     o     o     |  |
|  o      \   /      o  |
|   \      \ /      /   |
|    \      o      /    |
|     \    / \    /     |
|      o--o   o--o      |
|                       |
|                       |
|                       |
o-----------------------@
                         \
o-----------------------o \
|                       |  \
|                       |   \
|                       |    \
|      o--o   o--o      |     \
|     /````\ /    \     |      \
|    /``````o      \    |       \
|   /``du``/`\  dv  \   |        \
|  o``````/```\      o  |         \
|  |`````o`````o     |  @          \
|  |`````|`````|     |  |\          \
|  |`````o`````o     |  | \          \
|  o``````\```/      o  |  \          \
|   \``````\`/      /   |   \          \
|    \``````o      /    |    \          \
|     \````/ \    /     |     \          \
|      o--o   o--o      |      \          \
|                       |       \          \
|                       |        \          \
|                       |         \          \
o-----------------------o          \          \
                                    \          \
o-----------------------@   o--------\----------\---o   o-----------------------o
|                       |\  |         \          \  |   |                       |
|                       | \ |          \          @ |   |                       |
|                       |  \|           \           |   |                       |
|      o--o   o--o      |   \      o--o  \o--o      |   |      o--o   o--o      |
|     /    \ /````\     |   |\    /    \ /\   \     |   |     /````\ /````\     |
|    /      o``````\    |   | \  /      o  @   \    |   |    /``````o``````\    |
|   /  du  /`\``dv``\   |   |  \/  du  /`\  dv  \   |   |   /``du``/ \``dv``\   |
|  o      /```\``````o  |   |  o\     /```\      o  |   |  o``````/   \``````o  |
|  |     o`````o`````|  |   |  | \   o`````o     |  |   |  |`````o     o`````|  |
|  |     |`````|`````|  |   |  |  @  |``@--|-----|------@  |`````|     |`````|  |
|  |     o`````o`````|  |   |  |     o`````o     |  |   |  |`````o     o`````|  |
|  o      \```/``````o  |   |  o      \```/      o  |   |  o``````\   /``````o  |
|   \      \`/``````/   |   |   \      \`/      /   |   |   \``````\ /``````/   |
|    \      o``````/    |   |    \      o      /    |   |    \``````o``````/    |
|     \    / \````/     |   |     \    / \    /     |   |     \````/ \````/     |
|      o--o   o--o      |   |      o--o   o--o      |   |      o--o   o--o      |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
|                       |   |                       |   |                       |
o-----------------------o   o-----------------------o   o-----------------------o
 \                     /     \                     /     \                     /
  \        dJ         /        \        J        /        \        dJ         /
   \                 /           \             /           \                 /
    \               /   o----------\---------/----------o   \               /
     \             /    |            \     /            |    \             /
      \           /     |              \ /              |     \           /
       \         /      |         o-----o-----o         |      \         /
        \       /       |        /`````````````\        |       \       /
         \     /        |       /```````````````\       |        \     /
   o------\---/------o  |      /`````````````````\      |  o------\---/------o
   |       \ /       |  |     /```````````````````\     |  |       \ /       |
   |     o--o--o     |  |    /`````````````````````\    |  |     o--o--o     |
   |    /```````\    |  |   o```````````````````````o   |  |    /```````\    |
   |   /`````````\   |  |   |```````````````````````|   |  |   /`````````\   |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |  |````dx`````|  @----@ |```````````x`````@-----|------@  |``` dx ````|  |
   |  o```````````o  |  |   |```````````````````````|   |  |  o```````````o  |
   |   \`````````/   |  |   |```````````````````````|   |  |   \`````````/   |
   |    \```````/    |  |   o```````````````````````o   |  |    \```````/    |
   |     o-----o     |  |    \`````````````````````/    |  |     o-----o     |
   |                 |  |     \```````````````````/     |  |                 |
   o-----------------o  |      \`````````````````/      |  o-----------------o
                        |       \```````````````/       |
                        |        \`````````````/        |
                        |         o-----------o         |
                        |                               |
                        |                               |
                        o-------------------------------o

