Differential Logic and Dynamic Systems • Appendices

Author: Jon Awbrey

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• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Appendices • References • Document History •

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Appendices

Appendix 1. Propositional Forms and Differential Expansions

Table A1. Propositional Forms on Two Variables

\(\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\)

\(\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}\)
	\(x\colon\)	\(1~1~0~0\)
	\(y\colon\)	\(1~0~1~0\)
\(\begin{matrix} f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7} \end{matrix}\)	\(\begin{matrix} f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111} \end{matrix}\)	\(\begin{matrix} 0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1 \end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ \texttt{)} \\ \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ \texttt{(} x \texttt{)} ~ ~ ~ ~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ ~ ~ ~ ~ \texttt{(} y \texttt{)} \\ \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{(} x ~ ~ ~ y \texttt{)} \end{matrix}\)	\(\begin{matrix} \text{false} \\ \text{neither}~ x ~\text{nor}~ y \\ y ~\text{without}~ x \\ \text{not}~ x \\ x ~\text{without}~ y \\ \text{not}~ y \\ x ~\text{not equal to}~ y \\ \text{not both}~ x ~\text{and}~ y \end{matrix}\)	\(\begin{matrix} 0 \\ \lnot x \land \lnot y \\ \lnot x \land y \\ \lnot x \\ x \land \lnot y \\ \lnot y \\ x \ne y \\ \lnot x \lor \lnot y \end{matrix}\)
\(\begin{matrix} f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15} \end{matrix}\)	\(\begin{matrix} f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111} \end{matrix}\)	\(\begin{matrix} 1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1 \end{matrix}\)	\(\begin{matrix} x ~ ~ ~ y \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \\ ~ ~ ~ ~ y \\ ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\ x ~ ~ ~ ~ \\ \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\ \texttt{((} x \texttt{)(} y \texttt{))} \\ \texttt{((} ~~ \texttt{))} \end{matrix}\)	\(\begin{matrix} x ~\text{and}~ y \\ x ~\text{equal to}~ y \\ y \\ \text{not}~ x ~\text{without}~ y \\ x \\ \text{not}~ y ~\text{without}~ x \\ x ~\text{or}~ y \\ \text{true} \end{matrix}\)	\(\begin{matrix} x \land y \\ x = y \\ y \\ x \Rightarrow y \\ x \\ x \Leftarrow y \\ x \lor y \\ 1 \end{matrix}\)

Table A2. Propositional Forms on Two Variables

\(\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\)

\(\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}\)
	\(x\colon\)	\(1~1~0~0\)
	\(y\colon\)	\(1~0~1~0\)
\(f_{0}\)	\(f_{0000}\)	\(0~0~0~0\)	\(\texttt{(} ~~ \texttt{)}\)	\(\text{false}\)	\(0\)
\(\begin{matrix} f_{1}\\f_{2}\\f_{4}\\f_{8} \end{matrix}\)	\(\begin{matrix} f_{0001}\\f_{0010}\\f_{0100}\\f_{1000} \end{matrix}\)	\(\begin{matrix} 0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0 \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} \text{neither}~ x ~\text{nor}~ y \\ y ~\text{without}~ x \\ x ~\text{without}~ y \\ x ~\text{and}~ y \end{matrix}\)	\(\begin{matrix} \lnot x \land \lnot y \\ \lnot x \land y \\ x \land \lnot y \\ x \land y \end{matrix}\)
\(\begin{matrix} f_{3}\\f_{12} \end{matrix}\)	\(\begin{matrix} f_{0011}\\f_{1100} \end{matrix}\)	\(\begin{matrix} 0~0~1~1\\1~1~0~0 \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix} \text{not}~ x \\ x \end{matrix}\)	\(\begin{matrix} \lnot x \\ x \end{matrix}\)
\(\begin{matrix} f_{6}\\f_{9} \end{matrix}\)	\(\begin{matrix} f_{0110}\\f_{1001} \end{matrix}\)	\(\begin{matrix} 0~1~1~0\\1~0~0~1 \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} x ~\text{not equal to}~ y \\ x ~\text{equal to}~ y \end{matrix}\)	\(\begin{matrix} x \ne y \\ x = y \end{matrix}\)
\(\begin{matrix} f_{5}\\f_{10} \end{matrix}\)	\(\begin{matrix} f_{0101}\\f_{1010} \end{matrix}\)	\(\begin{matrix} 0~1~0~1\\1~0~1~0 \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} \text{not}~ y \\ y \end{matrix}\)	\(\begin{matrix} \lnot y \\ y \end{matrix}\)
\(\begin{matrix} f_{7}\\f_{11}\\f_{13}\\f_{14} \end{matrix}\)	\(\begin{matrix} f_{0111}\\f_{1011}\\f_{1101}\\f_{1110} \end{matrix}\)	\(\begin{matrix} 0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0 \end{matrix}\)	\(\begin{matrix} \texttt{(} x ~ ~ ~ y \texttt{)} \\ ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} \text{not both}~ x ~\text{and}~ y \\ \text{not}~ x ~\text{without}~ y \\ \text{not}~ y ~\text{without}~ x \\ x ~\text{or}~ y \end{matrix}\)	\(\begin{matrix} \lnot x \lor \lnot y \\ x \Rightarrow y \\ x \Leftarrow y \\ x \lor y \end{matrix}\)
\(f_{15}\)	\(f_{1111}\)	\(1~1~1~1\)	\(\texttt{((} ~~ \texttt{))}\)	\(\text{true}\)	\(1\)

