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Directory:Jon Awbrey/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 21:21, 4 October 2017

92,533 bytes added ,  21:21, 4 October 2017
fix more \underlines
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| align="center" | <math>\text{Map}\!</math>
 
| align="center" | <math>\text{Map}\!</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Trope}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Enlargement}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Difference}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Differential}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}\!</math>
 
\end{array}\!</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Remainder}\\\text{operator}\end{matrix}\!</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Radius}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Chord}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tangent}\\\text{functor}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math>
 
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math>
 
|-
 
|-
−
| align="center" | <math>\underline\text{Operand}\!</math>
+
| align="center" | <math>\underline{\text{Operand}}\!</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Trope}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Enlargement}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Difference}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Differential}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}\!</math>
 
\end{array}\!</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Remainder}\\\text{operator}\end{matrix}\!</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Radius}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Chord}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tangent}\\\text{functor}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
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<br>
 
<br>
 +
 +
===Appendix 2. Differential Forms===
 +
 +
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.
 +
 +
Table A7 expands the differential forms that result over a ''logical basis'':
 +
 +
{| align="center" cellpadding="6" style="text-align:center"
 +
|
 +
<math>\{~ \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 ~\}.\!</math>
 +
|}
 +
 +
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:
 +
 +
{| align="center" cellpadding="6" style="text-align:center"
 +
|
 +
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math>
 +
|}
 +
 +
Table A8 expands the differential forms that result over an ''algebraic basis'':
 +
 +
{| align="center" cellpadding="6" style="text-align:center"
 +
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
 +
|}
 +
 +
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====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| &nbsp;
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
 +
| <math>\mathrm{d}f~\!</math>
 +
|-
 +
| <math>f_{0}\!</math>
 +
| style="border-right:none" | <math>\texttt{(~)}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\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}</math>
 +
|-
 +
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\partial x
 +
\\
 +
\partial x
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\begin{matrix}
 +
\partial x & + & \partial y
 +
\\
 +
\partial x & + & \partial y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\begin{matrix}
 +
\partial y
 +
\\
 +
\partial y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\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}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| style="border-right:none" | <math>\texttt{((~))}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A8. Differential Forms Expanded on an Algebraic Basis====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| &nbsp;
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
 +
| <math>\mathrm{d}f~\!</math>
 +
|-
 +
| <math>f_{0}\!</math>
 +
| style="border-right:none" | <math>\texttt{(~)}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\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}</math>
 +
|-
 +
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}x
 +
\\
 +
\mathrm{d}x
 +
\end{matrix}\!</math>
 +
| <math>\begin{matrix}
 +
\mathrm{d}x
 +
\\
 +
\mathrm{d}x
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\\
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\\
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}y
 +
\\
 +
\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y
 +
\\
 +
\mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\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}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| style="border-right:none" | <math>\texttt{((~))}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A9. Tangent Proposition as Pointwise Linear Approximation====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}f =
 +
\\[2pt]
 +
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}^2\!f =
 +
\\[2pt]
 +
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\mathrm{d}f|_{x \, y}</math>
 +
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
 +
|-
 +
| style="border-right:none" | <math>f_0\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\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}</math>
 +
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}\!</math>
 +
| <math>\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}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>f_{15}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" |
 +
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
| <math>\mathrm{d}f|_{x \, y}</math>
 +
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
 +
|-
 +
| style="border-right:none" | <math>f_0\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\begin{matrix}
 +
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\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}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>f_{15}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A11. Partial Differentials and Relative Differentials====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math>
 +
|- style="background:ghostwhite; height:50px"
 +
| &nbsp;
 +
| <math>f\!</math>
 +
| <math>\frac{\partial f}{\partial x}\!</math>
 +
| <math>\frac{\partial f}{\partial y}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}f =
 +
\\[2pt]
 +
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math>
 +
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>\texttt{(~)}\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\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}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\end{matrix}</math>
 +
|
 +
<math>\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}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A12. Detail of Calculation for the Difference Map====
 +
 +
<br>
 +
 +
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math>
 +
|- style="background:ghostwhite"
 +
| style="width:6%" | &nbsp;
 +
| style="width:14%; border-left:1px solid black"  | <math>f\!</math>
 +
| style="width:20%; border-left:4px double black" |
 +
<math>\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}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\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}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\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}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\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}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{0}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{1}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{2}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{4}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{8}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{3}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{(} x \texttt{)}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{12}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>x\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{6}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{9}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{5}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{(} y \texttt{)}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{10}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>y\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{7}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{11}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{13}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{14}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{15}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
|}
 +
 +
<br>
 +
 +
===Appendix 3. Computational Details===
 +
 +
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math>
 +
|
 +
<math>\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}\!</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\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}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\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}\!</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\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}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math>
 +
|
 +
<math>\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}\!</math>
 +
|}
 +
 +
=====Computation of d''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation Summary for Conjunction=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\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}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of d''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation Summary for Equality=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\boldsymbol\varepsilon f_{11}
 +
& = && f_{11}(u, v)
 +
\\[4pt]
 +
& = && \texttt{(} u \texttt{(} v \texttt{))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{11}(1, 1)
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ }
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ }
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{11}
 +
& = && \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]
 +
&& + & \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]
 +
&& + & \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]
 +
&& + & \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{~}
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{E}f_{11}
 +
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{(}
 +
\\
 +
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\\
 +
&&& \texttt{(}
 +
\\
 +
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\
 +
&&& \texttt{))}
 +
\\[4pt]
 +
& = &&
 +
u v
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[4pt]
 +
& = &&
 +
u v
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[20pt]
 +
\mathrm{E}f_{11}
 +
& = && \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{~}
 +
\\[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{~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{D}f_{11}
 +
& = && \mathrm{E}f_{11}
 +
& + &  \boldsymbol\varepsilon f_{11}
 +
\\[4pt]
 +
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
& + &  f_{11}(u, v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{(} v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & 0
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
 +
& + &  \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
 +
& + &  0
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &  \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{c*{9}{l}}
 +
\mathrm{D}f_{11}
 +
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{11}
 +
& = & u v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 1
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}f_{11}
 +
& = &
 +
u v
 +
\cdot
 +
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\cdot
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\cdot
 +
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\cdot
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = &
 +
u v
 +
\cdot
 +
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\cdot
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\cdot
 +
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\cdot
 +
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of d''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\Downarrow
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\\[20pt]
 +
\mathrm{r}f_{11}
 +
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation Summary for Implication=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon f_{11}
 +
& = & u v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 1
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}f_{11}
 +
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[6pt]
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\\[6pt]
 +
\mathrm{r}f_{11}
 +
& = & 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}</math>
 +
|}
 +
 +
<br>
 +
 +
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{E}f_{14}
 +
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{((}
 +
\\
 +
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\\
 +
&&& \texttt{)(}
 +
\\
 +
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\
 +
&&& \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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of d''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation Summary for Disjunction=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
 +
|
 +
<math>\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}</math>
 +
|}
 +
 +
<br>
 +
 +
===Appendix 4. Source Materials===
 +
 +
===Appendix 5. Various Definitions of the Tangent Vector===
    
==References==
 
==References==
Jon Awbrey
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