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

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

            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             oXXXXXo
        ////////\                           /X\XXX/X\
       //////////\                         /XXX\X/XXX\
      o///////////o                       oXXXXXoXXXXXo
     / \////////// \                     / \XXX/X\XXX/ \
    /   \////////   \                   /   \X/XXX\X/   \
   /     \//////     \                 o     oXXXXXo     o
  /       \////       \               / \   / \XXX/ \   / \
 /         \//         \             /   \ /   \X/   \ /   \
o           o           o           o     o     o     o     o
|\         / \         /|           |\   / \   / \   / \   /|
| \       /   \       / |           | \ /   \ /   \ /   \ / |
|  \     /     \     /  |           |  o     o     o     o  |
|   \   /       \   /   |           |  |\   / \   / \   /|  |
| u  \ /         \ /  v |           |u | \ /   \ /   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $e$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $e$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $e$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o

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

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

            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             oXXXXXo
        ////////\                           //\XXX//\
       //////////\                         ////\X////\
      o///////////o                       o/////o/////o
     / \////////// \                     /\\/////\////\\
    /   \////////   \                   /\\\\/////\//\\\\
   /     \//////     \                 o\\\\\o/////o\\\\\o
  /       \////       \               / \\\\/ \//// \\\\/ \
 /         \//         \             /   \\/   \//   \\/   \
o           o           o           o     o     o     o     o
|\         / \         /|           |\   / \   /\\   / \   /|
| \       /   \       / |           | \ /   \ /\\\\ /   \ / |
|  \     /     \     /  |           |  o     o\\\\\o     o  |
|   \   /       \   /   |           |  |\   / \\\\/ \   /|  |
| u  \ /         \ /  v |           |u | \ /   \\/   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $E$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $E$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $E$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o

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

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

            o                                   o
           //\                                 //\
          ////\                               ////\
         //////\                             o/////o
        ////////\                           /X\////X\
       //////////\                         /XXX\//XXX\
      o///////////o                       oXXXXXoXXXXXo
     / \////////// \                     /\\XXX/X\XXX/\\
    /   \////////   \                   /\\\\X/XXX\X/\\\\
   /     \//////     \                 o\\\\\oXXXXXo\\\\\o
  /       \////       \               / \\\\/ \XXX/ \\\\/ \
 /         \//         \             /   \\/   \X/   \\/   \
o           o           o           o     o     o     o     o
|\         / \         /|           |\   / \   /\\   / \   /|
| \       /   \       / |           | \ /   \ /\\\\ /   \ / |
|  \     /     \     /  |           |  o     o\\\\\o     o  |
|   \   /       \   /   |           |  |\   / \\\\/ \   /|  |
| u  \ /         \ /  v |           |u | \ /   \\/   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $D$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $D$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $D$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o

