User:Jon Awbrey/APPENDICES

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Contents

Appendices (Version 1)

Appendix 1-A. Operator Maps for the Disjunction “f”

Table A1. Computation of “εf”

Table A2. Computation of “Ef”

Table A3. Computation of “Df” (1)

Table A4. Computation of “Df” (2)

Table A5. Computation of “df”

Table A6. Computation of “rf”

Table A7. Computation Summary for Disjunction


\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!

\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f
& = & u \!\cdot\! 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
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \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
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \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
& = & u \!\cdot\! 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
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
\end{array}


Appendix 1-B. Operator Maps for the Equality “g”

Table B1. Computation of “εg”

Table B2. Computation of “Eg”

Table B3. Computation of “Dg” (1)

Table B4. Computation of “Dg” (2)

Table B5. Computation of “dg”

Table B6. Computation of “rg”

Table B7. Computation Summary for Equality


\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!

\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon g
& = & u \!\cdot\! 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}g
& = & u \!\cdot\! 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}g
& = & u \!\cdot\! 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}g
& = & u \!\cdot\! 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}g
& = & u \!\cdot\! v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}


Appendix 2

Table C9. Tangent Proposition as Pointwise Linear Approximation


\text{Table C9.} ~~ \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
\\
\texttt{~} y \texttt{~} \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
\\
\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}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}
\texttt{~} y \texttt{~} \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
\\
\texttt{~} y \texttt{~} \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 C10. Taylor Series Expansion Df = df + d2f


\text{Table C10.} ~~ \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 & + &
\mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \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}\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}\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}\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{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \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\!


Table C11. Partial Differentials and Relative Differentials


\text{Table C11.} ~~ \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\! (~)\! 0\! 0\! 0\! 0\! 0\!

\begin{matrix}
f_{1} \\
f_{2} \\
f_{4} \\
f_{8} \\
\end{matrix}

\begin{matrix}
(x) & (y) \\
(x) &  y  \\
 x  & (y) \\
 x  &  y  \\
\end{matrix}

\begin{matrix}
(y) \\
 y  \\
(y) \\
 y  \\
\end{matrix}

\begin{matrix}
(x) \\
(x) \\
 x  \\
 x  \\
\end{matrix}

\begin{matrix}
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{3}  \\
f_{12} \\
\end{matrix}

\begin{matrix}
(x) \\
 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}
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{6} \\
f_{9} \\
\end{matrix}

\begin{matrix}
 (x, & y)  \\
((x, & y)) \\
\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}
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{5}  \\
f_{10} \\
\end{matrix}

\begin{matrix}
(y) \\
 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}
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{7}  \\
f_{11} \\
f_{13} \\
f_{14} \\
\end{matrix}

\begin{matrix}
 (x  &  y)  \\
 (x  & (y)) \\
((x) &  y)  \\
((x) & (y)) \\
\end{matrix}

\begin{matrix}
 y  \\
(y) \\
 y  \\
(y) \\
\end{matrix}

\begin{matrix}
 x  \\
 x  \\
(x) \\
(x) \\
\end{matrix}

\begin{matrix}
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

f_{15}\! ((~))\! 0\! 0\! 0\! 0\! 0\!


Table C12. Detail of Calculation for Df = Ef + f


\text{Table C12.} ~~ \text{Detail of Calculation for}~ {\mathrm{D}f = \mathrm{E}f + f}\!
 

\begin{array}{cr}
  & \mathrm{E}f|_{\mathrm{d}x\ \mathrm{d}y} \\
+ &           f|_{\mathrm{d}x\ \mathrm{d}y} \\
= & \mathrm{D}f|_{\mathrm{d}x\ \mathrm{d}y}
\end{array}

\begin{array}{cr}
  & \mathrm{E}f|_{\mathrm{d}x\ (\mathrm{d}y)} \\
+ &           f|_{\mathrm{d}x\ (\mathrm{d}y)} \\
= & \mathrm{D}f|_{\mathrm{d}x\ (\mathrm{d}y)}
\end{array}

\begin{array}{cr}
  & \mathrm{E}f|_{(\mathrm{d}x)\ \mathrm{d}y} \\
+ &           f|_{(\mathrm{d}x)\ \mathrm{d}y} \\
= & \mathrm{D}f|_{(\mathrm{d}x)\ \mathrm{d}y}
\end{array}

\begin{array}{cr}
  & \mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)} \\
+ &           f|_{(\mathrm{d}x)(\mathrm{d}y)} \\
= & \mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}
\end{array}

f_{0}\! 0 + 0 = 0\! 0 + 0 = 0\! 0 + 0 = 0\! 0 + 0 = 0\!
f_{1}\!
f_{2}\!
f_{4}\!
f_{8}\!

\begin{smallmatrix}
  &   x  &  y   & \mathrm{d}x & \mathrm{d}y \\
+ &  (x) & (y)  & \mathrm{d}x & \mathrm{d}y \\
= & ((x, &  y)) & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  x  & (y) & \mathrm{d}x & \mathrm{d}y \\
+ & (x) &  y  & \mathrm{d}x & \mathrm{d}y \\
= & (x, &  y) & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}
  & (x) &  y  & \mathrm{d}x & \mathrm{d}y \\
+ &  x  & (y) & \mathrm{d}x & \mathrm{d}y \\
= & (x, &  y) & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  (x) & (y)  & \mathrm{d}x & \mathrm{d}y \\
+ &   x  &  y   & \mathrm{d}x & \mathrm{d}y \\
= & ((x, &  y)) & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  x  & (y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x) & (y) & \mathrm{d}x & (\mathrm{d}y) \\
= &     & (y) & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  &  x  &  y  & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x) &  y  & \mathrm{d}x & (\mathrm{d}y) \\
= &     &  y  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x) & (y) & \mathrm{d}x & (\mathrm{d}y) \\
+ &  x  & (y) & \mathrm{d}x & (\mathrm{d}y) \\
= &     & (y) & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x) &  y  & \mathrm{d}x & (\mathrm{d}y) \\
+ &  x  &  y  & \mathrm{d}x & (\mathrm{d}y) \\
= &     &  y  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x) &  y  & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x) & (y) & (\mathrm{d}x) & \mathrm{d}y \\
= & (x) &     & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & (x) & (y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x) &  y  & (\mathrm{d}x) & \mathrm{d}y \\
= & (x) &     & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  x  &  y  & (\mathrm{d}x) & \mathrm{d}y \\
+ &  x  & (y) & (\mathrm{d}x) & \mathrm{d}y \\
= &  x  &     & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  x  & (y) & (\mathrm{d}x) & \mathrm{d}y \\
+ &  x  &  y  & (\mathrm{d}x) & \mathrm{d}y \\
= &  x  &     & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & (x) (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x) (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= &    0    & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x)\ y  & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x)\ y  & (\mathrm{d}x) & (\mathrm{d}y) \\
= &    0    & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  &  x\ (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ &  x\ (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= &    0    & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  &  x\  y  & (\mathrm{d}x) & (\mathrm{d}y) \\
+ &  x\  y  & (\mathrm{d}x) & (\mathrm{d}y) \\
= &    0    & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