Table A3. Ef Expanded Over Differential Features

\(\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\)

	\(f\)	\(\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}\)	\(\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}\)	\(\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}\)	\(\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}\)
\(f_{0}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(\begin{matrix} f_{1}\\f_{2}\\f_{4}\\f_{8} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} x ~ ~ ~ y \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ \texttt{(} x \texttt{)(} y \texttt{)} \end{matrix}\)	\(\begin{matrix} ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \\ \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} ~~ y ~~ \\ \texttt{(} x \texttt{)(} y \texttt{)} \\ x ~ ~ ~ y \\ ~~ x ~~ \texttt{(} y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)
\(\begin{matrix} f_{3}\\f_{12} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix} x \\ \texttt{(} x \texttt{)} \end{matrix}\)	\(\begin{matrix} x \\ \texttt{(} x \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)
\(\begin{matrix} f_{6}\\f_{9} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{((} x \texttt{,} ~~ y \texttt{))} \\ \texttt{(} x \texttt{,} ~~ y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{((} x \texttt{,} ~~ y \texttt{))} \\ \texttt{(} x \texttt{,} ~~ y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)
\(\begin{matrix} f_{5}\\f_{10} \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} y \\ \texttt{(} y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} y \\ \texttt{(} y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)
\(\begin{matrix} f_{7}\\f_{11}\\f_{13}\\f_{14} \end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{((} x \texttt{)(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \\ \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)
\(f_{15}\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\text{Fixed Point Total}\)		\(4\)	\(4\)	\(4\)	\(16\)

Table A4. Df Expanded Over Differential Features

\(\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\)

	\(f\)	\(\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\)	\(\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\)	\(\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\)	\(\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\)
\(f_{0}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(\begin{matrix} f_{1}\\f_{2}\\f_{4}\\f_{8} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} \texttt{((} x \texttt{,} ~~ y \texttt{))} \\ \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \\ \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ \texttt{(} x \texttt{)} \\ x \\ x \end{matrix}\)	\(\begin{matrix}0\\0\\0\\0\end{matrix}\)
\(\begin{matrix}f_{3}\\f_{12}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)
\(\begin{matrix}f_{6}\\f_{9}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)
\(\begin{matrix}f_{5}\\f_{10}\end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)
\(\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{((} x \texttt{,} ~~ y \texttt{))} \\ \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} y \\ \texttt{(} y \texttt{)} \\ y \\ \texttt{(} y \texttt{)} \end{matrix}\)	\(\begin{matrix} x \\ x \\ \texttt{(} x \texttt{)} \\ \texttt{(} x \texttt{)} \end{matrix}\)	\(\begin{matrix}0\\0\\0\\0\end{matrix}\)
\(f_{15}\)	\(1\)	\(0\)	\(0\)	\(0\)	\(0\)

Table A5. Ef Expanded Over Ordinary Features

\(\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\)

	\(f\)	\(\mathrm{E}f|_{xy}\)	\(\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\)	\(\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\)	\(\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\)
\(f_{0}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(\begin{matrix} f_{1}\\f_{2}\\f_{4}\\f_{8} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ \texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ \texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ \texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \end{matrix}\)
\(\begin{matrix} f_{3}\\f_{12} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \texttt{(} \mathrm{d}x \texttt{)} \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \texttt{(} \mathrm{d}x \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{)} \\ \mathrm{d}x \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{)} \\ \mathrm{d}x \end{matrix}\)
\(\begin{matrix} f_{6}\\f_{9} \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \\ \texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))} \\ \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))} \\ \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \\ \texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))} \end{matrix}\)
\(\begin{matrix} f_{5}\\f_{10} \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \texttt{(} \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \texttt{(} \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}y \end{matrix}\)
\(\begin{matrix} f_{7}\\f_{11}\\f_{13}\\f_{14} \end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)} \\ \texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))} \\ \texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)} \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)} \\ \texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))} \\ \texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)} \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)} \\ \texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))} \\ \texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)} \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \end{matrix}\)
\(f_{15}\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)

Table A6. Df Expanded Over Ordinary Features

\(\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\)

	\(f\)	\(\mathrm{D}f|_{xy}\)	\(\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\)	\(\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\)	\(\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\)
\(f_{0}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \end{matrix}\)	\(\begin{matrix} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \end{matrix}\)
\(\begin{matrix}f_{3}\\f_{12}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \end{matrix}\)
\(\begin{matrix}f_{6}\\f_{9}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)} \end{matrix}\)
\(\begin{matrix}f_{5}\\f_{10}\end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \end{matrix}\)
\(\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \end{matrix}\)	\(\begin{matrix} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \end{matrix}\)	\(\begin{matrix} \mathrm{d}x ~ ~ ~ \mathrm{d}y \\ ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)} \\ \texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \\ \texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))} \end{matrix}\)
\(f_{15}\)	\(1\)	\(0\)	\(0\)	\(0\)	\(0\)

Appendix 2. Differential Forms

The actions of the difference operator \(\mathrm{D}\) and the tangent operator \(\mathrm{d}\) on the 16 bivariate propositions are shown in Tables A7 and A8.