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

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

            o                                   o
           //\                                 //\
          ////\                               ////\
         //////\                             o/////o
        ////////\                           /X\////X\
       //////////\                         /XXX\//XXX\
      o///////////o                       oXXXXXoXXXXXo
     / \////////// \                     /\\XXX//\XXX/\\
    /   \////////   \                   /\\\\X////\X/\\\\
   /     \//////     \                 o\\\\\o/////o\\\\\o
  /       \////       \               / \\\\/\\////\\\\\/ \
 /         \//         \             /   \\/\\\\//\\\\\/   \
o           o           o           o     o\\\\\o\\\\\o     o
|\         / \         /|           |\   / \\\\/ \\\\/ \   /|
| \       /   \       / |           | \ /   \\/   \\/   \ / |
|  \     /     \     /  |           |  o     o     o     o  |
|   \   /       \   /   |           |  |\   / \   / \   /|  |
| u  \ /         \ /  v |           |u | \ /   \ /   \ / | v|
o-----o           o-----o           o--+--o     o     o--+--o
       \         /                     |   \   / \   /   |
        \       /                      | du \ /   \ / dv |
         \     /                       o-----o     o-----o
          \   /                               \   /
           \ /                                 \ /
            o                                   o
                 U%          $T$          $E$U%
                    o------------------>o
                    |                   |
                    |                   |
                    |                   |
                    |                   |
                 J  |                   | $T$J
                    |                   |
                    |                   |
                    |                   |
                    v                   v
                    o------------------>o
                 X%          $T$          $E$X%
            o                                   o
           //\                                 /X\
          ////\                               /XXX\
         //////\                             /XXXXX\
        ////////\                           /XXXXXXX\
       //////////\                         /XXXXXXXXX\
      ////////////o                       oXXXXXXXXXXXo
     ///////////// \                     //\XXXXXXXXX/\\
    /////////////   \                   ////\XXXXXXX/\\\\
   /////////////     \                 //////\XXXXX/\\\\\\
  /////////////       \               ////////\XXX/\\\\\\\\
 /////////////         \             //////////\X/\\\\\\\\\\
o////////////           o           o///////////o\\\\\\\\\\\o
|\//////////           /            |\////////// \\\\\\\\\\/|
| \////////           /             | \////////   \\\\\\\\/ |
|  \//////           /              |  \//////     \\\\\\/  |
|   \////           /               |   \////       \\\\/   |
| x  \//           /                | x  \//         \\/ dx |
o-----o           /                 o-----o           o-----o
       \         /                         \         /
        \       /                           \       /
         \     /                             \     /
          \   /                               \   /
           \ /                                 \ /
            o                                   o

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

Formula Display 11

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

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

Table 58.  Cast of Characters:  Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation                | Description      | Type                       |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| U%   | = [u, v]                | Source Universe  | [B^n]                      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| X%   | = [x, y]                | Target Universe  | [B^k]                      |
|      | = [f, g]                |                  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EU%  | = [u, v, du, dv]        | Extended         | [B^n x D^n]                |
|      |                         | Source Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| EX%  | = [x, y, dx, dy]        | Extended         | [B^k x D^k]                |
|      | = [f, g, df, dg]        | Target Universe  |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| F    | F = <f, g> : U% -> X%   | Transformation,  | [B^n] -> [B^k]             |
|      |                         | or Mapping       |                            |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
|      | f, g : U -> B           | Proposition,     | B^n -> B                   |
|      |                         |   special case   |                            |
| f    | f : U -> [x] c X%       |   of a mapping,  | c (B^n, B^n -> B)          |
|      |                         |   or component   |                            |
| g    | g : U -> [y] c X%       |   of a mapping.  | = (B^n +-> B) = [B^n]      |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| W    | W :                     | Operator         |                            |
|      | U% -> EU%,              |                  | [B^n] -> [B^n x D^n],      |
|      | X% -> EX%,              |                  | [B^k] -> [B^k x D^k],      |
|      | (U%->X%)->(EU%->EX%),   |                  | ([B^n] -> [B^k])           |
|      | for each W among:       |                  | ->                         |
|      | !e!, !h!, E, D, d       |                  | ([B^n x D^n]->[B^k x D^k]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| !e!  |                         | Tacit Extension Operator   !e!                |
| !h!  |                         | Trope Extension Operator   !h!                |
|  E   |                         | Enlargement Operator        E                 |
|  D   |                         | Difference Operator         D                 |
|  d   |                         | Differential Operator       d                 |
|      |                         |                                               |
o------o-------------------------o------------------o----------------------------o
|      |                         |                  |                            |
| $W$  | $W$ :                   | Operator         |                            |
|      | U% -> $T$U% = EU%,      |                  | [B^n] -> [B^n x D^n],      |
|      | X% -> $T$X% = EX%,      |                  | [B^k] -> [B^k x D^k],      |
|      | (U%->X%)->($T$U%->$T$X%)|                  | ([B^n] -> [B^k])           |
|      | for each $W$ among:     |                  | ->                         |
|      | $e$, $E$, $D$, $T$      |                  | ([B^n x D^n]->[B^k x D^k]) |
|      |                         |                  |                            |
o------o-------------------------o------------------o----------------------------o
|      |                         |                                               |
| $e$  |                         | Radius Operator        $e$  =  <!e!, !h!>     |
| $E$  |                         | Secant Operator        $E$  =  <!e!,  E >     |
| $D$  |                         | Chord Operator         $D$  =  <!e!,  D >     |
| $T$  |                         | Tangent Functor        $T$  =  <!e!,  d >     |
|      |                         |                                               |
o------o-------------------------o-----------------------------------------------o

Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes

Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
|              | Operator             | Proposition        | Transformation       |
|              |    or                |    or              |    or                |
|              | Operand              | Component          | Mapping              |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| 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
|              |                      |                    |                      |
| Tacit        | !e! :                | !e!F_i :           | !e!F :               |
| 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]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Enlargement  | E :                  | EF_i :             | EF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Difference   | D :                  | DF_i :             | DF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Differential | d :                  | dF_i :             | dF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Remainder    | r :                  | rF_i :             | rF :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
|              | (U%->X%)->(EU%->dX%) | B^n x D^n -> D     | [B^n x D^n]->[D^k]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Radius       | $e$ = <!e!, !h!> :   |                    | $e$F :               |
| Operator     |                      |                    |                      |
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
|              |                      |                    |                      |
|              |                      |                    | [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

Formula Display 12

o-----------------------------------------------------------o
|                                                           |
|         x   =   f(u, v)   =   ((u)(v))                    |
|                                                           |
|         y   =   g(u, v)   =   ((u, v))                    |
|                                                           |
o-----------------------------------------------------------o

Formula Display 13

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

Table 60. Propositional Transformation

Table 60.  Propositional Transformation
o-------------o-------------o-------------o-------------o
|      u      |      v      |      f      |      g      |
o-------------o-------------o-------------o-------------o
|             |             |             |             |
|      0      |      0      |      0      |      1      |
|             |             |             |             |
|      0      |      1      |      1      |      0      |
|             |             |             |             |
|      1      |      0      |      1      |      0      |
|             |             |             |             |
|      1      |      1      |      1      |      1      |
|             |             |             |             |
o-------------o-------------o-------------o-------------o
|             |             |  ((u)(v))   |  ((u, v))   |
o-------------o-------------o-------------o-------------o

Figure 61. Propositional Transformation

             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
| 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 61.  Propositional Transformation

Figure 62. Propositional Transformation (Short Form)

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)

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
| 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

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

Table 65. Induced Transformation on Propositions

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

Formula Display 14

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

Formula Display 15

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

Formula Display 16

o-------------------------------------------------o
|                                                 |
|   Ef  =  ((u + du)(v + dv))                     |
|                                                 |
|   Eg  =  ((u + du, v + dv))                     |
|                                                 |
o-------------------------------------------------o

Formula Display 17

o-------------------------------------------------o
|                                                 |
|   Df  =  ((u)(v))  +  ((u + du)(v + dv))        |
|                                                 |
|   Dg  =  ((u, v))  +  ((u + du, v + dv))        |
|                                                 |
o-------------------------------------------------o

Table 66-i. Computation Summary for f‹u, v› = ((u)(v))

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

Table 66-ii. Computation Summary for g‹u, v› = ((u, v))

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

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

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

Table 68. Computation of an Analytic Series in Symbolic Terms

Table 68.  Computation of an Analytic Series in Symbolic Terms
o-----o-----o------------o----------o----------o----------o----------o----------o
| u v | f g |     Df     |    Dg    |    df    |    dg    |    rf    |    rf    |
o-----o-----o------------o----------o----------o----------o----------o----------o
|     |     |            |          |          |          |          |          |
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
| 0 1 | 1 0 |  (du) dv   | (du, dv) |    dv    | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
| 1 0 | 1 0 |   du (dv)  | (du, dv) |    du    | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
| 1 1 | 1 1 |   du  dv   | (du, dv) |    ()    | (du, dv) |  du  dv  |    ()    |
|     |     |            |          |          |          |          |          |
o-----o-----o------------o----------o----------o----------o----------o----------o

