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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 06:02, 18 June 2008

10,632 bytes added ,  06:02, 18 June 2008
→‎Appendices @ PlanetMath : TeX Format: update
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=Examples @ PlanetMath : TeX Format=
 
=Examples @ PlanetMath : TeX Format=
   −
=Appendices @ PlanetMath : TeX Format=
+
=Appendix 1 @ PlanetMath : TeX Format=
    
<pre>
 
<pre>
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$(~)$    \\
 
$(~)$    \\
 
\hline
 
\hline
 +
\end{tabular}\end{quote}
 +
</pre>
 +
 +
=Appendix 2 @ PlanetMath : TeX Format=
 +
 +
<pre>
 +
\PMlinkescapephrase{image}
 +
\PMlinkescapephrase{Image}
 +
\PMlinkescapephrase{mode}
 +
\PMlinkescapephrase{Mode}
 +
 +
\textbf{Note.}  The following Tables are best viewed in the Page Image mode.
 +
 +
\tableofcontents
 +
 +
\subsection{Table A7.  Detail of Calculation for the Difference Map}
 +
 +
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
 +
\multicolumn{5}{c}{Table A7.  Detail of Calculation for $\operatorname{D}f = \operatorname{E}f + f$} \\[6pt]
 +
\hline\hline
 +
&
 +
$\begin{array}{cr}
 +
  & \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\
 +
+ &                f|_{\operatorname{d}x\ \operatorname{d}y} \\
 +
\end{array}$
 +
&
 +
$\begin{array}{cr}
 +
  & \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
 +
+ &                f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
 +
\end{array}$
 +
&
 +
$\begin{array}{cr}
 +
  & \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
 +
+ &                f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
 +
\end{array}$
 +
&
 +
$\begin{array}{cr}
 +
  & \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
 +
+ &                f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
 +
\end{array}$ \\[6pt]
 +
\hline\hline
 +
$f_{0}$ & $0 + 0$ & $0 + 0$ & $0 + 0$ & $0 + 0$ \\[6pt]
 +
\hline\hline
 +
$f_{1}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  &  y  & \operatorname{d}x & \operatorname{d}y \\
 +
+ & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) &  y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & (x) & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{2}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & (y) & \operatorname{d}x & \operatorname{d}y \\
 +
+ & (x) &  y  & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  &  y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & (x) &  y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & (x) &  y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & (x) &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{4}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) &  y  & \operatorname{d}x & \operatorname{d}y \\
 +
+ &  x  & (y) & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ &  x  & (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  &  y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ &  x  & (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ &  x  & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{8}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\
 +
+ &  x  &  y  & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) &  y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ &  x  &  y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ &  x  &  y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ &  x  &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline\hline
 +
$f_{3}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & & \operatorname{d}x & \operatorname{d}y \\
 +
+ & (x) & & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & (x) & & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{12}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & & \operatorname{d}x & \operatorname{d}y \\
 +
+ &  x  & & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & (x) & & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ &  x  & & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ &  x  & & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  x  & & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ &  x  & & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline\hline
 +
$f_{6}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x, &  y)  & \operatorname{d}x & \operatorname{d}y \\
 +
+ &  (x, &  y)  & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x, &  y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ &  (x, &  y)  & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x, &  y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ &  (x, &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x, &  y)  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ &  (x, &  y)  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{9}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x, &  y)) & \operatorname{d}x & \operatorname{d}y \\
 +
+ & ((x, &  y)) & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x, &  y)  & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & ((x, &  y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x, &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & ((x, &  y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x, &  y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & ((x, &  y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline\hline
 +
$f_{5}$
 +
&
 +
$\begin{smallmatrix}
 +
  & &  y  & \operatorname{d}x & \operatorname{d}y \\
 +
+ & & (y) & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & &  y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & & (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{10}$
 +
&
 +
$\begin{smallmatrix}
 +
  & & (y) & \operatorname{d}x & \operatorname{d}y \\
 +
+ & &  y  & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & &  y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & &  y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & & (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & &  y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline\hline
 +
$f_{7}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\
 +
+ &  (x  &  y)  & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) &  y)  & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ &  (x  &  y)  & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ &  (x  &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  &  y)  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ &  (x  &  y)  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{11}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) &  y)  & \operatorname{d}x & \operatorname{d}y \\
 +
+ &  (x  & (y)) & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ &  (x  & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ &  (x  & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ &  (x  & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{13}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  & (y)) & \operatorname{d}x & \operatorname{d}y \\
 +
+ & ((x) &  y)  & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  &  y)  & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & ((x) &  y)  & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & ((x) &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) &  y)  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & ((x) &  y)  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline
 +
$f_{14}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  &  y)  & \operatorname{d}x & \operatorname{d}y \\
 +
+ & ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  &  (x  & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
+ & ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
+ & ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
  & ((x) & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
+ & ((x) & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
\end{smallmatrix}$ \\[6pt]
 +
\hline\hline
 +
$f_{15}$ & $1 + 1$ & $1 + 1$ & $1 + 1$ & $1 + 1$ \\[6pt]
 +
\hline\hline
 
\end{tabular}\end{quote}
 
\end{tabular}\end{quote}
 
</pre>
 
</pre>
Jon Awbrey
12,080

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