differential propositional calculus : appendix 4

Contents

• Detail of Calculation for the Difference Map

Detail of Calculation for the Difference Map

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

Contributors to this entry (in most recent order):

• Jon Awbrey

As of this snapshot date, this entry was owned by Jon Awbrey.