# User:Jon Awbrey/DIFF

## 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 : AB) Linear functions (Bn)* = Bn
A^ (AB) Boolean functions BnB
A

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

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

(Bn, (BnB))
(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
(RnR) → R Tangent Vector (BnB) → B
Rn → ((RnR)→R) Vector Field Bn → ((BnB)→B)
(Rn × (RnR)) → R ditto (Bn × (BnB)) → B
((RnR) × Rn) → R ditto ((BnB) × Bn) → B
(RnR) → (RnR) Derivation (BnB) → (BnB)
Rn → Rm Basic Transformation Bn → Bm
(RnR) → (RmR) Function Transformation (BnB) → (BmB)
... ... ...

### 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 : XK)
isomorphic to Kn

X*
(hom : XB)
isomorphic to Bn

A*
(hom : AB)
isomorphic to Bn

X^
(XK)
isomorphic to:
(KnK)

X^
(XB)
isomorphic to:
(BnB)

A^
(AB)
isomorphic to:
(BnB)

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

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

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

Linear functions
on dA

(Dn)* = Dn
dA^ (dAB)

Boolean functions
on dA

DnB
dA

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

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

(Dn, (DnB))
(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


 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

 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         oo         o   |     |   o         oo         o   |
|   |    u    ||    v    |   |     |   |    u    ||    v    |   |
|   o         oo         o   |     |   o         oo         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         oo         o   |     |   o         oo         o   |
|   |    u    ||    v    |   |     |   |    u    ||    v    |   |
|   o         oo         o   |     |   o         oo         o   |
|    \         \/         /    |     |    \         \/         /    |
|     \         o         /     |     |     \         o         /     |
|      \       / \       /      |     |      \       / \       /      |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
\                             /       \                             /
\                         /           \                         /
\                     /               \          J          /
\                 /                   \                 /
\             /                       \             /
o----------\---------/----------o     o----------\---------/----------o
|            \     /            |     |            \     /            |
|              \ /              |     |              \ /              |
|         o-----@-----o         |     |         o-----@-----o         |
|        /\        |     |        /\        |
|       /\       |     |       /\       |
|      /\      |     |      /\      |
|     oo     |     |     oo     |
|     ||     |     |     ||     |
|     | J |     |     |     | x |     |
|     ||     |     |     ||     |
|     oo     |     |     oo     |
|      \/      |     |      \/      |
|       \/       |     |       \/       |
|        \/        |     |        \/        |
|         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         oo         o  |     |oo-----oo|
|  |    u    ||    v    |  |     |||     |||
o--o---------o-----o---------o--o     || u |     | v ||
|||     |||     |||     |||
|oo     oo|     |oo     oo|
|\\   //|     |\\   //|
|\\ //|     |\\ //|
|\o/|     |\o/|
|\/\/|     |\/\/|
|o-----oo-----o|     |o-----oo-----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-----oo-----o|
|      /\ /\      |     |/       \/       \|
|     /o\     |     |/         o         \|
|    //\\    |     |/         /\         \|
|   //\\   |     |/         /\         \|
|  ooo o  |     |o         oo         o|
|  ||||  |     ||         ||         ||
|  | u || v |  |     ||    u    ||    v    ||
|  ||||  |     ||         ||         ||
|  oooo  |     |o         oo         o|
|   \\//   |     |\         \/         /|
|    \\//    |     |\         \/         /|
|     \o/     |     |\         o         /|
|      \/ \/      |     |\       /\       /|
|       o-----o   o-----o       |     |o-----oo-----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-----oo-----o       |
|/ \\ // \|     |      /\     \/     /\      |
|/   \o/   \|     |     /\     o     /\     |
|/     \/\/     \|     |    /\   /\   /\    |
|/       \/\/       \|     |   /\ /\ /\   |
|o         o-----o         o|     |  oo-----oo  |
||         |     |         ||     |  ||     ||  |
||    u    |     |    v    ||     |  | u |     | v |  |
||         |     |         ||     |  ||     ||  |
|o         o     o         o|     |  oo     oo  |
|\         \   /         /|     |   \\   //   |
|\         \ /         /|     |    \\ //    |
|\         o         /|     |     \o/     |
|\       /\       /|     |      \/ \/      |
|o-----oo-----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 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 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