Table A7 expands the differential forms that result over a logical basis:

\(\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\)

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:

\(\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\)     and     \(\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\)

Table A8 expands the differential forms that result over an algebraic basis:

\(\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\)

This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.

Table A7. Differential Forms Expanded on a Logical Basis

\(\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\)

	\(f\)	\(\mathrm{D}f\)	\(\mathrm{d}f\)
\(f_{0}\)	\(\texttt{(} ~~ \texttt{)}\)	\(0\)	\(0\)
\(\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y \\ y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y \\ \texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y \\ y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y & + & \texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} ~\partial x & + & \texttt{(} x \texttt{)} ~\partial y \\ ~~ y ~~ ~\partial x & + & \texttt{(} x \texttt{)} ~\partial y \\ \texttt{(} y \texttt{)} ~\partial x & + & ~~ x ~~ ~\partial y \\ ~~ y ~~ ~\partial x & + & ~~ x ~~ ~\partial y \end{matrix}\)
\(\begin{matrix}f_{3}\\f_{12}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y \\ \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \partial x \\ \partial x \end{matrix}\)
\(\begin{matrix}f_{6}\\f_{9}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y \\ \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \partial x & + & \partial y \\ \partial x & + & \partial y \end{matrix}\)
\(\begin{matrix}f_{5}\\f_{10}\end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y \\ \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \partial y \\ \partial y \end{matrix}\)
\(\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y \\ \texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y & + & \texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y \\ y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y \\ \texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} ~~ y ~~ ~\partial x & + & ~~ x ~~ ~\partial y \\ \texttt{(} y \texttt{)} ~\partial x & + & ~~ x ~~ ~\partial y \\ ~~ y ~~ ~\partial x & + & \texttt{(} x \texttt{)} ~\partial y \\ \texttt{(} y \texttt{)} ~\partial x & + & \texttt{(} x \texttt{)} ~\partial y \end{matrix}\)
\(f_{15}\)	\(\texttt{((} ~~ \texttt{))}\)	\(0\)	\(0\)

Table A8. Differential Forms Expanded on an Algebraic Basis

\(\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\)

	\(f\)	\(\mathrm{D}f\)	\(\mathrm{d}f\)
\(f_{0}\)	\(\texttt{(} ~~ \texttt{)}\)	\(0\)	\(0\)
\(\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \\ ~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \\ \texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \\ ~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y \\ ~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y \\ \texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y \\ ~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y \end{matrix}\)
\(\begin{matrix}f_{3}\\f_{12}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \end{matrix}\)
\(\begin{matrix}f_{6}\\f_{9}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} \mathrm{d}x & + & \mathrm{d}y \\ \mathrm{d}x & + & \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x & + & \mathrm{d}y \\ \mathrm{d}x & + & \mathrm{d}y \end{matrix}\)
\(\begin{matrix}f_{5}\\f_{10}\end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \end{matrix}\)
\(\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} ~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \\ \texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \\ ~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \\ \texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y \end{matrix}\)	\(\begin{matrix} ~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y \\ \texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y \\ ~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y \\ \texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y \end{matrix}\)
\(f_{15}\)	\(\texttt{((} ~~ \texttt{))}\)	\(0\)	\(0\)

Table A9. Tangent Proposition as Pointwise Linear Approximation

\(\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\)

\(f\)	\(\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}^2\!f = \\[2pt] \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\mathrm{d}f|_{x \, y}\)	\(\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}\)	\(\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}\)	\(\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\)
\(f_0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}\)	\(\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}\)
\(\begin{matrix}f_{3}\\f_{12}\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}\)
\(\begin{matrix}f_{6}\\f_{9}\end{matrix}\)	\(\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}\)
\(\begin{matrix}f_{5}\\f_{10}\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\)
\(\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}\)	\(\begin{matrix} ~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}\)	\(\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}\)
\(f_{15}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)

Table A10. Taylor Series Expansion Df = df + d²f

\(\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\)