f_{3}\!
f_{12}\!

\begin{smallmatrix}
  &  x  & & \mathrm{d}x & \mathrm{d}y \\
+ & (x) & & \mathrm{d}x & \mathrm{d}y \\
= &  1  & & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & (x) & & \mathrm{d}x & \mathrm{d}y \\
+ &  x  & & \mathrm{d}x & \mathrm{d}y \\
= &  1  & & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  x  & & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x) & & \mathrm{d}x & (\mathrm{d}y) \\
= &  1  & & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x) & & \mathrm{d}x & (\mathrm{d}y) \\
+ &  x  & & \mathrm{d}x & (\mathrm{d}y) \\
= &  1  & & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x) & & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x) & & (\mathrm{d}x) & \mathrm{d}y \\
= &  0  & & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  x  & & (\mathrm{d}x) & \mathrm{d}y \\
+ &  x  & & (\mathrm{d}x) & \mathrm{d}y \\
= &  0  & & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & (x) & & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x) & & (\mathrm{d}x) & (\mathrm{d}y) \\
= &  0  & & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  &  x  & & (\mathrm{d}x) & (\mathrm{d}y) \\
+ &  x  & & (\mathrm{d}x) & (\mathrm{d}y) \\
= &  0  & & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

f_{6}\!
f_{9}\!

\begin{smallmatrix}
  &  (x , y)  & \mathrm{d}x & \mathrm{d}y \\
+ &  (x , y)  & \mathrm{d}x & \mathrm{d}y \\
= &     0     & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & ((x , y)) & \mathrm{d}x & \mathrm{d}y \\
+ & ((x , y)) & \mathrm{d}x & \mathrm{d}y \\
= &     0     & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & ((x , y)) & \mathrm{d}x & (\mathrm{d}y) \\
+ &  (x , y)  & \mathrm{d}x & (\mathrm{d}y) \\
= &     1     & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  &  (x , y)  & \mathrm{d}x & (\mathrm{d}y) \\
+ & ((x , y)) & \mathrm{d}x & (\mathrm{d}y) \\
= &     1     & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & ((x , y)) & (\mathrm{d}x) & \mathrm{d}y \\
+ &  (x , y)  & (\mathrm{d}x) & \mathrm{d}y \\
= &     1     & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  (x , y)  & (\mathrm{d}x) & \mathrm{d}y \\
+ & ((x , y)) & (\mathrm{d}x) & \mathrm{d}y \\
= &     1     & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  (x , y)  & (\mathrm{d}x) & (\mathrm{d}y) \\
+ &  (x , y)  & (\mathrm{d}x) & (\mathrm{d}y) \\
= &     0     & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & ((x , y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & ((x , y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
= &     0     & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

f_{5}\!
f_{10}\!

\begin{smallmatrix}
  & &  y  & \mathrm{d}x & \mathrm{d}y \\
+ & & (y) & \mathrm{d}x & \mathrm{d}y \\
= & &  1  & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & & (y) & \mathrm{d}x & \mathrm{d}y \\
+ & &  y  & \mathrm{d}x & \mathrm{d}y \\
= & &  1  & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}
  & & (y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & & (y) & \mathrm{d}x & (\mathrm{d}y) \\
= & &  0  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & &  y  & \mathrm{d}x & (\mathrm{d}y) \\
+ & &  y  & \mathrm{d}x & (\mathrm{d}y) \\
= & &  0  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & &  y  & (\mathrm{d}x) & \mathrm{d}y \\
+ & & (y) & (\mathrm{d}x) & \mathrm{d}y \\
= & &  1  & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & & (y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & &  y  & (\mathrm{d}x) & \mathrm{d}y \\
= & &  1  & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & & (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & & (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & &  0  & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & &  y  & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & &  y  & (\mathrm{d}x) & (\mathrm{d}y) \\
= & &  0  & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

f_{7}\!
f_{11}\!
f_{13}\!
f_{14}\!

