PlanetPhysics/Differential Propositional Calculus Appendix 4

Detail of Calculation for the Difference Map

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\multicolumn{5}{Detail of Calculation for Failed to parse (syntax error): {\displaystyle \operatorname{D f = \operatorname{E}f + f} }} \Failed to parse (syntax error): {\displaystyle 6pt] \hline\hline & <math>\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}<math> & <math>\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}<math> & <math>\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}<math> & <math>\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}} \Failed to parse (syntax error): {\displaystyle 6pt] \hline\hline <math>f_{0}} & $0+0=0$ & $0+0=0$ & $0+0=0$ & $0+0=0$ \Failed to parse (syntax error): {\displaystyle 6pt] \hline\hline <math>f_{1}} & </math>\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}</math> \Failed to parse (syntax error): {\displaystyle 6pt] \hline <math>f_{2}} & $\begin{array}{cccc}& x(y)& \mathrm{d}x& \mathrm{d}y\\ +& (x)y& \mathrm{d}x& \mathrm{d}y\\ =& (x,y)& \mathrm{d}x& \mathrm{d}y\end{array}$ & $\begin{array}{cccc}& xy& \mathrm{d}x& (\mathrm{d}y)\\ +& (x)y& \mathrm{d}x& (\mathrm{d}y)\\ =& y& \mathrm{d}x& (\mathrm{d}y)\end{array}$ & $\begin{array}{cccc}& (x)(y)& (\mathrm{d}x)& \mathrm{d}y\\ +& (x)y& (\mathrm{d}x)& \mathrm{d}y\\ =& (x)& (\mathrm{d}x)& \mathrm{d}y\end{array}$ & Failed to parse (syntax error): {\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}<math> \<math>6pt] \hline <math>f_{4}} & </math>\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}</math> \Failed to parse (syntax error): {\displaystyle 6pt] \hline <math>f_{8}} & $\begin{array}{cccc}& (x)(y)& \mathrm{d}x& \mathrm{d}y\\ +& xy& \mathrm{d}x& \mathrm{d}y\\ =& ((x,y))& \mathrm{d}x& \mathrm{d}y\end{array}$ & $\begin{array}{cccc}& (x)y& \mathrm{d}x& (\mathrm{d}y)\\ +& xy& \mathrm{d}x& (\mathrm{d}y)\\ =& y& \mathrm{d}x& (\mathrm{d}y)\end{array}$ & $\begin{array}{cccc}& x(y)& (\mathrm{d}x)& \mathrm{d}y\\ +& xy& (\mathrm{d}x)& \mathrm{d}y\\ =& x& (\mathrm{d}x)& \mathrm{d}y\end{array}$ & Failed to parse (syntax error): {\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}<math> \<math>6pt] \hline\hline <math>f_{3}} & </math>\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}</math> \Failed to parse (syntax error): {\displaystyle 6pt] \hline <math>f_{12}} & $\begin{array}{cccc}& (x)& \mathrm{d}x& \mathrm{d}y\\ +& x& \mathrm{d}x& \mathrm{d}y\\ =& 1& \mathrm{d}x& \mathrm{d}y\end{array}$ & $\begin{array}{cccc}& (x)& \mathrm{d}x& (\mathrm{d}y)\\ +& x& \mathrm{d}x& (\mathrm{d}y)\\ =& 1& \mathrm{d}x& (\mathrm{d}y)\end{array}$ & $\begin{array}{cccc}& x& (\mathrm{d}x)& \mathrm{d}y\\ +& x& (\mathrm{d}x)& \mathrm{d}y\\ =& 0& (\mathrm{d}x)& \mathrm{d}y\end{array}$ & Failed to parse (syntax error): {\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}<math> \<math>6pt] \hline\hline <math>f_{6}} & </math>\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}</math> \Failed