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[parent] differential propositional calculus : appendix 4 (Application)


Contents

Detail of Calculation for the Difference Map

Detail of Calculation for $\operatorname{D}f = \operatorname{E}f + f$
  \begin{displaymath}\begin{array}{cr} & \operatorname{E}f\vert _{\operatorname{d}... ...}f\vert _{\operatorname{d}x\ \operatorname{d}y} \ \end{array}\end{displaymath} \begin{displaymath}\begin{array}{cr} & \operatorname{E}f\vert _{\operatorname{d}... ...\vert _{\operatorname{d}x\ (\operatorname{d}y)} \ \end{array}\end{displaymath} \begin{displaymath}\begin{array}{cr} & \operatorname{E}f\vert _{(\operatorname{d... ...\vert _{(\operatorname{d}x)\ \operatorname{d}y} \ \end{array}\end{displaymath} \begin{displaymath}\begin{array}{cr} & \operatorname{E}f\vert _{(\operatorname{d... ...\vert _{(\operatorname{d}x)(\operatorname{d}y)} \ \end{array}\end{displaymath}
$f_{0}$ $0 + 0 = 0$ $0 + 0 = 0$ $0 + 0 = 0$ $0 + 0 = 0$
$f_{1}$ $\begin{smallmatrix} & x\ y & \operatorname{d}x & \operatorname{d}y \ + & (x)(... ... \ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & x\ (y) & \operatorname{d}x & (\operatorname{d}y) \ + & ... ...}y) \ = & (y) & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & (x)\ y & (\operatorname{d}x) & \operatorname{d}y \ + & ... ...d}y \ = & (x) & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \ + ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{2}$ $\begin{smallmatrix} & x\ (y) & \operatorname{d}x & \operatorname{d}y \ + & (x... ...}y \ = & (x, y) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & x\ y & \operatorname{d}x & (\operatorname{d}y) \ + & (x... ...{d}y) \ = & y & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \ + &... ...d}y \ = & (x) & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \ + ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{4}$ $\begin{smallmatrix} & (x)\ y & \operatorname{d}x & \operatorname{d}y \ + & x\... ...}y \ = & (x, y) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \ + &... ...}y) \ = & (y) & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & x\ y & (\operatorname{d}x) & \operatorname{d}y \ + & x\... ...e{d}y \ = & x & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \ + ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{8}$ $\begin{smallmatrix} & (x)(y) & \operatorname{d}x & \operatorname{d}y \ + & x\... ... \ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x)\ y & \operatorname{d}x & (\operatorname{d}y) \ + & ... ...{d}y) \ = & y & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & x\ (y) & (\operatorname{d}x) & \operatorname{d}y \ + & ... ...e{d}y \ = & x & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \ + & ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{3}$ $\begin{smallmatrix} & x & \operatorname{d}x & \operatorname{d}y \ + & (x) & \... ...ame{d}y \ = & 1 & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & x & \operatorname{d}x & (\operatorname{d}y) \ + & (x) &... ...{d}y) \ = & 1 & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & (x) & (\operatorname{d}x) & \operatorname{d}y \ + & (x)... ...e{d}y \ = & 0 & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x) & (\operatorname{d}x) & (\operatorname{d}y) \ + & (... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{12}$ $\begin{smallmatrix} & (x) & \operatorname{d}x & \operatorname{d}y \ + & x & \... ...ame{d}y \ = & 1 & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x) & \operatorname{d}x & (\operatorname{d}y) \ + & x &... ...{d}y) \ = & 1 & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & x & (\operatorname{d}x) & \operatorname{d}y \ + & x & (... ...e{d}y \ = & 0 & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & x & (\operatorname{d}x) & (\operatorname{d}y) \ + & x &... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{6}$ $\begin{smallmatrix} & (x, y) & \operatorname{d}x & \operatorname{d}y \ + & (x... ...ame{d}y \ = & 0 & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \ + ... ...{d}y) \ = & 1 & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \ + ... ...e{d}y \ = & 1 & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \ + ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{9}$ $\begin{smallmatrix} & ((x, y)) & \operatorname{d}x & \operatorname{d}y \ + & ... ...ame{d}y \ = & 0 & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x, y) & \operatorname{d}x & (\operatorname{d}y) \ + & ... ...{d}y) \ = & 1 & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & (x, y) & (\operatorname{d}x) & \operatorname{d}y \ + & ... ...e{d}y \ = & 1 & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \ ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{5}$ $\begin{smallmatrix} & y & \operatorname{d}x & \operatorname{d}y \ + & (y) & \... ...ame{d}y \ = & 1 & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (y) & \operatorname{d}x & (\operatorname{d}y) \ + & (y)... ...{d}y) \ = & 0 & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & y & (\operatorname{d}x) & \operatorname{d}y \ + & (y) &... ...e{d}y \ = & 1 & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (y) & (\operatorname{d}x) & (\operatorname{d}y) \ + & (... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{10}$ $\begin{smallmatrix} & (y) & \operatorname{d}x & \operatorname{d}y \ + & y & \... ...ame{d}y \ = & 1 & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & y & \operatorname{d}x & (\operatorname{d}y) \ + & y & \... ...{d}y) \ = & 0 & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & (y) & (\operatorname{d}x) & \operatorname{d}y \ + & y &... ...e{d}y \ = & 1 & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & y & (\operatorname{d}x) & (\operatorname{d}y) \ + & y &... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{7}$ $\begin{smallmatrix} & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \ + & ... ... \ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \ + ... ...{d}y) \ = & y & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \ + ... ...e{d}y \ = & x & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \ + ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{11}$ $\begin{smallmatrix} & ((x)\ y) & \operatorname{d}x & \operatorname{d}y \ + & ... ...}y \ = & (x, y) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \ +... ...}y) \ = & (y) & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & (x\ y) & (\operatorname{d}x) & \operatorname{d}y \ + & ... ...e{d}y \ = & x & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \ ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{13}$ $\begin{smallmatrix} & (x\ (y)) & \operatorname{d}x & \operatorname{d}y \ + & ... ...}y \ = & (x, y) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x\ y) & \operatorname{d}x & (\operatorname{d}y) \ + & ... ...{d}y) \ = & y & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \ +... ...d}y \ = & (x) & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \ ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{14}$ $\begin{smallmatrix} & (x\ y) & \operatorname{d}x & \operatorname{d}y \ + & ((... ... \ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \ + ... ...}y) \ = & (y) & \operatorname{d}x & (\operatorname{d}y) \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \ + ... ...d}y \ = & (x) & (\operatorname{d}x) & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \ ... ...}y) \ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \ \end{smallmatrix}$
$f_{15}$ $1 + 1 = 0$ $1 + 1 = 0$ $1 + 1 = 0$ $1 + 1 = 0$



"differential propositional calculus : appendix 4" is owned by Jon Awbrey.

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See Also: differential logic, minimal negation operator


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This is version 1 of differential propositional calculus : appendix 4, born on 2009-05-26.
Object id is 781, canonical name is DifferentialPropositionalCalculusAppendix4.
Accessed 384 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
 02.10.Ab (Logic and set theory)
 02.10.Ox (Combinatorics; graph theory)
 02.10.Ud (Linear algebra)
 02.20.-a (Group theory )
 02.30.-f (Function theory, analysis)
 02.40.-k (Geometry, differential geometry, and topology )
 02.40.Yy (Geometric mechanics )
 02.50.Tt (Inference methods)
 02.70.-c (Computational techniques )
 02.70.Bf (Finite-difference methods)
 02.70.Wz (Symbolic computation )
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