\begin{smallmatrix}
  & ((x) & (y)) & \mathrm{d}x & \mathrm{d}y \\
+ &  (x  &  y)  & \mathrm{d}x & \mathrm{d}y \\
= & ((x, &  y)) & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & ((x) &  y)  & \mathrm{d}x & \mathrm{d}y \\
+ &  (x  & (y)) & \mathrm{d}x & \mathrm{d}y \\
= &  (x, &  y)  & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  (x  & (y)) & \mathrm{d}x & \mathrm{d}y \\
+ & ((x) &  y)  & \mathrm{d}x & \mathrm{d}y \\
= &  (x, &  y)  & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  &  (x  &  y)  & \mathrm{d}x & \mathrm{d}y \\
+ & ((x) & (y)) & \mathrm{d}x & \mathrm{d}y \\
= & ((x, &  y)) & \mathrm{d}x & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & ((x) &  y) & \mathrm{d}x & (\mathrm{d}y) \\
+ &  (x  &  y) & \mathrm{d}x & (\mathrm{d}y) \\
= &      &  y  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & ((x) & (y)) & \mathrm{d}x & (\mathrm{d}y) \\
+ &  (x  & (y)) & \mathrm{d}x & (\mathrm{d}y) \\
= &      & (y)  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  &  (x  &  y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & ((x) &  y) & \mathrm{d}x & (\mathrm{d}y) \\
= &      &  y  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  &  (x  & (y)) & \mathrm{d}x & (\mathrm{d}y) \\
+ & ((x) & (y)) & \mathrm{d}x & (\mathrm{d}y) \\
= &      & (y)  & \mathrm{d}x & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x  & (y)) & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x  &  y)  & (\mathrm{d}x) & \mathrm{d}y \\
= &  x  &      & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & (x  &  y)  & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x  & (y)) & (\mathrm{d}x) & \mathrm{d}y \\
= &  x  &      & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & ((x) & (y)) & (\mathrm{d}x) & \mathrm{d}y \\
+ & ((x) &  y)  & (\mathrm{d}x) & \mathrm{d}y \\
= &  (x) &      & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}
  & ((x) &  y)  & (\mathrm{d}x) & \mathrm{d}y \\
+ & ((x) & (y)) & (\mathrm{d}x) & \mathrm{d}y \\
= &  (x) &      & (\mathrm{d}x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
  & (x\  y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x\  y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= &    0    & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & (x\ (y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x\ (y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
= &    0     & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & ((x)\ y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & ((x)\ y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= &     0    & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

\begin{smallmatrix}
  & ((x) (y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & ((x) (y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
= &     0     & (\mathrm{d}x) & (\mathrm{d}y)
\end{smallmatrix}

f_{15}\! 1 + 1 = 0\! 1 + 1 = 0\! 1 + 1 = 0\! 1 + 1 = 0\!


Appendix 3

Appendix 4

Appendices (Version 2)

InterSciWiki • Differential Propositional Calculus

PlanetMath • Differential Propositional Calculus

Appendix 1

Table A1. Propositional Forms on Two Variables


\text{Table A1.}~~\text{Propositional Forms on Two Variables}
\mathcal{L}_1 \mathcal{L}_2 \mathcal{L}_3 \mathcal{L}_4 \mathcal{L}_5 \mathcal{L}_6
 
x\! :
y\! :
1 1 0 0
1 0 1 0
     

f_{0}\!

f_{1}\!

f_{2}\!

f_{3}\!

f_{4}\!

f_{5}\!

f_{6}\!

f_{7}\!

f_{0000}\!

f_{0001}~\!

f_{0010}\!

f_{0011}\!

f_{0100}\!

f_{0101}\!

f_{0110}\!

f_{0111}\!

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

(~)\!

(x)(y)\!

(x)~y\!

(x)\!

x~(y)\!

(y)\!

(x,~y)\!

(x~y)\!

\operatorname{false}

\operatorname{neither}~ x ~\operatorname{nor}~ y

y ~\operatorname{without}~ x

\operatorname{not}~ x

x ~\operatorname{without}~ y

\operatorname{not}~ y

x ~\operatorname{not~equal~to}~ y

\operatorname{not~both}~ x ~\operatorname{and}~ y

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

f_{8}\!

f_{9}\!

f_{10}\!

f_{11}\!

f_{12}\!

f_{13}\!

f_{14}\!

f_{15}\!

f_{1000}\!

f_{1001}\!

f_{1010}\!

f_{1011}\!

f_{1100}\!

f_{1101}\!

f_{1110}\!

f_{1111}\!

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

x~y\!

((x,~y))\!

y\!

(x~(y))\!

x\!

((x)~y)\!

((x)(y))\!

((~))\!

x ~\operatorname{and}~ y

x ~\operatorname{equal~to}~ y

y\!

\operatorname{not}~ x ~\operatorname{without}~ y

x\!

\operatorname{not}~ y ~\operatorname{without}~ x

x ~\operatorname{or}~ y

\operatorname{true}

x \land y

x = y\!

y\!

x \Rightarrow y

x\!

x \Leftarrow y

x \lor y

1\!


Table A2. Propositional Forms on Two Variables


\text{Table A2.}~~\text{Propositional Forms on Two Variables}
\mathcal{L}_1 \mathcal{L}_2 \mathcal{L}_3 \mathcal{L}_4 \mathcal{L}_5 \mathcal{L}_6
 
x\! :
y\! :
1 1 0 0
1 0 1 0
     

f_{0}\!

f_{0000}\!

0 0 0 0

(~)\!

\operatorname{false}

1\!

f_{1}\!

f_{2}\!

f_{4}\!

f_{8}\!

f_{0001}~\!

f_{0010}\!

f_{0100}\!

f_{1000}\!

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

(x)(y)\!

(x)~y\!

x~(y)\!

x~y\!

\operatorname{neither}~ x ~\operatorname{nor}~ y

y ~\operatorname{without}~ x

x ~\operatorname{without}~ y

x ~\operatorname{and}~ y

\lnot x \land \lnot y

\lnot x \land y

x \land \lnot y

x \land y

f_{3}\!

f_{12}\!

f_{0011}\!

f_{1100}\!

0 0 1 1

1 1 0 0

(x)\!

x\!

\operatorname{not}~ x

x\!

\lnot x

x\!

f_{6}\!

f_{9}\!

f_{0110}\!

f_{1001}\!

0 1 1 0

1 0 0 1

(x,~y)\!

((x,~y))\!

x ~\operatorname{not~equal~to}~ y

x ~\operatorname{equal~to}~ y

x \ne y

x = y\!

f_{5}\!

f_{10}\!

f_{0101}\!

f_{1010}\!

0 1 0 1

1 0 1 0

(y)\!

y\!

\operatorname{not}~ y

y\!

\lnot y

y\!

f_{7}\!

f_{11}\!

f_{13}\!

f_{14}\!

f_{0111}\!

f_{1011}\!

f_{1101}\!

f_{1110}\!

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0

(x~y)\!

(x~(y))\!

((x)~y)\!

((x)(y))\!

