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Revision as of 18:24, 19 September 2013
, 18:24, 19 September 2013update
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−====Table A10. Taylor Series Expansion====
\\+= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y+\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y+|+<math>\begin{matrix}+|+<math>\begin{matrix}
−|
−<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
+
+
+
+\end{matrix}</math>+|+<math>\begin{matrix}+|
−<math>\begin{matrix}
−|
−<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 & + &
−0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y+| <math>0\!</math>+| <math>0\!</math>+| <math>0\!</math>+| <math>0\!</math>+|}+ +<br>+ +====Table A11. Partial Differentials and Relative Differentials====+
−<br>
−
−|+ 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>
<p><math>\operatorname{d}f =</math></p>+<p><math>\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y</math></p>+| <math>\left. \frac{\partial x}{\partial y} \right| f</math>+| <math>\left. \frac{\partial y}{\partial x} \right| f</math>
−|- style="height:36px"
−| <math>f_0\!</math>
−| <math>(~)\!</math>
−| <math>0\!</math>
−| <math>0\!</math>
−| <math>0\!</math>
−| <math>0\!</math>
−| <math>0\!</math>
−|-
f_{1} \\+f_{2} \\+f_{4} \\
−f_{8} \\
−|}
(x) & (y) \\+(x) & y \\+ x & (y) \\+ x & y \\+|}+|+|+<math>\begin{smallmatrix}+ y \\+ y \\+|}+(x) \\+(x) \\+ x \\
− x \\
−|}
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\+ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\+(y) & \operatorname{d}x & + & x & \operatorname{d}y \\+ y & \operatorname{d}x & + & x & \operatorname{d}y \\+|+
−~ \\+~ \\+
−~ \\+
−|}+
+~ \\+~ \\+~ \\+{| align="center"+|+f_{3} \\+f_{12} \\+|}+|+{| align="center"+ x \\+|}+1 \\+1 \\+0 \\+0 \\+~ \\+~ \\
−|
−~ \\
−~ \\
−|}
f_{6} \\+f_{9} \\+|}+|+{| align="center"+ (x, & y) \\+|}+{| align="center"+|+1 \\+1 \\+|}+|+|+<math>\begin{smallmatrix}+1 \\+1 \\+|}+|+|+<math>\begin{smallmatrix}+|
−{| align="center"
−|
−~ \\
−~ \\
−|
−{| align="center"
−|
−~ \\
−~ \\
−|}
−{| align="center"
−|
−f_{5} \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−|
− y \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−|
−0 \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−|
−1 \\
−1 \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−|
−\operatorname{d}y \\
−\operatorname{d}y \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−|
−~ \\
−~ \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−|
−~ \\
−~ \\
−\end{smallmatrix}</math>
{| align="center"+f_{11} \\+f_{13} \\+f_{14} \\+\end{smallmatrix}</math>+|+{| align="center"+ (x & y) \\+ (x & (y)) \\+((x) & y) \\+((x) & (y)) \\+\end{smallmatrix}</math>+|+{| align="center"+ y \\
−(y) \\
− y \\
−(y) \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−|
−<math>\begin{smallmatrix}
− x \\
− x \\
−(x) \\
−(x) \\
−\end{smallmatrix}</math>
−|}
−|
−{| align="center"
− y & \operatorname{d}x & + & x & \operatorname{d}y \\
−(y) & \operatorname{d}x & + & x & \operatorname{d}y \\
− y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
−(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
−\end{smallmatrix}</math>
−|
−{| align="center"
−\end{smallmatrix}</math>
−|
−{| align="center"
−\end{smallmatrix}</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>
−+ & f|_{\mathrm{d}x ~ \mathrm{d}y}+= & \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}
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}+\\[4pt]+\end{array}</math>
−| style="width:20%; border-left:1px solid black" |
−<math>\begin{array}{cr}
−\\[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}
−\\[4pt]
−+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
−\\[4pt]
−= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
−|-
−| 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>
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math>
−| style="border-top:4px double black; border-left:4px double black" |
| 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>
−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+===Appendix 2. Computational Details===
−=====Computation of “Ef”=====
−======Computation of “εg”======+
−=====Computation of “Eg”=====
−
<br>
<br>
−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.
<br>
<br>
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<br>
<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>
<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>
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| 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>
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| <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" |
Line 10,350:
Line 10,292:
~ & \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=====
Line 10,449:
Line 10,730:
====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=====
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<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==
