Talk:PlanetPhysics/Differential Propositional Calculus Appendix 4

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: differential propositional calculus : appendix 4 %%% Primary Category Code: 02. %%% Filename: DifferentialPropositionalCalculusAppendix4.tex %%% Version: 1 %%% Owner: Jon Awbrey %%% Author(s): Jon Awbrey %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\subsection{Detail of Calculation for the Difference Map}

\begin{center}\begin{tabular}{||c||c|c|c|c||} \multicolumn{5}{c}{\textbf{Detail of Calculation for $\operatorname{D}f = \operatorname{E}f + f$}} \\[6pt] \hline\hline & $\begin{array}{cr} & \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\ + &                f|_{\operatorname{d}x\ \operatorname{d}y} \\ = & \operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y} \\ \end{array}$ & $\begin{array}{cr} & \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\ + &                f|_{\operatorname{d}x\ (\operatorname{d}y)} \\ = & \operatorname{D}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\ \end{array}$ & $\begin{array}{cr} & \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\ + &                f|_{(\operatorname{d}x)\ \operatorname{d}y} \\ = & \operatorname{D}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\ \end{array}$ & $\begin{array}{cr} & \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\ + &                f|_{(\operatorname{d}x)(\operatorname{d}y)} \\ = & \operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\ \end{array}$ \\[6pt] \hline\hline $f_{0}$ & $0 + 0 = 0$ & $0 + 0 = 0$ & $0 + 0 = 0$ & $0 + 0 = 0$ \\[6pt] \hline\hline $f_{1}$ & $\begin{smallmatrix} &  x\ y   & \operatorname{d}x & \operatorname{d}y \\ + & (x)(y)  & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\ = &    (y) & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x)\ y & (\operatorname{d}x) & \operatorname{d}y \\ + & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)    & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{2}$ & $\begin{smallmatrix} & x\ (y) & \operatorname{d}x & \operatorname{d}y \\ + & (x)\ y & \operatorname{d}x & \operatorname{d}y \\ = & (x, y) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x\  y  & \operatorname{d}x & (\operatorname{d}y) \\ + & (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\ = &     y  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x)\ y & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)    & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0   & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{4}$ & $\begin{smallmatrix} & (x)\ y & \operatorname{d}x & \operatorname{d}y \\ + & x\ (y) & \operatorname{d}x & \operatorname{d}y \\ = & (x, y) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\ + & x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\ = &    (y) & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x\  y  & (\operatorname{d}x) & \operatorname{d}y \\ + & x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\ = & x      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{8}$ & $\begin{smallmatrix} & (x)(y)  & \operatorname{d}x & \operatorname{d}y \\ + &  x\ y   & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\ + & x\  y & \operatorname{d}x & (\operatorname{d}y) \\ = &     y & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\ + & x\  y  & (\operatorname{d}x) & \operatorname{d}y \\ = & x      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ + & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline\hline $f_{3}$ & $\begin{smallmatrix} & x  & \operatorname{d}x & \operatorname{d}y \\ + & (x) & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x  & \operatorname{d}x & (\operatorname{d}y) \\ + & (x) & \operatorname{d}x & (\operatorname{d}y) \\ = & 1  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x) & (\operatorname{d}x) & \operatorname{d}y \\ = & 0  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x) & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{12}$ & $\begin{smallmatrix} & (x) & \operatorname{d}x & \operatorname{d}y \\ + & x  & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x) & \operatorname{d}x & (\operatorname{d}y) \\ + & x  & \operatorname{d}x & (\operatorname{d}y) \\ = & 1  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x  & (\operatorname{d}x) & \operatorname{d}y \\ + & x  & (\operatorname{d}x) & \operatorname{d}y \\ = & 0  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & x & (\operatorname{d}x) & (\operatorname{d}y) \\ + & x & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline\hline $f_{6}$ & $\begin{smallmatrix} & (x, y) & \operatorname{d}x & \operatorname{d}y \\ + & (x, y) & \operatorname{d}x & \operatorname{d}y \\ = &  0    & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x, y)  & \operatorname{d}x & (\operatorname{d}y) \\ = &   1     & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x, y)  & (\operatorname{d}x) & \operatorname{d}y \\ = &   1     & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{9}$ & $\begin{smallmatrix} & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ + & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ = &   0     & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x, y)  & \operatorname{d}x & (\operatorname{d}y) \\ + & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\ = &   1     & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x, y)  & (\operatorname{d}x) & \operatorname{d}y \\ + & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\ = &   1     & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0     & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline\hline $f_{5}$ & $\begin{smallmatrix} & y  & \operatorname{d}x & \operatorname{d}y \\ + & (y) & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (y) & \operatorname{d}x & (\operatorname{d}y) \\ + & (y) & \operatorname{d}x & (\operatorname{d}y) \\ = & 0  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & y  & (\operatorname{d}x) & \operatorname{d}y \\ + & (y) & (\operatorname{d}x) & \operatorname{d}y \\ = & 1  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{10}$ & $\begin{smallmatrix} & (y) & \operatorname{d}x & \operatorname{d}y \\ + & y  & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & y  & \operatorname{d}x & (\operatorname{d}y) \\ + & y  & \operatorname{d}x & (\operatorname{d}y) \\ = & 0  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (y) & (\operatorname{d}x) & \operatorname{d}y \\ + & y  & (\operatorname{d}x) & \operatorname{d}y \\ = & 1  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & y  & (\operatorname{d}x) & (\operatorname{d}y) \\ + & y  & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline\hline $f_{7}$ & $\begin{smallmatrix} & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\ + & (x\ y)  & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x\  y) & \operatorname{d}x & (\operatorname{d}y) \\ = &      y  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x\ y)  & (\operatorname{d}x) & \operatorname{d}y \\ = & x       & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{11}$ & $\begin{smallmatrix} & ((x)\ y) & \operatorname{d}x & \operatorname{d}y \\ + & (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\ = & (x,  y)  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\ = &     (y)  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x\ y)  & (\operatorname{d}x) & \operatorname{d}y \\ + & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\ = & x       & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0     & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{13}$ & $\begin{smallmatrix} & (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\ + & ((x)\ y) & \operatorname{d}x & \operatorname{d}y \\ = & (x,  y)  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x\  y) & \operatorname{d}x & (\operatorname{d}y) \\ + & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\ = &      y  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\ + & ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &    0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline $f_{14}$ & $\begin{smallmatrix} & (x\ y)  & \operatorname{d}x & \operatorname{d}y \\ + & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\ + & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\ = &     (y)  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \\ + & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0     & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$ \\[6pt] \hline\hline $f_{15}$ & $1 + 1 = 0$ & $1 + 1 = 0$ & $1 + 1 = 0$ & $1 + 1 = 0$ \\[6pt] \hline\hline \end{tabular}\end{center}

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