\operatorname{not~both}~ x ~\operatorname{and}~ y

\operatorname{not}~ x ~\operatorname{without}~ y

\operatorname{not}~ y ~\operatorname{without}~ x

x ~\operatorname{or}~ y

\lnot x \lor \lnot y

x \Rightarrow y

x \Leftarrow y

x \lor y

f_{15}\!

f_{1111}\!

1 1 1 1

((~))\!

\operatorname{true}

1\!


Table A3. Ef Expanded Over Differential Features


\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}
  f\!
\operatorname{T}_{11}f
\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}
\operatorname{T}_{10}f
\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}
\operatorname{T}_{01}f
\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}
\operatorname{T}_{00}f
\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}
f_{0}\! (~)\! (~)\! (~)\! (~)\! (~)\!
f_{1}\!
f_{2}\!
f_{4}\!
f_{8}\!
(x)(y)\!
(x)\ y\!
x\ (y)\!
x\ y\!
x\ y\!
x\ (y)\!
(x)\ y\!
(x)(y)\!
x\ (y)\!
x\ y\!
(x)(y)\!
(x)\ y\!
(x)\ y\!
(x)(y)\!
x\ y\!
x\ (y)\!
(x)(y)\!
(x)\ y\!
x\ (y)\!
x\ y\!
f_{3}\!
f_{12}\!
(x)\!
x\!
x\!
(x)\!
x\!
(x)\!
(x)\!
x\!
(x)\!
x\!
f_{6}\!
f_{9}\!
(x,\ y)\!
((x,\ y))\!
(x,\ y)\!
((x,\ y))\!
((x,\ y))\!
(x,\ y)\!
((x,\ y))\!
(x,\ y)\!
(x,\ y)\!
((x,\ y))\!
f_{5}\!
f_{10}\!
(y)\!
y\!
y\!
(y)\!
(y)\!
y\!
y\!
(y)\!
(y)\!
y\!
f_{7}\!
f_{11}\!
f_{13}\!
f_{14}\!
(x\ y)\!
(x\ (y))\!
((x)\ y)\!
((x)(y))\!
((x)(y))\!
((x)\ y)\!
(x\ (y))\!
(x\ y)\!
((x)\ y)\!
((x)(y))\!
(x\ y)\!
(x\ (y))\!
(x\ (y))\!
(x\ y)\!
((x)(y))\!
((x)\ y)\!
(x\ y)\!
(x\ (y))\!
((x)\ y)\!
((x)(y))\!
f_{15}\! ((~))\! ((~))\! ((~))\! ((~))\! ((~))\!
Fixed Point Total : 4\! 4\! 4\! 16\!


Table A4. Df Expanded Over Differential Features


\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}
  f\! \operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y} \operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)} \operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y} \operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}
f_{0}\! (~)\! (~)\! (~)\! (~)\! (~)\!
f_{1}\!
f_{2}\!
f_{4}\!
f_{8}\!
(x)(y)\!
(x)\ y\!
x\ (y)\!
x\ y\!
((x,\ y))\!
(x,\ y)\!
(x,\ y)\!
((x,\ y))\!
(y)\!
y\!
(y)\!
y\!
(x)\!
(x)\!
x\!
x\!
(~)\!
(~)\!
(~)\!
(~)\!
f_{3}\!
f_{12}\!
(x)\!
x\!
((~))\!
((~))\!
((~))\!
((~))\!
(~)\!
(~)\!
(~)\!
(~)\!
f_{6}\!
f_{9}\!
(x,\ y)\!
((x,\ y))\!
(~)\!
(~)\!
((~))\!
((~))\!
((~))\!
((~))\!
(~)\!
(~)\!
f_{5}\!
f_{10}\!
(y)\!
y\!
((~))\!
((~))\!
(~)\!
(~)\!
((~))\!
((~))\!
(~)\!
(~)\!
f_{7}\!
f_{11}\!
f_{13}\!
f_{14}\!
(x\ y)\!
(x\ (y))\!
((x)\ y)\!
((x)(y))\!
((x,\ y))\!
(x,\ y)\!
(x,\ y)\!
((x,\ y))\!
y\!
(y)\!
y\!
(y)\!
x\!
x\!
(x)\!
(x)\!
(~)\!
(~)\!
(~)\!
(~)\!
f_{15}\! ((~))\! (~)\! (~)\! (~)\! (~)\!


Table A5. Ef Expanded Over Ordinary Features


\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}
  f\! \operatorname{E}f|_{xy} \operatorname{E}f|_{x(y)} \operatorname{E}f|_{(x)y} \operatorname{E}f|_{(x)(y)}
f_{0}\! (~)\! (~)\! (~)\! (~)\! (~)\!
f_{1}\!
f_{2}\!
f_{4}\!
f_{8}\!
(x)(y)\!
(x)\ y\!
x\ (y)\!
x\ y\!
\operatorname{d}x\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
(\operatorname{d}x)\ \operatorname{d}y
(\operatorname{d}x) (\operatorname{d}y)
\operatorname{d}x\ (\operatorname{d}y)
\operatorname{d}x\ \operatorname{d}y
(\operatorname{d}x) (\operatorname{d}y)
(\operatorname{d}x)\ \operatorname{d}y
(\operatorname{d}x)\ \operatorname{d}y
(\operatorname{d}x) (\operatorname{d}y)
\operatorname{d}x\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
(\operatorname{d}x) (\operatorname{d}y)
(\operatorname{d}x)\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
\operatorname{d}x\ \operatorname{d}y
f_{3}\!
f_{12}\!
(x)\!
x\!
\operatorname{d}x
(\operatorname{d}x)
\operatorname{d}x
(\operatorname{d}x)
(\operatorname{d}x)
\operatorname{d}x
(\operatorname{d}x)
\operatorname{d}x
f_{6}\!
f_{9}\!
(x,\ y)\!
((x,\ y))\!
(\operatorname{d}x,\ \operatorname{d}y)
((\operatorname{d}x,\ \operatorname{d}y))
((\operatorname{d}x,\ \operatorname{d}y))
(\operatorname{d}x,\ \operatorname{d}y)
((\operatorname{d}x,\ \operatorname{d}y))
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
((\operatorname{d}x,\ \operatorname{d}y))
f_{5}\!
f_{10}\!
(y)\!
y\!
\operatorname{d}y
(\operatorname{d}y)
(\operatorname{d}y)
\operatorname{d}y
\operatorname{d}y
(\operatorname{d}y)
(\operatorname{d}y)
\operatorname{d}y
f_{7}\!
f_{11}\!
f_{13}\!
f_{14}\!
(x\ y)\!
(x\ (y))\!
((x)\ y)\!
((x)(y))\!
((\operatorname{d}x)(\operatorname{d}y))
((\operatorname{d}x)\ \operatorname{d}y)
(\operatorname{d}x\ (\operatorname{d}y))
(\operatorname{d}x\ \operatorname{d}y)
((\operatorname{d}x)\ \operatorname{d}y)
((\operatorname{d}x)(\operatorname{d}y))
(\operatorname{d}x\ \operatorname{d}y)
(\operatorname{d}x\ (\operatorname{d}y))
(\operatorname{d}x\ (\operatorname{d}y))
(\operatorname{d}x\ \operatorname{d}y)
((\operatorname{d}x)(\operatorname{d}y))
((\operatorname{d}x)\ \operatorname{d}y)
(\operatorname{d}x\ \operatorname{d}y)
(\operatorname{d}x\ (\operatorname{d}y))
((\operatorname{d}x)\ \operatorname{d}y)
((\operatorname{d}x)(\operatorname{d}y))
f_{15}\! ((~))\! ((~))\! ((~))\! ((~))\! ((~))\!