### 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             oo             o  |
|  |             ||             |  |
|  |      u      ||      v      |  |
|  |             ||             |  |
|  o             oo             o  |
|   \             \/             /   |
|    \             \/             /    |
|     \             o             /     |
|      \           / \           /      |
|       o---------o   o---------o       |
|                                       |
|                                       |
o---------------------------------------o
\                                     /
\                                 /
\                             /
\            J            /
\                     /
\                 /
\             /
o--------------\---------/--------------o
|                \     /                |
|                  \ /                  |
|            o------@------o            |
|           /\           |
|          /\          |
|         /\         |
|        /\        |
|       oo       |
|       ||       |
|       | x |       |
|       ||       |
|       oo       |
|        \/        |
|         \/         |
|          \/          |
|           \/           |
|            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


### 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             oo             o   |   |     \       / \       /     |
|   |             |@|             |   |   |      o-----o   o-----o      |
|   |             |\|             |   |   |                             |
|   |             |\|             |   |   o-----------------------------o
|   |      u      |\|      v      |   |
|   |             |\|             |   |   o-----------------------------o
|   |             ||             |   |   |                             |
|   |             ||\            |   |   |      o-----o   o-----o      |
|   o             oo \           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-----oo-----o|
\   |/\/\|
\  |/o\|
\ |//\\|
\|oooo|
@|du||dv||
|oooo|
|\\//|
|\o/|
|\/\/|
|o-----oo-----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 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


### 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             oo             o   |   |     \%%%%%%%/ \       /     |
|   |             |@|             |   |   |      o-----o   o-----o      |
|   |             |\|             |   |   |                             |
|   |             |\|             |   |   o-----------------------------o
|   |      u      |\|      v      |   |
|   |             |\|             |   |   o-----------------------------o
|   |             ||             |   |   |                             |
|   |             ||\            |   |   |      o-----o   o-----o      |
|   o             oo \           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 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 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


### 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


### 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             oo             o   |   |     \%%%%%%%/ \       /     |
|   |             |@|             |   |   |      o-----o   o-----o      |
|   |             |\|             |   |   |                             |
|   |             |\|             |   |   o-----------------------------o
|   |      u      |\|      v      |   |
|   |             |\|             |   |   o-----------------------------o
|   |             ||             |   |   |                             |
|   |             ||\            |   |   |      o-----o   o-----o      |
|   o             oo \           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


### 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             oo             o   |   |     \%%%%%%%/ \       /     |
|   |             |@|             |   |   |      o-----o   o-----o      |
|   |             |\|             |   |   |                             |
|   |             |\|             |   |   o-----------------------------o
|   |      u      |\|      v      |   |
|   |             |\|             |   |   o-----------------------------o
|   |             ||             |   |   |                             |
|   |             ||\            |   |   |      o-----o   o-----o      |
|   o             oo \           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


### 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             oo             o   |   |     \       / \       /     |
|   |             |@|             |   |   |      o-----o   o-----o      |
|   |             |\|             |   |   |                             |
|   |             |\|             |   |   o-----------------------------o
|   |      u      |\|      v      |   |
|   |             |\|             |   |   o-----------------------------o
|   |             ||             |   |   |                             |
|   |             ||\            |   |   |      o-----o   o-----o      |
|   o             oo \           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 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


