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Revision as of 18:24, 19 September 2013
, 18:24, 19 September 2013update
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But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix 1 and a summary of the results is presented in Tables 66-i and 66-ii.
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix 3 and a summary of the results is presented in Tables 66-i and 66-ii.
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<br>
====Differential Forms====
===Appendix 2. Differential Forms===
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables A7 and A8.
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.
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<br>
====Table A9. Differential = Pointwise Linear Approximation to the Difference====
====Table A9. Tangent Proposition as Pointwise Linear Approximation====
<br>
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
|+ 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>
<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="background:ghostwhite; height:40px"
| style="border-right:none" | <math>f\!</math>
| style="border-right:none" | <math>f\!</math>
| style="border-left:4px double black" |
| style="border-left:4px double black" |
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{D}f
\mathrm{d}f =
\\[2pt]
= & \mathrm{d}f & + & \mathrm{d}^2\!f
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\\
\end{matrix}</math>
|
\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 \, y}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| style="border-right:none" | <math>f_0\!</math>
| style="border-right:none" | <math>f_0\!</math>
| style="border-left:4px double black" | <math>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>
| <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-right:none" |
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math>
| style="border-left:4px double black" |
| style="border-left:4px double black" |
<math>\begin{matrix}
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\\
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\\
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\\
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
\\
\mathrm{d}x\;\mathrm{d}y
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
\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_{3}\\f_{12}\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" |
| style="border-left:4px double black" |
<math>\begin{matrix}
<math>\begin{matrix}
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
\\
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\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>
\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}
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\mathrm{D}f
\\
= & \mathrm{d}f & + & \mathrm{d}^2\!f
\\
\mathrm{d}x\\\mathrm{d}x
= & \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>
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
\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_{6}\\f_{9}\end{matrix}</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" |
| style="border-left:4px double black" |
<math>\begin{matrix}
<math>\begin{matrix}
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\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{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math>
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-left:4px double black" |
| style="border-left:4px double black" |
<math>\begin{matrix}
<math>\begin{matrix}
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\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}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}\!</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\mathrm{d}x\\\mathrm{d}x
\end{matrix}\!</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\mathrm{d}x\\\mathrm{d}x
\end{matrix}\!</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\mathrm{d}x\\\mathrm{d}x
\end{matrix}\!</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\mathrm{d}x\\\mathrm{d}x
\end{matrix}\!</math>
\end{matrix}</math>
|-
|-
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-left:4px double black" |
| style="border-left:4px double black" |
<math>\begin{matrix}
<math>\begin{matrix}
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\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{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| style="border-right:none" | <math>f_{15}\!</math>
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-left:4px double black" | <math>0\!</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>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
|
{| align="center"
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
|
<math>\begin{smallmatrix}
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
\end{smallmatrix}</math>
|
|
{| align="center"
<math>\begin{matrix}
|
\mathrm{d}y\\\mathrm{d}y
<math>\begin{smallmatrix}
\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}
\end{smallmatrix}</math>
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
{| align="center"
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
(y) \\
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
(y) \\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\end{smallmatrix}</math>
\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>
|
|
{| align="center"
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
|
|
<math>\begin{smallmatrix}
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
\end{smallmatrix}</math>
|
|
{| align="center"