Table A6. Df Expanded Over Ordinary Features


\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}
  f\! \operatorname{D}f|_{xy} \operatorname{D}f|_{x(y)} \operatorname{D}f|_{(x)y} \operatorname{D}f|_{(x)(y)}
f_{0}\! (~)\! (~)\! (~)\! (~)\! (~)\!
f_{1}\!
f_{2}\!
f_{4}\!
f_{8}\!
(x)(y)\!
(x)\ y\!
x\ (y)\!
x\ y\!
\operatorname{d}x\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
(\operatorname{d}x)\ \operatorname{d}y
((\operatorname{d}x)(\operatorname{d}y))
\operatorname{d}x\ (\operatorname{d}y)
\operatorname{d}x\ \operatorname{d}y
((\operatorname{d}x)(\operatorname{d}y))
(\operatorname{d}x)\ \operatorname{d}y
(\operatorname{d}x)\ \operatorname{d}y
((\operatorname{d}x)(\operatorname{d}y))
\operatorname{d}x\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
((\operatorname{d}x)(\operatorname{d}y))
(\operatorname{d}x)\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
\operatorname{d}x\ \operatorname{d}y
f_{3}\!
f_{12}\!
(x)\!
x\!
\operatorname{d}x
\operatorname{d}x
\operatorname{d}x
\operatorname{d}x
\operatorname{d}x
\operatorname{d}x
\operatorname{d}x
\operatorname{d}x
f_{6}\!
f_{9}\!
(x,\ y)\!
((x,\ y))\!
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
(\operatorname{d}x,\ \operatorname{d}y)
f_{5}\!
f_{10}\!
(y)\!
y\!
\operatorname{d}y
\operatorname{d}y
\operatorname{d}y
\operatorname{d}y
\operatorname{d}y
\operatorname{d}y
\operatorname{d}y
\operatorname{d}y
f_{7}\!
f_{11}\!
f_{13}\!
f_{14}\!
(x\ y)\!
(x\ (y))\!
((x)\ y)\!
((x)(y))\!
((\operatorname{d}x)(\operatorname{d}y))
(\operatorname{d}x)\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
\operatorname{d}x\ \operatorname{d}y
(\operatorname{d}x)\ \operatorname{d}y
((\operatorname{d}x)(\operatorname{d}y))
\operatorname{d}x\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
\operatorname{d}x\ (\operatorname{d}y)
\operatorname{d}x\ \operatorname{d}y
((\operatorname{d}x)(\operatorname{d}y))
(\operatorname{d}x)\ \operatorname{d}y
\operatorname{d}x\ \operatorname{d}y
\operatorname{d}x\ (\operatorname{d}y)
(\operatorname{d}x)\ \operatorname{d}y
((\operatorname{d}x)(\operatorname{d}y))
f_{15}\! ((~))\! (~)\! (~)\! (~)\! (~)\!


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}\! (~)\! 0\! 0\!

\begin{smallmatrix}
f_{1} \\
f_{2} \\
f_{4} \\
f_{8} \\
\end{smallmatrix}

\begin{smallmatrix}
(x) & (y) \\
(x) &  y  \\
 x  & (y) \\
 x  &  y  \\
\end{smallmatrix}

\begin{smallmatrix}
    (y)  &  \mathrm{d}x ~(\mathrm{d}y) & + &
 (x)     & (\mathrm{d}x)~ \mathrm{d}y  & + &
((x, y)) &  \mathrm{d}x ~ \mathrm{d}y  \\
     y   &  \mathrm{d}x ~(\mathrm{d}y) & + &
 (x)     & (\mathrm{d}x)~ \mathrm{d}y  & + &
 (x, y)  &  \mathrm{d}x ~ \mathrm{d}y  \\
    (y)  &  \mathrm{d}x ~(\mathrm{d}y) & + &
  x      & (\mathrm{d}x)~ \mathrm{d}y  & + &
 (x, y)  &  \mathrm{d}x ~ \mathrm{d}y  \\
     y   &  \mathrm{d}x ~(\mathrm{d}y) & + &
  x      & (\mathrm{d}x)~ \mathrm{d}y  & + &
((x, y)) &  \mathrm{d}x ~ \mathrm{d}y  \\
\end{smallmatrix}

\begin{smallmatrix}
(y) & \partial x & + & (x) & \partial y \\
 y  & \partial x & + & (x) & \partial y \\
(y) & \partial x & + &  x  & \partial y \\
 y  & \partial x & + &  x  & \partial y \\
\end{smallmatrix}

\begin{smallmatrix}
f_{3}  \\
f_{12} \\
\end{smallmatrix}

\begin{smallmatrix}
(x) \\
 x  \\
\end{smallmatrix}

\begin{smallmatrix}
\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{smallmatrix}

