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

Revision as of 17:04, 18 June 2008

3,166 bytes added ,  17:04, 18 June 2008
→‎Appendix 2 @ PlanetMath : TeX Format: update
Line 985: Line 985:     
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
 
\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]
+
\multicolumn{5}{c}{\textbf{Table A7.  Detail of Calculation for} $\operatorname{D}f = \operatorname{E}f + f$} \\[6pt]
 
\hline\hline
 
\hline\hline
 
&
 
&
Line 991: Line 991:  
   & \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\
 
   & \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\
 
+ &                f|_{\operatorname{d}x\ \operatorname{d}y} \\
 
+ &                f|_{\operatorname{d}x\ \operatorname{d}y} \\
 +
= & \operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y} \\
 
\end{array}$
 
\end{array}$
 
&
 
&
Line 996: Line 997:  
   & \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
 
   & \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
 
+ &                f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
 
+ &                f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
 +
= & \operatorname{D}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
 
\end{array}$
 
\end{array}$
 
&
 
&
Line 1,001: Line 1,003:  
   & \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
 
   & \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
 
+ &                f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
 
+ &                f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
 +
= & \operatorname{D}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
 
\end{array}$
 
\end{array}$
 
&
 
&
Line 1,006: Line 1,009:  
   & \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
 
   & \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
 
+ &                f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
 
+ &                f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
 +
= & \operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
 
\end{array}$ \\[6pt]
 
\end{array}$ \\[6pt]
 
\hline\hline
 
\hline\hline
−
$f_{0}$ & $0 + 0$ & $0 + 0$ & $0 + 0$ & $0 + 0$ \\[6pt]
+
$f_{0}$ & $0 + 0 = 0$ & $0 + 0 = 0$ & $0 + 0 = 0$ & $0 + 0 = 0$ \\[6pt]
 
\hline\hline
 
\hline\hline
 
$f_{1}$
 
$f_{1}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & x &  y & \operatorname{d}x & \operatorname{d}y \\
+
   &   x\ y   & \operatorname{d}x & \operatorname{d}y \\
−
+ & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\
+
+ & (x)(y)  & \operatorname{d}x & \operatorname{d}y \\
 +
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x & (y) & \operatorname{d}x & (\operatorname{d}y) \\
+
   &  x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &     (y) & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) &  y  & (\operatorname{d}x) & \operatorname{d}y \\
+
   & (x)\ y  & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
= & (x)     & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & (x) & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,035: Line 1,043:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x & (y) & \operatorname{d}x & \operatorname{d}y \\
+
   &  x\ (y) & \operatorname{d}x & \operatorname{d}y \\
−
+ & (x) &  y & \operatorname{d}x & \operatorname{d}y \\
+
+ & (x)\ y  & \operatorname{d}x & \operatorname{d}y \\
 +
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x & y  & \operatorname{d}x & (\operatorname{d}y) \\
+
   &  x\ y  & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & (x) & y  & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & (x)\ y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &     y  & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\
+
   & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & (x) & y & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & (x)\ y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
= & (x)    & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) &  y & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & (x) & y & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &    0  & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,057: Line 1,069:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) &  y  & \operatorname{d}x & \operatorname{d}y \\
+
   & (x)\ y  & \operatorname{d}x & \operatorname{d}y \\
−
+ &  x & (y) & \operatorname{d}x & \operatorname{d}y \\
+
+ &  x\ (y) & \operatorname{d}x & \operatorname{d}y \\
 +
= & (x,  y) & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\
+
   & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ &  x & (y) & \operatorname{d}x & (\operatorname{d}y) \\
+
+ &  x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &     (y) & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x & y  & (\operatorname{d}x) & \operatorname{d}y \\
+
   &  x\ y  & (\operatorname{d}x) & \operatorname{d}y \\
−
+ &  x & (y) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ &  x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  x      & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & x & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & x & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,079: Line 1,095:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\
+
   & (x)(y) & \operatorname{d}x & \operatorname{d}y \\
−
+ & x & y & \operatorname{d}x & \operatorname{d}y \\
+
+ &   x\ y  & \operatorname{d}x & \operatorname{d}y \\
 +
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) &  y & \operatorname{d}x & (\operatorname{d}y) \\
+
   & (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\
−
+ &  x  & y & \operatorname{d}x & (\operatorname{d}y) \\
+
+ &  x\ y & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &      y & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x & (y) & (\operatorname{d}x) & \operatorname{d}y \\
+
   &  x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\
−
+ &  x  & y  & (\operatorname{d}x) & \operatorname{d}y \\
+
+ &  x\ y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
= & x      & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & x &  y & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & x & y  & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0 & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline\hline
 