Formula Display 18

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

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

o-----------------------------------o o-----------------------------------o
| U                                 | |`U`````````````````````````````````|
|                                   | |```````````````````````````````````|
|                 ^                 | |```````````````````````````````````|
|                 |                 | |```````````````````````````````````|
|       o-------o | o-------o       | |```````o-------o```o-------o```````|
| ^    /`````````\|/`````````\    ^ | | ^ ```/      ^  \`/  ^      \``` ^ |
|  \  /```````````|```````````\  /  | |``\``/        \  o  /        \``/``|
|   \/`````u`````/|\`````v`````\/   | |```\/     u    \/`\/    v     \/```|
|   /\``````````/`|`\``````````/\   | |```/\          /\`/\          /\```|
|  o``\````````o``@``o````````/``o  | |``o  \        o``@``o        /  o``|
|  |```\```````|`````|```````/```|  | |``|   \       |`````|       /   |``|
|  |````@``````|`````|``````@````|  | |``|    @-------->`<--------@    |``|
|  |```````````|`````|```````````|  | |``|           |`````|           |``|
|  o```````````o` ^ `o```````````o  | |``o           o`````o           o``|
|   \```````````\`|`/```````````/   | |```\           \```/           /```|
|    \```` ^ ````\|/```` ^ ````/    | |````\     ^     \`/     ^     /````|
|     \`````\`````|`````/`````/     | |`````\     \     o     /     /`````|
|      \`````\```/|\```/`````/      | |``````\     \   /`\   /     /``````|
|       o-----\-o | o-/-----o       | |```````o-----\-o```o-/-----o```````|
|              \  |  /              | |``````````````\`````/``````````````|
|               \ | /               | |```````````````\```/```````````````|
|                \|/                | |````````````````\`/````````````````|
|                 @                 | |`````````````````@`````````````````|
o-----------------------------------o o-----------------------------------o
 \                                 /   \                                 /
   \                             /       \                             /
     \         ((u)(v))        /           \        ((u, v))         /
       \                     /               \                     /
         \                 /                   \                 /
o----------\-------------/-----------------------\-------------/----------o
| X          \         /                           \         /            |
|              \     /                               \     /              |
|                \ /                                   \ /                |
|                 o----------------o   o----------------o                 |
|                /                  \ /                  \                |
|               /                    o                    \               |
|              /                    / \                    \              |
|             /                    /   \                    \             |
|            /                    /     \                    \            |
|           /                    /       \                    \           |
|          /                    /         \                    \          |
|         o                    o           o                    o         |
|         |                    |           |                    |         |
|         |                    |           |                    |         |
|         |         f          |           |          g         |         |
|         |                    |           |                    |         |
|         |                    |           |                    |         |
|         o                    o           o                    o         |
|          \                    \         /                    /          |
|           \                    \       /                    /           |
|            \                    \     /                    /            |
|             \                    \   /                    /             |
|              \                    \ /                    /              |
|               \                    o                    /               |
|                \                  / \                  /                |
|                 o----------------o   o----------------o                 |
|                                                                         |
|                                                                         |
|                                                                         |
o-------------------------------------------------------------------------o
Figure 69.  Difference Map of F = <f, g> = <((u)(v)), ((u, v))>

Formula Display 19

o-------------------------------------------------------------------------------o
|                                                                               |
|  df  =  uv.   0      +  u(v). du       +  (u)v.     dv   +  (u)(v).(du, dv)   |
|                                                                               |
|  dg  =  uv.(du, dv)  +  u(v).(du, dv)  +  (u)v.(du, dv)  +  (u)(v).(du, dv)   |
|                                                                               |
o-------------------------------------------------------------------------------o