\begin{smallmatrix}
\partial x \\
\partial x \\
\end{smallmatrix}

\begin{smallmatrix}
f_{6} \\
f_{9} \\
\end{smallmatrix}

\begin{smallmatrix}
 (x, & y)  \\
((x, & y)) \\
\end{smallmatrix}

\begin{smallmatrix}
 \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{smallmatrix}

\begin{smallmatrix}
\partial x & + & \partial y \\
\partial x & + & \partial y \\
\end{smallmatrix}

\begin{smallmatrix}
f_{5}  \\
f_{10} \\
\end{smallmatrix}

\begin{smallmatrix}
(y) \\
 y  \\
\end{smallmatrix}

\begin{smallmatrix}
(\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{smallmatrix}

\begin{smallmatrix}
\partial y \\
\partial y \\
\end{smallmatrix}

\begin{smallmatrix}
f_{7}  \\
f_{11} \\
f_{13} \\
f_{14} \\
\end{smallmatrix}

\begin{smallmatrix}
 (x  &  y)  \\
 (x  & (y)) \\
((x) &  y)  \\
((x) & (y)) \\
\end{smallmatrix}

\begin{smallmatrix}
     y   &  \mathrm{d}x ~(\mathrm{d}y) & + &
  x      & (\mathrm{d}x)~ \mathrm{d}y  & + &
((x, y)) &  \mathrm{d}x ~ \mathrm{d}y  \\
    (y)  &  \mathrm{d}x ~(\mathrm{d}y) & + &
  x      & (\mathrm{d}x)~ \mathrm{d}y  & + &
 (x, y)  &  \mathrm{d}x ~ \mathrm{d}y  \\
     y   &  \mathrm{d}x ~(\mathrm{d}y) & + &
 (x)     & (\mathrm{d}x)~ \mathrm{d}y  & + &
 (x, y)  &  \mathrm{d}x ~ \mathrm{d}y  \\
    (y)  &  \mathrm{d}x ~(\mathrm{d}y) & + &
 (x)     & (\mathrm{d}x)~ \mathrm{d}y  & + &
((x, y)) &  \mathrm{d}x ~ \mathrm{d}y  \\
\end{smallmatrix}

\begin{smallmatrix}
 y  & \partial x & + &  x  & \partial y \\
(y) & \partial x & + &  x  & \partial y \\
 y  & \partial x & + & (x) & \partial y \\
(y) & \partial x & + & (x) & \partial y \\
\end{smallmatrix}

f_{15}\! ((~))\! 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}\! (~)\! 0\! 0\!

\begin{smallmatrix}
f_{1} \\
f_{2} \\
f_{4} \\
f_{8} \\
\end{smallmatrix}

\begin{smallmatrix}
(x) & (y) \\
(x) &  y  \\
 x  & (y) \\
 x  &  y  \\
\end{smallmatrix}

\begin{smallmatrix}
(y) & \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 \\
(y) & \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{smallmatrix}

\begin{smallmatrix}
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y \\
\end{smallmatrix}

\begin{smallmatrix}
f_{3}  \\
f_{12} \\
\end{smallmatrix}

\begin{smallmatrix}
(x) \\
 x  \\
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
\mathrm{d}x \\
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
\mathrm{d}x \\
\end{smallmatrix}

\begin{smallmatrix}
f_{6} \\
f_{9} \\
\end{smallmatrix}

\begin{smallmatrix}
 (x, & y)  \\
((x, & y)) \\
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x & + & \mathrm{d}y \\
\mathrm{d}x & + & \mathrm{d}y \\
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x & + & \mathrm{d}y \\
\mathrm{d}x & + & \mathrm{d}y \\
\end{smallmatrix}

\begin{smallmatrix}
f_{5}  \\
f_{10} \\
\end{smallmatrix}

\begin{smallmatrix}
(y) \\
 y  \\
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}y \\
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}y \\
\end{smallmatrix}

\begin{smallmatrix}
f_{7}  \\
f_{11} \\
f_{13} \\
f_{14} \\
\end{smallmatrix}

\begin{smallmatrix}
 (x  &  y)  \\
 (x  & (y)) \\
((x) &  y)  \\
((x) & (y)) \\
\end{smallmatrix}

\begin{smallmatrix}
 y  & \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 \\
 y  & \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{smallmatrix}

\begin{smallmatrix}
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
\end{smallmatrix}

f_{15}\! ((~))\! 0\! 0\!


Appendix 3

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
\\
\texttt{~} y \texttt{~} \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
\\
\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}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}
\texttt{~} y \texttt{~} \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
\\
\texttt{~} y \texttt{~} \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}\!
 

\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 \;(y)}\! \mathrm{d}f|_{(x)\; y}\! \mathrm{d}f|_{(x)(y)}\!
f_0\! 0\! 0\! 0\! 0\! 0\! 0\!
\begin{smallmatrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{smallmatrix}\!

\begin{smallmatrix}
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
\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{smallmatrix}

\begin{smallmatrix}
0 \\
\mathrm{d}x \\
\mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
0 \\
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y \\
0 \\
\mathrm{d}x
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}y \\
\mathrm{d}x \\
0
\end{smallmatrix}

\begin{smallmatrix}f_{3}\\f_{12}\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
\mathrm{d}x
\end{smallmatrix}

\begin{smallmatrix}
0 \\
0
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
\mathrm{d}x
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
\mathrm{d}x
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
\mathrm{d}x
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
\mathrm{d}x
\end{smallmatrix}

\begin{smallmatrix}f_{6}\\f_{9}\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
0 \\
0
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}f_{5}\\f_{10}\end{smallmatrix}\!

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}
0 \\
0
\end{smallmatrix}\!