\(f\)	\(\begin{matrix} \mathrm{D}f \\ = & \mathrm{d}f & + & \mathrm{d}^2\!f \\ = & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\mathrm{d}f|_{x \, y}\)	\(\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}\)	\(\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}\)	\(\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\)
\(f_0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\begin{matrix} 0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0 \end{matrix}\)
\(\begin{matrix}f_{3}\\f_{12}\end{matrix}\)	\(\begin{matrix} ~~ 1 ~~ \cdot \mathrm{d}x & + & ~~ 0 ~~ \cdot \mathrm{d}y & + & ~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ ~~ 1 ~~ \cdot \mathrm{d}x & + & ~~ 0 ~~ \cdot \mathrm{d}y & + & ~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}\)
\(\begin{matrix}f_{6}\\f_{9}\end{matrix}\)	\(\begin{matrix} ~~ 1 ~~ \cdot \mathrm{d}x & + & ~~ 1 ~~ \cdot \mathrm{d}y & + & ~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ ~~ 1 ~~ \cdot \mathrm{d}x & + & ~~ 1 ~~ \cdot \mathrm{d}y & + & ~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}\)
\(\begin{matrix}f_{5}\\f_{10}\end{matrix}\)	\(\begin{matrix} ~~ 0 ~~ \cdot \mathrm{d}x & + & ~~ 1 ~~ \cdot \mathrm{d}y & + & ~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ ~~ 0 ~~ \cdot \mathrm{d}x & + & ~~ 1 ~~ \cdot \mathrm{d}y & + & ~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}\)
\(\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}\)	\(\begin{matrix} ~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & ~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}\)	\(\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0 \end{matrix}\)	\(\begin{matrix} \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x \end{matrix}\)	\(\begin{matrix} \mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y \end{matrix}\)	\(\begin{matrix} 0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}\)
\(f_{15}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)

Table A11. Partial Differentials and Relative Differentials

\(\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\)

	\(f\)	\(\frac{\partial f}{\partial x}\)	\(\frac{\partial f}{\partial y}\)	\(\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}\)	\(\left. \frac{\partial x}{\partial y} \right| f\)	\(\left. \frac{\partial y}{\partial x} \right| f\)
\(f_0\)	\(\texttt{(} ~ \texttt{)}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)} ~~ y ~~ \\ ~~ x ~~ \texttt{(} y \texttt{)} \\ x ~ ~ ~ y \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \\ \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ \texttt{(} x \texttt{)} \\ x \\ x \end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}\)
\(\begin{matrix}f_{3}\\f_{12}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{)} \\ x \end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\end{matrix}\)
\(\begin{matrix}f_{6}\\f_{9}\end{matrix}\)	\(\begin{matrix} \texttt{(} x \texttt{,} ~~ y \texttt{)} \\ \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\end{matrix}\)
\(\begin{matrix}f_{5}\\f_{10}\end{matrix}\)	\(\begin{matrix} \texttt{(} y \texttt{)} \\ y \end{matrix}\)	\(\begin{matrix}0\\0\end{matrix}\)	\(\begin{matrix}1\\1\end{matrix}\)	\(\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\end{matrix}\)
\(\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}\)	\(\begin{matrix} \texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)} \\ \texttt{(} ~~ x ~~ \texttt{(} y \texttt{))} \\ \texttt{((} x \texttt{)} ~~ y ~~ \texttt{)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}\)	\(\begin{matrix} y \\ \texttt{(} y \texttt{)} \\ y \\ \texttt{(} y \texttt{)} \end{matrix}\)	\(\begin{matrix} x \\ x \\ \texttt{(} x \texttt{)} \\ \texttt{(} x \texttt{)} \end{matrix}\)	\(\begin{matrix} ~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y \\ ~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}\)	\(\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}\)
\(f_{15}\)	\(\texttt{((} ~ \texttt{))}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)

Table A12. Detail of Calculation for the Difference Map

\(\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\)

