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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}
\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}
\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}
\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}
\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}
\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}
\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}
\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}
\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}
\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}
| 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}
\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}
\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}
\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}
\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}
\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}
\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}
\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}
\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}
\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}
\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}
<br>
<br>
==References==
===Appendix 2. Differential Forms===
===Works Cited===
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 A7 and A8.
{| cellpadding=3
Table A7 expands the differential forms that result over a ''logical basis'':
| valign=top | [AuM]
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.
{| align="center" cellpadding="6" style="text-align:center"
|-
|
| valign=top | [BiG]
<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>
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.
|}
|-
| valign=top | [Boo]
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:
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.
|-
{| align="center" cellpadding="6" style="text-align:center"
| valign=top | [BoT]
|
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> and <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math>
|-
|}
| valign=top | [dCa]
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.
Table A8 expands the differential forms that result over an ''algebraic basis'':
|-
| valign=top | [Che46]
{| align="center" cellpadding="6" style="text-align:center"
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
|-
|}
| valign=top | [Che56]
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.
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''.
|-
| valign=top | [Cho86]
====Table A7. Differential Forms Expanded on a Logical Basis====
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.
|-
<br>
| valign=top | [Cho93]
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.
{| 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>
| valign=top | [DoM]
|- style="background:ghostwhite; height:40px"
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.
|
|-
| style="border-right:none" | <math>f\!</math>
| valign=top | [Fuji]
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.
| <math>\mathrm{d}f~\!</math>
|-
|-
| valign=top | [Hic]
| <math>f_{0}\!</math>
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.
| style="border-right:none" | <math>\texttt{(~)}\!</math>
|-
| style="border-left:4px double black" | <math>0\!</math>
| valign=top | [Hir]
| <math>0\!</math>
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.
|-
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| valign=top | [How]
| style="border-right:none" |
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].
<math>\begin{matrix}
|-
\texttt{(} x \texttt{)(} y \texttt{)}
| valign=top | [JGH]
\\
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.
\texttt{(} x \texttt{)~} y \texttt{~}
|-
\\
| valign=top | [KoA]
\texttt{~} x \texttt{~(} y \texttt{)}
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.
\\
\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"
|
| 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"
|
| <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%" |
| 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 ε''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 ε''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 ε''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 ε''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==
===Works Cited===
{| cellpadding=3
| valign=top | [AuM]
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.
|-
| valign=top | [BiG]
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.
|-
| valign=top | [Boo]
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.
|-
| valign=top | [BoT]
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.
|-
| valign=top | [dCa]
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.
|-
| valign=top | [Che46]
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.
|-
| valign=top | [Che56]
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.
|-
| valign=top | [Cho86]
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.
|-
| valign=top | [Cho93]
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.
|-
| valign=top | [DoM]
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.
|-
| valign=top | [Fuji]
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.
|-
| valign=top | [Hic]
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.
|-
| valign=top | [Hir]
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.
|-
| valign=top | [How]
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].
|-
| valign=top | [JGH]
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.
|-
| valign=top | [KoA]
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.
|-
|-
| valign=top | [Koh]
| valign=top | [Koh]
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.
|}
|}
==Document History==
<p align="center"><math>\begin{array}{lcr}
& \text{Differential Logic and Dynamic Systems} &
\\
\text{Author:} & \text{Jon Awbrey} & \text{October 20, 1994}
\\
\text{Course:} & \text{Engineering 690, Graduate Project} & \text{Winter Term 1994}
\\
\text{Supervisor:} & \text{M.A. Zohdy} & \text{Oakland University}
\\
\text{Created:} && \text{16 Dec 1993}
\\
\text{Relayed:} && \text{31 Oct 1994}
\\
\text{Revised:} && \text{03 Jun 2003}
\\
\text{Recoded:} && \text{03 Jun 2007}
\end{array}</math></p>
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