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}y
\end{smallmatrix}\!

\begin{smallmatrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{smallmatrix}

\begin{smallmatrix}
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
\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{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}y \\
\mathrm{d}x \\
0
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y \\
0 \\
\mathrm{d}x
\end{smallmatrix}

\begin{smallmatrix}
\mathrm{d}x \\
0 \\
\mathrm{d}x + \mathrm{d}y \\
\mathrm{d}y
\end{smallmatrix}

\begin{smallmatrix}
0 \\
\mathrm{d}x \\
\mathrm{d}y \\
\mathrm{d}x + \mathrm{d}y
\end{smallmatrix}

f_{15}\! 0\! 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\! (~)\! 0\! 0\! 0\! 0\! 0\!

\begin{matrix}
f_{1} \\
f_{2} \\
f_{4} \\
f_{8} \\
\end{matrix}

\begin{matrix}
(x) & (y) \\
(x) &  y  \\
 x  & (y) \\
 x  &  y  \\
\end{matrix}

\begin{matrix}
(y) \\
 y  \\
(y) \\
 y  \\
\end{matrix}

\begin{matrix}
(x) \\
(x) \\
 x  \\
 x  \\
\end{matrix}

\begin{matrix}
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{3}  \\
f_{12} \\
\end{matrix}

\begin{matrix}
(x) \\
 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}
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{6} \\
f_{9} \\
\end{matrix}

\begin{matrix}
 (x, & y)  \\
((x, & y)) \\
\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}
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{5}  \\
f_{10} \\
\end{matrix}

\begin{matrix}
(y) \\
 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}
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
\end{matrix}

\begin{matrix}
f_{7}  \\
f_{11} \\
f_{13} \\
f_{14} \\
\end{matrix}

\begin{matrix}
 (x  &  y)  \\
 (x  & (y)) \\
((x) &  y)  \\
((x) & (y)) \\
\end{matrix}

\begin{matrix}
 y  \\
(y) \\
 y  \\
(y) \\
\end{matrix}

\begin{matrix}
 x  \\
 x  \\
(x) \\
(x) \\
\end{matrix}

\begin{matrix}
 y  & \mathrm{d}x & + &  x  & \mathrm{d}y \\
(y) & \mathrm{d}x & + &  x  & \mathrm{d}y \\
 y  & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

\begin{matrix}
~ \\
~ \\
~ \\
~ \\
\end{matrix}

f_{15}\! ((~))\! 0\! 0\! 0\! 0\! 0\!


Appendix 4

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}

\begin{smallmatrix}f_{0}\end{smallmatrix}

\begin{smallmatrix}0\end{smallmatrix}

\begin{smallmatrix}0 & + & 0 & = & 0\end{smallmatrix}

\begin{smallmatrix}0 & + & 0 & = & 0\end{smallmatrix}

\begin{smallmatrix}0 & + & 0 & = & 0\end{smallmatrix}

\begin{smallmatrix}0 & + & 0 & = & 0\end{smallmatrix}

\begin{smallmatrix}
f_{1}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{~(} x \texttt{)(} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{2}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{~(} x \texttt{)~} y \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{4}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{~~} x \texttt{~(} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{8}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{~~} x \texttt{~~} y \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{3}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{(} x \texttt{)}
\end{smallmatrix}\!

\begin{smallmatrix}
~ & x
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & x
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{12}
\end{smallmatrix}

\begin{smallmatrix}
x
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & x
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & x
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & x
\\[4pt]
+ & x
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
~ & x
\\[4pt]
+ & x
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{6}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\end{smallmatrix}\!

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{9}
\end{smallmatrix}\!

\begin{smallmatrix}
\texttt{((} x \texttt{,~} y \texttt{))}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{5}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{(} y \texttt{)}
\end{smallmatrix}\!

\begin{smallmatrix}
~ & y
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
~ & y
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{10}
\end{smallmatrix}

\begin{smallmatrix}
y
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & y
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & y
\\[4pt]
+ & y
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & y
\\[4pt]
= & 1
\end{smallmatrix}

\begin{smallmatrix}
~ & y
\\[4pt]
+ & y
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{7}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{~(} x \texttt{~~} y \texttt{)~}
\end{smallmatrix}\!

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{11}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{~(} x \texttt{~(} y \texttt{))}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{smallmatrix}\!

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{13}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{((} x \texttt{)~} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}
f_{14}
\end{smallmatrix}

\begin{smallmatrix}
\texttt{((} x \texttt{)(} y \texttt{))}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{smallmatrix}

\begin{smallmatrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & 0
\end{smallmatrix}

\begin{smallmatrix}f_{15}\end{smallmatrix}

\begin{smallmatrix}1\end{smallmatrix}

\begin{smallmatrix}1 & + & 1 & = & 0\end{smallmatrix}

\begin{smallmatrix}1 & + & 1 & = & 0\end{smallmatrix}

\begin{smallmatrix}1 & + & 1 & = & 0\end{smallmatrix}

\begin{smallmatrix}1 & + & 1 & = & 0\end{smallmatrix}


Appendices (Version 3)

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{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{(} x \texttt{)~~~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~~~(} y \texttt{)}
\\
\texttt{(} x \texttt{,~} y \texttt{)}
\\
\texttt{(} x \texttt{~~} 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}
\texttt{~~} x \texttt{~~} y \texttt{~~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\\
\texttt{~~~~~} y \texttt{~~}
\\
\texttt{~(} x \texttt{~(} y \texttt{))}
\\
\texttt{~~} x \texttt{~~~~~}
\\
\texttt{((} x \texttt{)~} y \texttt{)~}
\\
\texttt{((} x \texttt{)(} y \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{(~)}\! \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 \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\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{)}
\\
\texttt{~} x \texttt{~}
\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{)}
\\
\texttt{~} y \texttt{~}
\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 \texttt{~~} 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{((~))}\! \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 \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}
\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{(} x \texttt{)~} 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}
f_{3}\\f_{12}
\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{~} x \texttt{~}
\\
\texttt{(} x \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{~} x \texttt{~}
\\
\texttt{(} x \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\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{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}

\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\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{((} x \texttt{)(} 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}
\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{(~} x \texttt{~~} 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 \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}
\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{)}
\\
\texttt{~} y \texttt{~}
\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 \texttt{~~} 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 \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 \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{~}
\\
\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{)}
\\
\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{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\end{matrix}

\begin{matrix}
f_{3}\\f_{12}
\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~}
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~}
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~}
\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{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{~} \mathrm{d}y \texttt{~}
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}y \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{~} \mathrm{d}y \texttt{~}
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}y \texttt{~}
\end{matrix}