	\(f\)	\(\begin{array}{cr} ~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y} \\[4pt] + & f|_{\mathrm{d}x ~ \mathrm{d}y} \\[4pt] = & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y} \end{array}\)	\(\begin{array}{cr} ~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y} \\[4pt] + & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y} \\[4pt] = & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y} \end{array}\)	\(\begin{array}{cr} ~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}} \\[4pt] + & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}} \\[4pt] = & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}} \end{array}\)	\(\begin{array}{cr} ~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}} \\[4pt] + & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}} \\[4pt] = & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}} \end{array}\)
\(f_{0}\)	\(0\)	\(0 ~+~ 0 ~=~ 0\)	\(0 ~+~ 0 ~=~ 0\)	\(0 ~+~ 0 ~=~ 0\)	\(0 ~+~ 0 ~=~ 0\)
\(f_{1}\)	\(\texttt{(} x \texttt{)(} y \texttt{)}\)	\(\begin{matrix} ~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] + & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] = & \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} ~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] = & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] + & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] = & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] = & 0 \end{matrix}\)
\(f_{2}\)	\(\texttt{(} x \texttt{)} ~~ y ~~\)	\(\begin{matrix} ~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] = & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] + & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] = & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] = & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] + & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] = & 0 \end{matrix}\)
\(f_{4}\)	\(~~ x ~~ \texttt{(} y \texttt{)}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] + & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] = & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] + & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] = & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] + & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] = & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] + & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] = & 0 \end{matrix}\)
\(f_{8}\)	\(x ~ ~ ~ y\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~ \\[4pt] + & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] = & \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~ \\[4pt] + & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] = & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~ \\[4pt] + & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] = & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] + & ~ ~ ~ x ~ ~ ~ y ~ ~ ~ \\[4pt] = & 0 \end{matrix}\)
\(f_{3}\)	\(\texttt{(} x \texttt{)}\)	\(\begin{matrix} ~ & x \\[4pt] + & \texttt{(} x \texttt{)} \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & x \\[4pt] + & \texttt{(} x \texttt{)} \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{(} x \texttt{)} \\[4pt] + & \texttt{(} x \texttt{)} \\[4pt] = & 0 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{(} x \texttt{)} \\[4pt] + & \texttt{(} x \texttt{)} \\[4pt] = & 0 \end{matrix}\)
\(f_{12}\)	\(x\)	\(\begin{matrix} ~ & \texttt{(} x \texttt{)} \\[4pt] + & x \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{(} x \texttt{)} \\[4pt] + & x \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & ~~ x ~~ \\[4pt] + & x \\[4pt] = & 0 \end{matrix}\)	\(\begin{matrix} ~ & ~~ x ~~ \\[4pt] + & x \\[4pt] = & 0 \end{matrix}\)
\(f_{6}\)	\(\texttt{(} x \texttt{,} ~~ y \texttt{)}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] = & 0 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] + & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] + & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] = & 0 \end{matrix}\)
\(f_{9}\)	\(\texttt{((} x \texttt{,} ~~ y \texttt{))}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] + & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] = & 0 \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] + & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \\[4pt] + & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] + & \texttt{((} x \texttt{,} ~~ y \texttt{))} \\[4pt] = & 0 \end{matrix}\)
\(f_{5}\)	\(\texttt{(} y \texttt{)}\)	\(\begin{matrix} ~ & y \\[4pt] + & \texttt{(} y \texttt{)} \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{(} y \texttt{)} \\[4pt] + & \texttt{(} y \texttt{)} \\[4pt] = & 0 \end{matrix}\)	\(\begin{matrix} ~ & y \\[4pt] + & \texttt{(} y \texttt{)} \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{(} y \texttt{)} \\[4pt] + & \texttt{(} y \texttt{)} \\[4pt] = & 0 \end{matrix}\)
\(f_{10}\)	\(y\)	\(\begin{matrix} ~ & \texttt{(} y \texttt{)} \\[4pt] + & y \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & ~~ y ~~ \\[4pt] + & y \\[4pt] = & 0 \end{matrix}\)	\(\begin{matrix} ~ & \texttt{(} y \texttt{)} \\[4pt] + & y \\[4pt] = & 1 \end{matrix}\)	\(\begin{matrix} ~ & ~~ y ~~ \\[4pt] + & y \\[4pt] = & 0 \end{matrix}\)
\(f_{7}\)	\(\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] + & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] = & \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] = & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] + & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] = & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] = & 0 \end{matrix}\)
\(f_{11}\)	\(\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] = & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] + & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] = & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] + & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] = & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] + & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] = & 0 \end{matrix}\)
\(f_{13}\)	\(\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] + & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] = & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] + & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] = & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] + & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] = & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] + & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] = & 0 \end{matrix}\)
\(f_{14}\)	\(\texttt{((} x \texttt{)(} y \texttt{))}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~ \\[4pt] + & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] = & \texttt{((} x \texttt{,} ~~ y \texttt{))} \end{matrix}\)	\(\begin{matrix} ~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))} \\[4pt] + & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] = & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~ \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~ \\[4pt] + & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] = & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~ \end{matrix}\)	\(\begin{matrix} ~ & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] + & \texttt{((} x \texttt{)(} y \texttt{))} \\[4pt] = & 0 \end{matrix}\)
\(f_{15}\)	\(1\)	\(1 ~+~ 1 ~=~ 0\)	\(1 ~+~ 1 ~=~ 0\)	\(1 ~+~ 1 ~=~ 0\)	\(1 ~+~ 1 ~=~ 0\)

Appendix 3. Computational Details

Operator Maps for the Logical Conjunction f₈⟨u, v⟩

Computation of εf₈

\(\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}\)

\(\begin{array}{*{10}{l}} \boldsymbol\varepsilon f_{8} & = && f_{8} \langle u, v \rangle \\[4pt] & = && u v \\[4pt] & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & u v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & u v \cdot \mathrm{d}u ~ \mathrm{d}v \\[20pt] \boldsymbol\varepsilon f_{8} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \\[4pt] && + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~ \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v ~~ \end{array}\)

Computation of Ef₈

\(\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\)

\(\begin{array}{*{9}{l}} \mathrm{E}f_{8} & = & f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle \\[4pt] & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)} \\[4pt] & = & u v \cdot f_{8} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle & + & u \texttt{(} v \texttt{)} \cdot f_{8} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle & + & \texttt{(} u \texttt{)} v \cdot f_{8} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8} \langle \mathrm{d}u, \mathrm{d}v \rangle \\[4pt] & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \\[20pt] \mathrm{E}f_{8} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \\[4pt] &&& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v \\[4pt] &&&&& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] &&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

\(\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\)

\(\begin{array}{*{9}{c}} \mathrm{E}f_{8} & = & \{ u + \mathrm{d}u \} \cdot \{ v + \mathrm{d}v \} \\[6pt] & = & u \cdot v & + & u \cdot \mathrm{d}v & + & v \cdot \mathrm{d}u & + & \mathrm{d}u \cdot \mathrm{d}v \\[6pt] \mathrm{E}f_{8} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