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
|
<math>\begin{smallmatrix}
<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>
\end{smallmatrix}</math>
| style="border-left:4px double black" | <math>0\!</math>
|}
| <math>0\!</math>
| <math>0\!</math>
{| align="center"
| <math>0\!</math>
|
| <math>0\!</math>
<math>\begin{smallmatrix}
|}
~ \\
<br>
====Table A11. Partial Differentials and Relative Differentials====
\end{smallmatrix}</math>
<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>
|
|
{| align="center"
<math>\begin{matrix}
|
\mathrm{d}f =
<math>\begin{smallmatrix}
\\[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>
\end{smallmatrix}</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{)}
<math>\begin{smallmatrix}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\end{smallmatrix}</math>
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
|
|
<math>\begin{smallmatrix}
<math>\begin{matrix}
(x) \\
\texttt{(} y \texttt{)}
\\
\end{smallmatrix}</math>
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
|
|
{| align="center"
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
|
|
<math>\begin{smallmatrix}
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\end{smallmatrix}</math>
\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>
|
|
{| align="center"
<math>\begin{matrix}
|
\texttt{(} x \texttt{)}
<math>\begin{smallmatrix}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
\end{smallmatrix}</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>
|
|
{| align="center"
<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{smallmatrix}
<math>\begin{matrix}
\operatorname{d}x \\
\texttt{(} y \texttt{)}
\operatorname{d}x \\
\\
\end{smallmatrix}</math>
\texttt{~} y \texttt{~}
|}
\end{matrix}</math>
|
| <math>\begin{matrix}0\\0\end{matrix}</math>
{| align="center"
| <math>\begin{matrix}1\\1\end{matrix}</math>
|
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
<math>\begin{smallmatrix}
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
\end{smallmatrix}</math>
|}
{| align="center"
|
<math>\begin{smallmatrix}
\end{smallmatrix}</math>
|-
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
|
|
{| align="center"
<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{smallmatrix}
<math>\begin{matrix}
\texttt{~} y \texttt{~}
\\
\end{smallmatrix}</math>
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
|
|
<math>\begin{smallmatrix}
<math>\begin{matrix}
\texttt{~} x \texttt{~}
((x, & y)) \\
\\
\end{smallmatrix}</math>
\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
<math>\begin{smallmatrix}
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\end{smallmatrix}</math>
\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
{| align="center"
\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>
\end{smallmatrix}</math>
| <math>\texttt{((~))}\!</math>
| <math>0\!</math>
| <math>0\!</math>
{| align="center"
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
\operatorname{d}x & + & \operatorname{d}y \\
\operatorname{d}x & + & \operatorname{d}y \\
\end{smallmatrix}</math>
|}
<math>\begin{smallmatrix}
\end{smallmatrix}</math>
|}
<math>\begin{smallmatrix}
\end{smallmatrix}</math>
|-
|
<math>\begin{smallmatrix}
f_{10} \\
|}
<math>\begin{smallmatrix}
(y) \\
|}
<math>\begin{smallmatrix}
0 \\
|}
<math>\begin{smallmatrix}
|}
<math>\begin{smallmatrix}
|}
<math>\begin{smallmatrix}
|}
<math>\begin{smallmatrix}
|}
|}
<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>
<math>\begin{smallmatrix}
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math>
f_{7} \\
| 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{~~}
<math>\begin{smallmatrix}
\\[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{)~}
<math>\begin{smallmatrix}
|}
|
<math>\begin{smallmatrix}
|}
|
<math>\begin{smallmatrix}
~ \\
~ \\
~ \\
~ \\
|}
|
<math>\begin{smallmatrix}
~ \\
~ \\
~ \\
~ \\
|}
|- style="height:36px"
| <math>f_{15}\!</math>
| <math>((~))\!</math>
| <math>0\!</math>
|}
| style="width:20%; border-left:4px double black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}
\\[4pt]
\\[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]
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{matrix}</math>
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
\end{array}</math>
| style="border-top:4px double black; border-left:1px solid black" |
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
= & 0
\end{matrix}</math>
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{2}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
|-
| 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>
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
| style="border-top:1px solid black; border-left:4px double black" |
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}</math>
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{matrix}</math>
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\\[4pt]
= & 0
= & 0
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| style="border-top:4px double black" | <math>f_{15}\!</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: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: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>
| 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}{*{9}{l}}
\boldsymbol\varepsilon f_{8} & = & f_{8}(u, v)
\\[4pt]
& = & u \cdot v
\\[4pt]
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }
\end{array}\!</math>
|-
|
<math>\begin{array}{*{4}{l}}
\boldsymbol\varepsilon f_{8}
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}
\\[4pt]