\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\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{((} \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 \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{))}
\\
\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{~)}
\\
\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{))}
\\
\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 \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 \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{~}
\\
\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{))}
\\
\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{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\end{matrix}

\begin{matrix}f_{3}\\f_{12}\end{matrix}

\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\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{)}
\\
\texttt{~} y \texttt{~}
\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 \texttt{~~} 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 \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{))}
\\
\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{~}
\\
\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{)}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\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{(~)}\! 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 \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\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
\\
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\\
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\end{matrix}

\begin{matrix}f_{3}\\f_{12}\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\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{)}
\\
\texttt{~} y \texttt{~}
\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 \texttt{~~} 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}
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\\
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\end{matrix}

f_{15}\! \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{(~)}\! 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 \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{(} y \texttt{)}~\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
\\
\texttt{(} y \texttt{)}~\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}
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\end{matrix}

\begin{matrix}f_{3}\\f_{12}\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\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{)}
\\
\texttt{~} y \texttt{~}
\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 \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{~}~\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
\\
\texttt{~} y \texttt{~}~\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}
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\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{((~))}\! 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
\\
\texttt{~} y \texttt{~} \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
\\
\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}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}
\texttt{~} y \texttt{~} \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
\\
\texttt{~} y \texttt{~} \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 & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \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}
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \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}
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \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}
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \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}
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \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{(~)}\! 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 \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\\
\texttt{~} x \texttt{~}
\end{matrix}

\begin{matrix}
\texttt{(} y \texttt{)} \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
\\
\texttt{(} y \texttt{)} \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}
\begin{matrix}f_{3}\\f_{12}\end{matrix}

\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\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{)}
\\
\texttt{~} y \texttt{~}
\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 \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{~}
\\
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{~} x \texttt{~}
\\
\texttt{~} x \texttt{~}
\\
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\end{matrix}

\begin{matrix}
\texttt{~} y \texttt{~} \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
\\
\texttt{~} y \texttt{~} \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{((~))}\! 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}
~ & \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{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}

\begin{matrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \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 \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{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & 0
\end{matrix}

f_{4}\!

\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{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}

\begin{matrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & 0
\end{matrix}

f_{8}\!

\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{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[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 \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{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} 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{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}

\begin{matrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\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 \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}

\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \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 \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{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}

\begin{matrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \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}(u, v)
\\[4pt]
& = && uv
\\[4pt]
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + &  uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + &  uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + &  uv \cdot \mathrm{d}u ~ \mathrm{d}v
\\[20pt]
\boldsymbol\varepsilon f_{8}
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
\\[4pt]
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\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}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[4pt]
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
\\[4pt]
& = & \texttt{ } u \texttt{  } v \texttt{ } \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{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
\\[4pt]
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \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{(} u \texttt{) } v \texttt{ } \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}(u + \mathrm{d}u, v + \mathrm{d}v)
& + &  f_{8}(u, v)
\\[4pt]
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
& + &  uv
\\[20pt]
\mathrm{D}f_{8}
& = && 0
& + &  0
& + &  0
& + &  0
\\[4pt]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \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{~}
& + &  0
& + &  0
\\[4pt]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
& + &  0
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + &  0
\\[4pt]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \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{~}
\\[20pt]
\mathrm{D}f_{8}
& = && \texttt{ } u \texttt{  } v \texttt{ } \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{~}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \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 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}(u, v)
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[6pt]
& = & uv
& + & \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 \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + &   u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{E}f_{8}
& = & u \,\cdot\, 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{)} \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}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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}(u, v) = uv\!

\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{8}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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}(u, v)
\\[4pt]
& = && \texttt{((} u \texttt{,~} v \texttt{))}
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{9}(1, 1)
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)
\\[4pt]
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{9}
& = && \texttt{ } u \texttt{  } v \texttt{ } \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]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \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]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \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]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \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{~}
\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}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\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{) }
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\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}
& = && \texttt{ } u \texttt{  } v \texttt{ } \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
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \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{~}
& + &  0
\\[4pt]
&& + & 0
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \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{)}
& + &  0
\\[4pt]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \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{~}
\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}(u + \mathrm{d}u, v + \mathrm{d}v)
& + &  f_{9}(u, v)
\\[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]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \!\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{~}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\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]
&& + & \texttt{ } u \texttt{  } v \texttt{ } \!\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{)}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\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
& + &  0
& + &  0
& + &  0
\\[20pt]
\mathrm{D}f_{9}
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\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{)}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!

\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{9}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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 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}(u, v)
\\[4pt]
& = && \texttt{((} u \texttt{)(} v \texttt{))}
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{14}(1, 1)
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ }
& + &  \texttt{ } u \texttt{ (} v \texttt{)}
& + &  \texttt{(} u \texttt{) } v \texttt{ }
& + &  0
\\[20pt]
\boldsymbol\varepsilon f_{14}
& = && \texttt{ } u \texttt{  } v \texttt{ } \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{)}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + &  0
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \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{~}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
& + &  0
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \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{)}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + &  0
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \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{~}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
& + &  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}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\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{))}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\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}
& = && \texttt{ } u \texttt{  } v \texttt{ } \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{)}
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + &  0
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \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{~}
& + &  0
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
\\[4pt]
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + &  0
& + &  \texttt{(} u \texttt{) } v \texttt{ } \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
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \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{~}
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\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}(u + \mathrm{d}u, v + \mathrm{d}v)
& + &  f_{14}(u, v)
\\[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 \texttt{~~}
\\[4pt]
&& + & 0
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + &  0
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
\\[4pt]
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
& + &  0
& + &  0
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
\\[20pt]
\mathrm{D}f_{14}
& = && uv \!\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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[20pt]
\mathrm{r}f_{14}
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!

\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{14}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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}
& = & uv \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 4. Source Materials

Appendix 5. Various Definitions of the Tangent Vector