to parse (syntax error): {\displaystyle 6pt] \hline <math>f_{9}} & $\begin{array}{cccc}& ((x,y))& \mathrm{d}x& \mathrm{d}y\\ +& ((x,y))& \mathrm{d}x& \mathrm{d}y\\ =& 0& \mathrm{d}x& \mathrm{d}y\end{array}$ & $\begin{array}{cccc}& (x,y)& \mathrm{d}x& (\mathrm{d}y)\\ +& ((x,y))& \mathrm{d}x& (\mathrm{d}y)\\ =& 1& \mathrm{d}x& (\mathrm{d}y)\end{array}$ & $\begin{array}{cccc}& (x,y)& (\mathrm{d}x)& \mathrm{d}y\\ +& ((x,y))& (\mathrm{d}x)& \mathrm{d}y\\ =& 1& (\mathrm{d}x)& \mathrm{d}y\end{array}$ & Failed to parse (syntax error): {\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}<math> \<math>6pt] \hline\hline <math>f_{5}} & </math>\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}</math> \Failed to parse (syntax error): {\displaystyle 6pt] \hline <math>f_{10}} & $\begin{array}{cccc}& (y)& \mathrm{d}x& \mathrm{d}y\\ +& y& \mathrm{d}x& \mathrm{d}y\\ =& 1& \mathrm{d}x& \mathrm{d}y\end{array}$ & $\begin{array}{cccc}& y& \mathrm{d}x& (\mathrm{d}y)\\ +& y& \mathrm{d}x& (\mathrm{d}y)\\ =& 0& \mathrm{d}x& (\mathrm{d}y)\end{array}$ & $\begin{array}{cccc}& (y)& (\mathrm{d}x)& \mathrm{d}y\\ +& y& (\mathrm{d}x)& \mathrm{d}y\\ =& 1& (\mathrm{d}x)& \mathrm{d}y\end{array}$ & Failed to parse (syntax error): {\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}<math> \<math>6pt] \hline\hline <math>f_{7}} & </math>\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}</math> \Failed to parse (syntax error): {\displaystyle 6pt] \hline <math>f_{11}} & $\begin{array}{cccc}& ((x)y)& \mathrm{d}x& \mathrm{d}y\\ +& (x(y))& \mathrm{d}x& \mathrm{d}y\\ =& (x,y)& \mathrm{d}x& \mathrm{d}y\end{array}$ & $\begin{array}{cccc}& ((x)(y))& \mathrm{d}x& (\mathrm{d}y)\\ +& (x(y))& \mathrm{d}x& (\mathrm{d}y)\\ =& (y)& \mathrm{d}x& (\mathrm{d}y)\end{array}$ & $\begin{array}{cccc}& (xy)& (\mathrm{d}x)& \mathrm{d}y\\ +& (x(y))& (\mathrm{d}x)& \mathrm{d}y\\ =& x& (\mathrm{d}x)& \mathrm{d}y\end{array}$ & Failed to parse (syntax error): {\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}<math> \<math>6pt] \hline <math>f_{13}} & </math>\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}Failed to parse (syntax error): {\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}</math> \Failed to parse (syntax error): {\displaystyle 6pt] \hline <math>f_{14}} & $\begin{array}{cccc}& (xy)& \mathrm{d}x& \mathrm{d}y\\ +& ((x)(y))& \mathrm{d}x& \mathrm{d}y\\ =& ((x,y))& \mathrm{d}x& \mathrm{d}y\end{array}$ & $\begin{array}{cccc}& (x(y))& \mathrm{d}x& (\mathrm{d}y)\\ +& ((x)(y))& \mathrm{d}x& (\mathrm{d}y)\\ =& (y)& \mathrm{d}x& (\mathrm{d}y)\end{array}$ & $\begin{array}{cccc}& ((x)y)& (\mathrm{d}x)& \mathrm{d}y\\ +& ((x)(y))& (\mathrm{d}x)& \mathrm{d}y\\ =& (x)& (\mathrm{d}x)& \mathrm{d}y\end{array}$ & Failed to parse (syntax error): {\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}<math> \<math>6pt] \hline\hline <math>f_{15}} & $1+1=0$ & $1+1=0$ & $1+1=0$ & $1+1=0$ \<math>6pt] \hline\hline

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