Computation of Df₈

\(\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\)

\(\begin{array}{*{10}{l}} \mathrm{D}f_{8} & = && \mathrm{E}f_{8} & + & \boldsymbol\varepsilon f_{8} \\[4pt] & = && f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle & + & f_{8} \langle u, v \rangle \\[4pt] & = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)} & + & u v \\[20pt] \mathrm{D}f_{8} & = && 0 & + & 0 & + & 0 & + & 0 \\[4pt] && + & u v \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~ ~ ~ & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~ & + & 0 & + & 0 \\[4pt] && + & u v \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{} ~~ & + & 0 & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 \\[4pt] && + & u v \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v ~ ~ ~ & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v ~~ \\[20pt] \mathrm{D}f_{8} & = && u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~ & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v ~~ \end{array}\)

\(\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\)

\(\begin{array}{*{9}{l}} \mathrm{D}f_{8} & = & \boldsymbol\varepsilon f_{8} & + & \mathrm{E}f_{8} \\[6pt] & = & f_{8} \langle u, v \rangle & + & f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle \\[6pt] & = & u v & + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)} \\[6pt] & = & 0 & + & u \cdot \mathrm{d}v & + & v \cdot \mathrm{d}u & + & \mathrm{d}u ~ \mathrm{d}v \\[6pt] \mathrm{D}f_{8} & = & 0 & + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

\(\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\)

\(\begin{array}{c*{9}{l}} \mathrm{D}f_{8} & = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8} \\[20pt] \boldsymbol\varepsilon f_{8} & = & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} & + & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & ~~ u ~~ v ~~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & ~ ~ ~ u ~ ~ ~ v ~ ~ ~ \cdot \mathrm{d}u ~~ \mathrm{d}v \\[6pt] \mathrm{E}f_{8} & = & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} & + & ~~ u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)} v ~~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & ~~ \texttt{(} u \texttt{)(} v \texttt{)} ~~ \cdot \mathrm{d}u ~~ \mathrm{d}v \\[20pt] \mathrm{D}f_{8} & = & ~ ~ ~ 0 ~ ~ ~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} & + & ~ ~ ~ u ~ ~ ~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & ~ ~ ~ v ~ ~ ~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{((} u \texttt{,} ~~ v \texttt{))} \cdot \mathrm{d}u ~~ \mathrm{d}v \end{array}\)

Computation of df₈

\(\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\)

\(\begin{array}{c*{8}{l}} \mathrm{D}f_{8} & = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \\[6pt] \Downarrow \\[6pt] \mathrm{d}f_{8} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 \end{array}\)

Computation of rf₈

\(\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\)

\(\begin{array}{c*{8}{l}} \mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8} \\[20pt] \mathrm{D}f_{8} & = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \\[6pt] \mathrm{d}f_{8} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 \\[20pt] \mathrm{r}f_{8} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

Computation Summary for Conjunction

\(\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8} \langle u, v \rangle = u v\)

\(\begin{array}{c*{8}{l}} \boldsymbol\varepsilon f_{8} & = & u v \cdot 1 & + & u \texttt{(} v \texttt{)} \cdot 0 & + & \texttt{(} u \texttt{)} v \cdot 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 \\[6pt] \mathrm{E}f_{8} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \\[6pt] \mathrm{D}f_{8} & = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \\[6pt] \mathrm{d}f_{8} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 \\[6pt] \mathrm{r}f_{8} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

Operator Maps for the Logical Equality f₉⟨u, v⟩

Computation of εf₉

\(\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\)

\(\begin{array}{*{10}{l}} \boldsymbol\varepsilon f_{9} & = && f_{9} \langle u, v \rangle \\[4pt] & = && \texttt{((} u \texttt{,} ~~ v \texttt{))} \\[4pt] & = && u v \cdot f_{9} \langle 1, 1 \rangle & + & u \texttt{(} v \texttt{)} \cdot f_{9} \langle 1, 0 \rangle & + & \texttt{(} u \texttt{)} v \cdot f_{9} \langle 0, 1 \rangle & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9} \langle 0, 0 \rangle \\[4pt] & = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \\[20pt] \boldsymbol\varepsilon f_{9} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \\[4pt] && + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v \end{array}\)

Computation of Ef₉

\(\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\)

\(\begin{array}{*{10}{l}} \mathrm{E}f_{9} & = && f_{9} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle \\[4pt] & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))} \\[4pt] & = && u v \cdot f_{9} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle & + & u \texttt{(} v \texttt{)} \cdot f_{9} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle & + & \texttt{(} u \texttt{)} v \cdot f_{9} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9} \langle \mathrm{d}u, \mathrm{d}v \rangle \\[4pt] & = && u v \cdot \texttt{((} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{))} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{))} \\[20pt] \mathrm{E}f_{9} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \\[4pt] && + & 0 & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & 0 \\[4pt] && + & 0 & + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v \end{array}\)

Computation of Df₉

\(\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\)