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

                o                                   o
               / \                                 / \
              /   \                               /   \
             /     \                             /  O  \
            /       \                           o  /@\  o
           /         \                         / \     / \
          /           \                       /   \   /   \
         /      O      \                     /  O  \ /  O  \
        o      /@\      o                   o  /@\  o  /@\  o
       / \             / \                 / \   \ / \   \ / \
      /   \           /   \               /   \   /   \   /   \
     /     \         /     \             /  O  \ /  O  \ /  O  \
    /       \       /       \           o  /@   o  /@\  o  /@   o
   /         \     /         \         / \   \ / \     / \   \ / \
  /           \   /           \       /   \   /   \   /   \   /   \
 /      O      \ /      O      \     /  O  \ /  O  \ /  O  \ /  O  \
o      /@       o      /@       o   o  /@   o  /@   o  /@   o  /@   o
|\             / \             /|   |\     / \ /   / \ /   / \     /|
| \           /   \           / |   | \   /   \   /   \   /   \   / |
|  \         /     \         /  |   |  \ /  O  \ /  O  \ /  O  \ /  |
|   \       /       \       /   |   |   o  /@   o   @\  o  /@   o   |
|    \     /         \     /    |   |   |\ / \ / \     / \ / \ /|   |
|     \   /           \   /     |   |   | \   /   \   /   \   / |   |
| u    \ /      O      \ /    v |   | u |  \ /  O  \ /  O  \ /  | v |
o-------o       @\      o-------o   o---+---o   @\  o   @\  o---+---o
         \             /                |    \ / \ / \ / \ /    |
          \           /                 |     \   /   \   /     |
           \         /                  | du   \ /  O  \ /   dv |
            \       /                   o-------o   @\  o-------o
             \     /                             \     /
              \   /                               \   /
               \ /                                 \ /
                o                                   o
                     U%          $T$          $E$U%
                        o------------------>o
                        |                   |
                        |                   |
                        |                   |
                        |                   |
                     F  |                   | $T$F
                        |                   |
                        |                   |
                        |                   |
                        v                   v
                        o------------------>o
                     X%          $T$          $E$X%
                o                                   o
               / \                                 / \
              /   \                               /   \
             /     \                             /  O  \
            /       \                           o  /@\  o
           /         \                         / \     / \
          /           \                       /   \   /   \
         /      O      \                     /  O  \ /  O  \
        o      /@\      o                   o  /@\  o  /@\  o
       / \             / \                 / \   \ / \ /   / \
      /   \           /   \               /   \   /   \   /   \
     /     \         /     \             /  O  \ /  O  \ /  O  \
    /       \       /       \           o  /@   o  /@\  o   @\  o
   /         \     /         \         / \   \ / \ / \ / \ /   / \
  /           \   /           \       /   \   /   \   /   \   /   \
 /      O      \ /      O      \     /  O  \ /  O  \ /  O  \ /  O  \
o      /@       o       @\      o   o  /@   o  /@   o   @\  o   @\  o
|\             / \             /|   |\     / \ / \ / \ / \ / \     /|
| \           /   \           / |   | \   /   \   /   \   /   \   / |
|  \         /     \         /  |   |  \ /  O  \ /  O  \ /  O  \ /  |
|   \       /       \       /   |   |   o  /@   o   @   o   @\  o   |
|    \     /         \     /    |   |   |\ /   / \ / \ / \   \ /|   |
|     \   /           \   /     |   |   | \   /   \   /   \   / |   |
| x    \ /      O      \ /    y |   | x |  \ /  O  \ /  O  \ /  | y |
o-------o       @       o-------o   o---+---o   @   o   @   o---+---o
         \             /                |    \ /   / \   \ /    |
          \           /                 |     \   /   \   /     |
           \         /                  | dx   \ /  O  \ /   dy |
            \       /                   o-------o   @   o-------o
             \     /                             \     /
              \   /                               \   /
               \ /                                 \ /
                o                                   o

