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# Differential Propositional Calculus • Appendices

Author: Jon Awbrey

## Appendices

### Appendix 1. Propositional Forms and Differential Expansions

#### Table A1. Propositional Forms on Two Variables

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

#### Table A2. Propositional Forms on Two Variables

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

#### Table A3. Ef Expanded Over Differential Features

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

#### Table A4. Df Expanded Over Differential Features

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

#### Table A5. Ef Expanded Over Ordinary Features

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

#### Table A6. Df Expanded Over Ordinary Features

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

### Appendix 2. Differential Forms

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

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

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

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:

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

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

 ${\displaystyle \{~1,~\mathrm {d} x,~\mathrm {d} y,~\mathrm {d} x~\mathrm {d} y~\}.}$

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

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

#### Table A8. Differential Forms Expanded on an Algebraic Basis

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

#### Table A9. Tangent Proposition as Pointwise Linear Approximation

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

#### Table A10. Taylor Series Expansion Df = df + d²f

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

#### Table A11. Partial Differentials and Relative Differentials

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