\(\begin{array}{*{10}{l}} \mathrm{D}f_{9} & = && \mathrm{E}f_{9} & + & \boldsymbol\varepsilon f_{9} \\[4pt] & = && f_{9} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle & + & f_{9} \langle u, v \rangle \\[4pt] & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))} & + & \texttt{((} u \texttt{,} v \texttt{))} \\[20pt] \mathrm{D}f_{9} & = && 0 & + & 0 & + & 0 & + & 0 \\[4pt] && + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] && + & 0 & + & 0 & + & 0 & + & 0 \\[20pt] \mathrm{D}f_{9} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)} \end{array}\)

\(\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\)

\(\begin{array}{*{9}{l}} \mathrm{D}f_{9} & = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v & + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \end{array}\)

Computation of df₉

\(\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\)

\(\begin{array}{c*{8}{l}} \mathrm{D}f_{9} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \\[6pt] \Downarrow \\[6pt] \mathrm{d}f_{9} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \end{array}\)

Computation of rf₉

\(\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\)

\(\begin{array}{c*{8}{l}} \mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9} \\[20pt] \mathrm{D}f_{9} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \\[6pt] \mathrm{d}f_{9} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \\[20pt] \mathrm{r}f_{9} & = & u v \cdot 0 & + & u \texttt{(} v \texttt{)} \cdot 0 & + & \texttt{(} u \texttt{)} v \cdot 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 \end{array}\)

Computation Summary for Equality

\(\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9} \langle u, v \rangle = \texttt{((} u \texttt{,} v \texttt{))}\)

\(\begin{array}{c*{8}{l}} \boldsymbol\varepsilon f_{9} & = & u v \cdot 1 & + & u \texttt{(} v \texttt{)} \cdot 0 & + & \texttt{(} u \texttt{)} v \cdot 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1 \\[6pt] \mathrm{E}f_{9} & = & u v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))} \\[6pt] \mathrm{D}f_{9} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \\[6pt] \mathrm{d}f_{9} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \\[6pt] \mathrm{r}f_{9} & = & u v \cdot 0 & + & u \texttt{(} v \texttt{)} \cdot 0 & + & \texttt{(} u \texttt{)} v \cdot 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 \end{array}\)

Operator Maps for the Logical Implication f₁₁⟨u, v⟩

Computation of εf₁₁

\(\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\)

\(\begin{array}{*{10}{l}} \boldsymbol\varepsilon f_{11} & = && f_{11} \langle u, v \rangle \\[4pt] & = && \texttt{(} u \texttt{(} v \texttt{))} \\[4pt] & = && u v \cdot f_{11} \langle 1, 1 \rangle & + & u \texttt{(} v \texttt{)} \cdot f_{11} \langle 1, 0 \rangle & + & \texttt{(} u \texttt{)} v \cdot f_{11} \langle 0, 1 \rangle & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11} \langle 0, 0 \rangle \\[4pt] & = && u v & + & 0 & + & \texttt{(} u \texttt{)} v & + & \texttt{(} u \texttt{)(} v \texttt{)} \\[20pt] \boldsymbol\varepsilon f_{11} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \\[4pt] && + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & 0 & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & 0 & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v \end{array}\)

Computation of Ef₁₁

\(\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\)

\(\begin{array}{*{10}{l}} \mathrm{E}f_{11} & = && f_{11} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle \\[4pt] & = && \texttt{(} \\ &&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \\ &&& \texttt{(} \\ &&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)} \\ &&& \texttt{))} \\[4pt] & = && u v \cdot \texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))} & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} \\[4pt] & = && u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} \\[20pt] \mathrm{E}f_{11} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \\[4pt] && + & 0 & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v \end{array}\)

Computation of Df₁₁

\(\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\)

\(\begin{array}{*{10}{l}} \mathrm{D}f_{11} & = && \mathrm{E}f_{11} & + & \boldsymbol\varepsilon f_{11} \\[4pt] & = && f_{11} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle & + & f_{11} \langle u, v \rangle \\[4pt] & = && \texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)} \texttt{))} & + & \texttt{(} u \texttt{(} v \texttt{))} \\[20pt] \mathrm{D}f_{11} & = && 0 & + & 0 & + & 0 & + & 0 \\[4pt] && + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & 0 & + & 0 \\[4pt] && + & 0 & + & u \texttt{(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] && + & 0 & + & u \texttt{(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~~ \mathrm{d}v & + & 0 \\[20pt] \mathrm{D}f_{11} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} \end{array}\)

\(\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\)

\(\begin{array}{c*{9}{l}} \mathrm{D}f_{11} & = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11} \\[20pt] \boldsymbol\varepsilon f_{11} & = & u v \cdot 1 & + & u \texttt{(} v \texttt{)} \cdot 0 & + & \texttt{(} u \texttt{)} v \cdot 1 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1 \\[6pt] \mathrm{E}f_{11} & = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~~ \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} \\[20pt] \mathrm{D}f_{11} & = & u v \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} \end{array}\)

Computation of df₁₁

\(\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\)

\(\begin{array}{c*{8}{l}} \mathrm{D}f_{11} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} \\[6pt] \Downarrow \\[6pt] \mathrm{d}f_{11} & = & u v \cdot \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \end{array}\)