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

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

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /////\ /////\     |  |     /XXXX\ /XXXX\     |  |     /\\\\\ /\\\\\     |
|    ///////o//////\    |  |    /XXXXXXoXXXXXX\    |  |    /\\\\\\o\\\\\\\    |
|   //////// \//////\   |  |   /XXXXXX/ \XXXXXX\   |  |   /\\\\\\/ \\\\\\\\   |
|  o///////   \//////o  |  |  oXXXXXX/   \XXXXXXo  |  |  o\\\\\\/   \\\\\\\o  |
|  |/////o     o/////|  |  |  |XXXXXo     oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  |/du//|     |//dv/|  |  |  |XXXXX|     |XXXXX|  |  |  |\du\\|     |\\dv\|  |
|  |/////o     o/////|  |  |  |XXXXXo     oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  o//////\   ///////o  |  |  oXXXXXX\   /XXXXXXo  |  |  o\\\\\\\   /\\\\\\o  |
|   \//////\ ////////   |  |   \XXXXXX\ /XXXXXX/   |  |   \\\\\\\\ /\\\\\\/   |
|    \//////o///////    |  |    \XXXXXXoXXXXXX/    |  |    \\\\\\\o\\\\\\/    |
|     \///// \/////     |  |     \XXXX/ \XXXX/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @ (u)(v)       o-----------------------o          dv' @ (u)(v)   =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /    \ /////\     |  |     /\\\\\ /XXXX\     |  |     /\\\\\ /\\\\\     |
|    /      o//////\    |  |    /\\\\\\oXXXXXX\    |  |    /\\\\\\o\\\\\\\    |
|   /      //\//////\   |  |   /\\\\\\//\XXXXXX\   |  |   /\\\\\\/ \\\\\\\\   |
|  o      ////\//////o  |  |  o\\\\\\////\XXXXXXo  |  |  o\\\\\\/   \\\\\\\o  |
|  |     o/////o/////|  |  |  |\\\\\o/////oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  | du  |/////|//dv/|  |  |  |\\\\\|/////|XXXXX|  |  |  |\du\\|     |\\dv\|  |
|  |     o/////o/////|  |  |  |\\\\\o/////oXXXXX|  |  |  |\\\\\o     o\\\\\|  |
|  o      \//////////o  |  |  o\\\\\\\////XXXXXXo  |  |  o\\\\\\\   /\\\\\\o  |
|   \      \/////////   |  |   \\\\\\\\//XXXXXX/   |  |   \\\\\\\\ /\\\\\\/   |
|    \      o///////    |  |    \\\\\\\oXXXXXX/    |  |    \\\\\\\o\\\\\\/    |
|     \    / \/////     |  |     \\\\\/ \XXXX/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @ (u) v        o-----------------------o          dv' @ (u) v    =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /////\ /    \     |  |     /XXXX\ /\\\\\     |  |     /\\\\\ /\\\\\     |
|    ///////o      \    |  |    /XXXXXXo\\\\\\\    |  |    /\\\\\\o\\\\\\\    |
|   /////////\      \   |  |   /XXXXXX//\\\\\\\\   |  |   /\\\\\\/ \\\\\\\\   |
|  o//////////\      o  |  |  oXXXXXX////\\\\\\\o  |  |  o\\\\\\/   \\\\\\\o  |
|  |/////o/////o     |  |  |  |XXXXXo/////o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  |/du//|/////|  dv |  |  |  |XXXXX|/////|\\\\\|  |  |  |\du\\|     |\\dv\|  |
|  |/////o/////o     |  |  |  |XXXXXo/////o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  o//////\////      o  |  |  oXXXXXX\////\\\\\\o  |  |  o\\\\\\\   /\\\\\\o  |
|   \//////\//      /   |  |   \XXXXXX\//\\\\\\/   |  |   \\\\\\\\ /\\\\\\/   |
|    \//////o      /    |  |    \XXXXXXo\\\\\\/    |  |    \\\\\\\o\\\\\\/    |
|     \///// \    /     |  |     \XXXX/ \\\\\/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @  u (v)       o-----------------------o          dv' @  u (v)   =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| dU                    |  | dU                    |  | dU                    |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|     /    \ /    \     |  |     /\\\\\ /\\\\\     |  |     /\\\\\ /\\\\\     |
|    /      o      \    |  |    /\\\\\\o\\\\\\\    |  |    /\\\\\\o\\\\\\\    |
|   /      / \      \   |  |   /\\\\\\/ \\\\\\\\   |  |   /\\\\\\/ \\\\\\\\   |
|  o      /   \      o  |  |  o\\\\\\/   \\\\\\\o  |  |  o\\\\\\/   \\\\\\\o  |
|  |     o     o     |  |  |  |\\\\\o     o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  | du  |     |  dv |  |  |  |\\\\\|     |\\\\\|  |  |  |\du\\|     |\\dv\|  |
|  |     o     o     |  |  |  |\\\\\o     o\\\\\|  |  |  |\\\\\o     o\\\\\|  |
|  o      \   /      o  |  |  o\\\\\\\   /\\\\\\o  |  |  o\\\\\\\   /\\\\\\o  |
|   \      \ /      /   |  |   \\\\\\\\ /\\\\\\/   |  |   \\\\\\\\ /\\\\\\/   |
|    \      o      /    |  |    \\\\\\\o\\\\\\/    |  |    \\\\\\\o\\\\\\/    |
|     \    / \    /     |  |     \\\\\/ \\\\\/     |  |     \\\\\/ \\\\\/     |
|      o--o   o--o      |  |      o--o   o--o      |  |      o--o   o--o      |
|                       |  |                       |  |                       |
o-----------------------o  o-----------------------o  o-----------------------o
 =      du' @  u v         o-----------------------o          dv' @  u v     =
  =                        | dU'                   |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

