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PlanetPhysics/Differential Propositional Calculus Appendix 4

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Detail of Calculation for the Difference Map

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\multicolumn{5}{Detail of Calculation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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}} &   &   &   &   \Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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}} &   &   &   & 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle 6pt] \hline <math>f_{8}} &   &   &   & 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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}} &   &   &   & 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle 6pt] \hline <math>f_{9}} &   &   &   & 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle 6pt] \hline <math>f_{10}} &   &   &   & 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle 6pt] \hline <math>f_{11}} &   &   &   & 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle 6pt] \hline <math>f_{14}} &   &   &   & 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}} &   &   &   &   \<math>6pt] \hline\hline

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