Computation of rf₁₁

\(\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\)

\(\begin{array}{c*{8}{l}} \mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11} \\[20pt] \mathrm{D}f_{11} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} \\[6pt] \mathrm{d}f_{11} & = & u v \cdot \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \\[20pt] \mathrm{r}f_{11} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

Computation Summary for Implication

\(\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11} \langle u, v \rangle = \texttt{(} u \texttt{(} v \texttt{))}\)

\(\begin{array}{c*{8}{l}} \boldsymbol\varepsilon f_{11} & = & u v \cdot 1 & + & u \texttt{(} v \texttt{)} \cdot 0 & + & \texttt{(} u \texttt{)} v \cdot 1 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1 \\[6pt] \mathrm{E}f_{11} & = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} \\[6pt] \mathrm{D}f_{11} & = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} \\[6pt] \mathrm{d}f_{11} & = & u v \cdot \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \\[6pt] \mathrm{r}f_{11} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

Operator Maps for the Logical Disjunction f₁₄⟨u, v⟩

Computation of εf₁₄

\(\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\)

\(\begin{array}{*{10}{l}} \boldsymbol\varepsilon f_{14} & = && f_{14} \langle u, v \rangle \\[4pt] & = && \texttt{((} u \texttt{)(} v \texttt{))} \\[4pt] & = && u v \cdot f_{14} \langle 1, 1 \rangle & + & u \texttt{(} v \texttt{)} \cdot f_{14} \langle 1, 0 \rangle & + & \texttt{(} u \texttt{)} v \cdot f_{14} \langle 0, 1 \rangle & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14} \langle 0, 0 \rangle \\[4pt] & = && u v & + & u \texttt{(} v \texttt{)} & + & \texttt{(} u \texttt{)} v & + & 0 \\[20pt] \boldsymbol\varepsilon f_{14} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & 0 \\[4pt] && + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & 0 \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & 0 \end{array}\)

Computation of Ef₁₄

\(\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\)

\(\begin{array}{*{10}{l}} \mathrm{E}f_{14} & = && f_{14} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle \\[4pt] & = && \texttt{((} \\ &&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \\ &&& \texttt{)(} \\ &&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)} \\ &&& \texttt{))} \\[4pt] & = && u v \cdot f_{14} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle & + & u \texttt{(} v \texttt{)} \cdot f_{14} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle & + & \texttt{(} u \texttt{)} v \cdot f_{14} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14} \langle \mathrm{d}u, \mathrm{d}v \rangle \\[4pt] & = && u v \cdot \texttt{(} \mathrm{d}u ~~ \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} \\[20pt] \mathrm{E}f_{14} & = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & 0 \\[4pt] && + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v \\[4pt] && + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] && + & 0 & + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v \end{array}\)

Computation of Df₁₄

\(\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\)

\(\begin{array}{*{10}{l}} \mathrm{D}f_{14} & = && \mathrm{E}f_{14} & + & \boldsymbol\varepsilon f_{14} \\[4pt] & = && f_{14} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle & + & f_{14} \langle u, v \rangle \\[4pt] & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))} & + & \texttt{((} u \texttt{)(} v \texttt{))} \\[20pt] \mathrm{D}f_{14} & = && 0 & + & 0 & + & 0 & + & 0 \\[4pt] && + & 0 & + & 0 & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v \\[4pt] && + & 0 & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)} \\[4pt] && + & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v \\[20pt] \mathrm{D}f_{14} & = && u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} \end{array}\)

\(\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\)

\(\begin{array}{*{9}{l}} \mathrm{D}f_{14} & = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} \end{array}\)

Computation of df₁₄

\(\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\)

\(\begin{array}{c*{8}{l}} \mathrm{D}f_{14} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} \\[6pt] \Downarrow \\[6pt] \mathrm{d}f_{14} & = & u v \cdot 0 & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \end{array}\)

Computation of rf₁₄

\(\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\)

\(\begin{array}{c*{8}{l}} \mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14} \\[20pt] \mathrm{D}f_{14} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} \\[6pt] \mathrm{d}f_{14} & = & u v \cdot 0 & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \\[20pt] \mathrm{r}f_{14} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

Computation Summary for Disjunction

\(\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14} \langle u, v \rangle = \texttt{((} u \texttt{)(} v \texttt{))}\)

\(\begin{array}{c*{8}{l}} \boldsymbol\varepsilon f_{14} & = & u v \cdot 1 & + & u \texttt{(} v \texttt{)} \cdot 1 & + & \texttt{(} u \texttt{)} v \cdot 1 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 \\[6pt] \mathrm{E}f_{14} & = & u v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)} & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} & + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} \\[6pt] \mathrm{D}f_{14} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} \\[6pt] \mathrm{d}f_{14} & = & u v \cdot 0 & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} \\[6pt] \mathrm{r}f_{14} & = & u v \cdot \mathrm{d}u ~ \mathrm{d}v & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\)

Appendix 5. Various Definitions of the Tangent Vector

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• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Appendices • References • Document History •

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