o-----------------------o  o-----------------------o  o-----------------------o
| U                     |  |\U\\\\\\\\\\\\\\\\\\\\\|  |\U\\\\\\\\\\\\\\\\\\\\\|
|      o--o   o--o      |  |\\\\\\o--o\\\o--o\\\\\\|  |\\\\\\o--o\\\o--o\\\\\\|
|     /////\ /////\     |  |\\\\\/////\\/////\\\\\\|  |\\\\\/    \\/    \\\\\\|
|    ///////o//////\    |  |\\\\///////o//////\\\\\|  |\\\\/      o      \\\\\|
|   /////////\//////\   |  |\\\////////X\//////\\\\|  |\\\/      /\\      \\\\|
|  o//////////\//////o  |  |\\o///////XXX\//////o\\|  |\\o      /\\\\      o\\|
|  |/////o/////o/////|  |  |\\|/////oXXXXXo/////|\\|  |\\|     o\\\\\o     |\\|
|  |//u//|/////|//v//|  |  |\\|//u//|XXXXX|//v//|\\|  |\\|  u  |\\\\\|  v  |\\|
|  |/////o/////o/////|  |  |\\|/////oXXXXXo/////|\\|  |\\|     o\\\\\o     |\\|
|  o//////\//////////o  |  |\\o//////\XXX///////o\\|  |\\o      \\\\/      o\\|
|   \//////\/////////   |  |\\\\//////\X////////\\\|  |\\\\      \\/      /\\\|
|    \//////o///////    |  |\\\\\//////o///////\\\\|  |\\\\\      o      /\\\\|
|     \///// \/////     |  |\\\\\\/////\\/////\\\\\|  |\\\\\\    /\\    /\\\\\|
|      o--o   o--o      |  |\\\\\\o--o\\\o--o\\\\\\|  |\\\\\\o--o\\\o--o\\\\\\|
|                       |  |\\\\\\\\\\\\\\\\\\\\\\\|  |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o  o-----------------------o  o-----------------------o
 =          u'             o-----------------------o              v'         =
  =                        | U'                    |                        =
   =                       |      o--o   o--o      |                       =
    =                      |     /////\ /\\\\\     |                      =
     =                     |    ///////o\\\\\\\    |                     =
      =                    |   ////////X\\\\\\\\   |                    =
       =                   |  o///////XXX\\\\\\\o  |                   =
        =                  |  |/////oXXXXXo\\\\\|  |                  =
         = = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
                           |  |/////oXXXXXo\\\\\|  |
                           |  o//////\XXX/\\\\\\o  |
                           |   \//////\X/\\\\\\/   |
                           |    \//////o\\\\\\/    |
                           |     \///// \\\\\/     |
                           |      o--o   o--o      |
                           |                       |
                           o-----------------------o

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