# PlanetPhysics/Differential Propositional Calculus Appendix 4

### Detail of Calculation for the Difference Map

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

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