### Formula Display 10

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


### 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


### 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--oo--o|
|     /    \ /    \     |   |\    /    \ /\   \     |   |/\/\|
|    /      o      \    |   | \  /      o  @   \    |   |/o\|
|   /  du  / \  dv  \   |   |  \/  du  /\  dv  \   |   |/du/\dv\|
|  o      /   \      o  |   |  o\     /\      o  |   |o/\o|
|  |     o     o     |  |   |  | \   oo     |  |   ||oo||
|  |     |     |     |  |   |  |  @  |@--|-----|------@|||||
|  |     o     o     |  |   |  |     oo     |  |   ||oo||
|  o      \   /      o  |   |  o      \/      o  |   |o\/o|
|   \      \ /      /   |   |   \      \/      /   |   |\\//|
|    \      o      /    |   |    \      o      /    |   |\o/|
|     \    / \    /     |   |     \    / \    /     |   |\/\/|
|      o--o   o--o      |   |      o--o   o--o      |   |o--oo--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     |
|    /\    |  |   oo   |  |    /\    |
|   /\   |  |   ||   |  |   /\   |
|  oo  |  |   ||   |  |  oo  |
|  |dx|  @----@ |x@-----|------@  | dx |  |
|  oo  |  |   ||   |  |  oo  |
|   \/   |  |   ||   |  |   \/   |
|    \/    |  |   oo   |  |    \/    |
|     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  |
|  |     oo     |  |
|  |     ||     |  |
|  |     oo     |  |
|  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--oo--o|
|     /    \ /\     |   |\    /    \ /\   \     |   |/    \/    \|
|    /      o\    |   | \  /      o  @   \    |   |/      o      \|
|   /  du  / \dv\   |   |  \/  du  /\  dv  \   |   |/  du  / \  dv  \|
|  o      /   \o  |   |  o\     /\      o  |   |o      /   \      o|
|  |     o     o|  |   |  | \   oo     |  |   ||     o     o     ||
|  |     |     ||  |   |  |  @  |@--|-----|------@|     |     |     ||
|  |     o     o|  |   |  |     oo     |  |   ||     o     o     ||
|  o      \   /o  |   |  o      \/      o  |   |o      \   /      o|
|   \      \ //   |   |   \      \/      /   |   |\      \ /      /|
|    \      o/    |   |    \      o      /    |   |\      o      /|
|     \    / \/     |   |     \    / \    /     |   |\    /\    /|
|      o--o   o--o      |   |      o--o   o--o      |   |o--oo--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     |
|    /\    |  |   oo   |  |    /\    |
|   /\   |  |   ||   |  |   /\   |
|  oo  |  |   ||   |  |  oo  |
|  |dx|  @----@ |x@-----|------@  | dx |  |
|  oo  |  |   ||   |  |  oo  |
|   \/   |  |   ||   |  |   \/   |
|    \/    |  |   oo   |  |    \/    |
|     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  |
|  |     oo     |  |
|  |     ||     |  |
|  |     oo     |  |
|  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|  |   |  | \   oo     |  |   |  |oo|  |
|  |     |     ||  |   |  |  @  |@--|-----|------@  ||||  |
|  |     o     o|  |   |  |     oo     |  |   |  |oo|  |
|  o      \   /o  |   |  o      \/      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     |
|    /\    |  |   oo   |  |    /\    |
|   /\   |  |   ||   |  |   /\   |
|  oo  |  |   ||   |  |  oo  |
|  |dx|  @----@ |x@-----|------@  | dx |  |
|  oo  |  |   ||   |  |  oo  |
|   \/   |  |   ||   |  |   \/   |
|    \/    |  |   oo   |  |    \/    |
|     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  |         \
|  |oo     |  @          \
|  |||     |  |\          \
|  |oo     |  | \          \
|  o\/      o  |  \          \
|   \\/      /   |   \          \
|    \o      /    |    \          \
|     \/ \    /     |     \          \
|      o--o   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  |
|  |     oo|  |   |  | \   oo     |  |   |  |o     o|  |
|  |     |||  |   |  |  @  |@--|-----|------@  ||     ||  |
|  |     oo|  |   |  |     oo     |  |   |  |o     o|  |
|  o      \/o  |   |  o      \/      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     |
|    /\    |  |   oo   |  |    /\    |
|   /\   |  |   ||   |  |   /\   |
|  oo  |  |   ||   |  |  oo  |
|  |dx|  @----@ |x@-----|------@  | dx |  |
|  oo  |  |   ||   |  |  oo  |
|   \/   |  |   ||   |  |   \/   |
|    \/    |  |   oo   |  |    \/    |
|     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---oo---o|
|     /'''''\ /'''''\     |   |  \      |         |   |/     \/     \|
|    /'''''''o'''''''\    |   |   \     |         |   |/       o       \|
|   /'''''''/'\'''''''\   |   |    \    |         |   |/       /\       \|
|  o'''''''o'''o'''''''o  |   |     \   |         |   |o       oo       o|
|  |'''u'''|'''|'''v'''|  |   |      \  |         |   ||   u   ||   v   ||
|  o'''''''o'''o'''''''o  |   |       \ |         |   |o       oo       o|
|   \'''''''\'/'''''''/   |   |        \|         |   |\       \/       /|
|    \'''''''o'''''''/    |   |         \         |   |\       o       /|
|     \'''''/ \'''''/     |   |         |\        |   |\     /\     /|
|      o---o   o---o      |   |         | \       |   |o---oo---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-------oo-------o|
| ^    /\|/\    ^ | | ^ /      ^  \/  ^      \ ^ |
|  \  /|\  /  | |\/        \  o  /        \/|
|   \/u/|\v\/   | |\/     u    \/\/    v     \/|
|   /\/|\/\   | |/\          /\/\          /\|
|  o\o@o/o  | |o  \        o@o        /  o|
|  |\||/|  | ||   \       ||       /   ||
|  |@||@|  | ||    @--------><--------@    ||
|  ||||  | ||           ||           ||
|  oo ^ oo  | |o           oo           o|
|   \\|//   | |\           \/           /|
|    \ ^ \|/ ^ /    | |\     ^     \/     ^     /|
|     \\|//     | |\     \     o     /     /|
|      \\/|\//      | |\     \   /\   /     /|
|       o-----\-o | o-/-----o       | |o-----\-oo-/-----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))>