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \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{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[4pt]
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot f_{8}(1\!+\!\mathrm{d}u, 1\!+\!\mathrm{d}v)
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot f_{8}(1\!+\!\mathrm{d}u, \mathrm{d}v)
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot f_{8}(\mathrm{d}u, 1\!+\!\mathrm{d}v)
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
\\[4pt]
& = &
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
& + &
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)
& + &
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})
& + &
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
\end{array}\!</math>
|-
|
<math>\begin{array}{*{9}{l}}
\mathrm{E}f_{8}
& = &
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&&& + &
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }
\\[4pt]
&&&&& + &
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&&&&&&& + &
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\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 F8.2-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{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + &
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }
& + &
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }
\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}{*{9}{l}}
\mathrm{D}f_{8}
& = & \mathrm{E}f_{8}
& + & \boldsymbol\varepsilon f_{8}
\\[6pt]
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
& + & f_{8}(u, v)
\\[6pt]
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
& + & u \cdot v
\end{array}\!</math>
|-
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{8}
& = &
u \cdot v \cdot \qquad 0
\\[6pt]
& + &
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
& + &
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
\\[6pt]
& + &
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
&&& + &
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
& + &
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}
&&&&& + &
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}
\end{array}</math>
|-
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{8}
& = &
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
& + &
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \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]
& = & u \cdot v
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
& = & 0
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u \cdot \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 \cdot \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}{*{5}{l}}
\mathrm{D}f_{8} & = & \boldsymbol\varepsilon f_{8} & + & \mathrm{E}f_{8}
\end{array}</math>
|-
|
<math>\begin{array}{*{9}{l}}
\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
\end{array}</math>
|-
|
<math>\begin{array}{*{9}{l}}
\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>
|}
<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 F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{8}
& = &
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v
& + &
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{8}
& = &
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + &
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + &
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}
<br>
=====Computation of r''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 F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math>
|
<math>\begin{array}{*{5}{l}}
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} & + & \mathrm{d}f_{8}
\end{array}</math>
|-
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{8}
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0
\end{array}</math>
|-
|
<math>\begin{array}{*{9}{l}}
\mathrm{r}f_{8}
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~~~~~~
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \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}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{8}
& = & u \!\cdot\! v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{E}f_{8}
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\mathrm{D}f_{8}
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{r}f_{8}
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
\end{array}</math>
|}
|}
<br>
<br>
====Operator Maps for the Logical Disjunction ''f''(u, v)====
====Operator Maps for the Logical Disjunction ''f''(u, v)====
=====Computation of “εf”=====
=====Computation of ε''f''=====
=====Computation of “Df” (1)=====
=====Computation of E''f''=====
=====Computation of “Df” (2)=====
=====Computation of D''f''=====
=====Computation of “df”=====
=====Computation of d''f''=====
=====Computation of “rf”=====
=====Computation of r''f''=====
=====Computation Summary for Disjunction=====
=====Computation Summary for Disjunction=====
====Operator Maps for the Logical Equality ''g''(u, v)====
====Operator Maps for the Logical Equality ''g''(u, v)====
=====Computation of ε''g''=====
=====Computation of “Dg” (1)=====
=====Computation of E''g''=====
=====Computation of “Dg” (2)=====
=====Computation of D''g''=====
=====Computation of “dg”=====
=====Computation of d''g''=====
=====Computation of “rg”=====
=====Computation of r''g''=====
=====Computation Summary for Equality=====
=====Computation Summary for Equality=====
<br>
<br>
===Appendix 3. Source Materials===
===Appendix 4. Source Materials===
===Appendix 4. Various Definitions of the Tangent Vector===
===Appendix 5. Various Definitions of the Tangent Vector===
==References==
==References==