\hline\hline
Line 1,101: Line 1,121:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x  & & \operatorname{d}x & \operatorname{d}y \\
+
   &  x  & \operatorname{d}x & \operatorname{d}y \\
−
+ & (x) & & \operatorname{d}x & \operatorname{d}y \\
+
+ & (x) & \operatorname{d}x & \operatorname{d}y \\
 +
= &  1  & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x  & & \operatorname{d}x & (\operatorname{d}y) \\
+
   &  x  & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & (x) & & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & (x) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &  1  & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
+
   & (x) & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & (x) & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  0  & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & (x) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & (x) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0  & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,123: Line 1,147:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & & \operatorname{d}x & \operatorname{d}y \\
+
   & (x) & \operatorname{d}x & \operatorname{d}y \\
−
+ &  x  & & \operatorname{d}x & \operatorname{d}y \\
+
+ &  x  & \operatorname{d}x & \operatorname{d}y \\
 +
= &  1  & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x) & & \operatorname{d}x & (\operatorname{d}y) \\
+
   & (x) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ &  x  & & \operatorname{d}x & (\operatorname{d}y) \\
+
+ &  x  & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &  1  & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  x  & & (\operatorname{d}x) & \operatorname{d}y \\
+
   &  x  & (\operatorname{d}x) & \operatorname{d}y \\
−
+ &  x  & & (\operatorname{d}x) & \operatorname{d}y \\
+
+ &  x  & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  0  & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & x & & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & x & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & x & & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & x & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline\hline
 
\hline\hline
Line 1,145: Line 1,173:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x, &  y) & \operatorname{d}x & \operatorname{d}y \\
+
   & (x, y) & \operatorname{d}x & \operatorname{d}y \\
−
+ & (x, & y)  & \operatorname{d}x & \operatorname{d}y \\
+
+ & (x, y) & \operatorname{d}x & \operatorname{d}y \\
 +
= &  0    & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x, &  y)) & \operatorname{d}x & (\operatorname{d}y) \\
+
   & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ &  (x, & y) & \operatorname{d}x & (\operatorname{d}y) \\
+
+ &  (x, y)  & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &    1    & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x, &  y)) & (\operatorname{d}x) & \operatorname{d}y \\
+
   & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\
−
+ &  (x, &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
+
+ &  (x, y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &    1    & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x, &  y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & (x, & y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,167: Line 1,199:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x, &  y)) & \operatorname{d}x & \operatorname{d}y \\
+
   & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
−
+ & ((x, &  y)) & \operatorname{d}x & \operatorname{d}y \\
+
+ & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
 +
= &    0    & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  (x, &  y)  & \operatorname{d}x & (\operatorname{d}y) \\
+
   &  (x, y)  & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & ((x, & y)) & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &    1    & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  (x, &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
+
   &  (x, y)  & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & ((x, &  y)) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &    1    & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x, &  y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & ((x, & y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &    0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline\hline
 
\hline\hline
Line 1,189: Line 1,225:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & &  y  & \operatorname{d}x & \operatorname{d}y \\
+
   &  y  & \operatorname{d}x & \operatorname{d}y \\
−
+ & & (y) & \operatorname{d}x & \operatorname{d}y \\
+
+ & (y) & \operatorname{d}x & \operatorname{d}y \\
 +
= &  1  & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
+
   & (y) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & (y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &  0  & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & &  y  & (\operatorname{d}x) & \operatorname{d}y \\
+
   &  y  & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & & (y) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & (y) & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  1  & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0  & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,211: Line 1,251:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & & (y) & \operatorname{d}x & \operatorname{d}y \\
+
   & (y) & \operatorname{d}x & \operatorname{d}y \\
−
+ & &  y & \operatorname{d}x & \operatorname{d}y \\
+
+ & y  & \operatorname{d}x & \operatorname{d}y \\
 +
= &  1 & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & &  y  & \operatorname{d}x & (\operatorname{d}y) \\
+
   &  y  & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & &  y & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & y  & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &  0 & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & & (y) & (\operatorname{d}x) & \operatorname{d}y \\
+
   & (y) & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & &  y & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & y  & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  1 & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   &  y  & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & &  y & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & y  & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0 & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline\hline
 
\hline\hline
Line 1,233: Line 1,277:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\
+
   & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\
−
+ &  (x  & y) & \operatorname{d}x & \operatorname{d}y \\
+
+ &  (x\ y) & \operatorname{d}x & \operatorname{d}y \\
 +
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) &  y) & \operatorname{d}x & (\operatorname{d}y) \\
+
   & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ &  (x  & y)  & \operatorname{d}x & (\operatorname{d}y) \\
+
+ &  (x\ y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &      y & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
+
   & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\
−
+ &  (x & y) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & (x\ y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
= & x      & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x &  y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & (x & y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,255: Line 1,303:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) &  y)  & \operatorname{d}x & \operatorname{d}y \\
+
   & ((x)\ y)  & \operatorname{d}x & \operatorname{d}y \\
−
+ &  (x  & (y)) & \operatorname{d}x & \operatorname{d}y \\
+
+ &  (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\
 +
= & (x,  y) & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
+
   & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ &  (x & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
+
+ &  (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &      (y) & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x & y)  & (\operatorname{d}x) & \operatorname{d}y \\
+
   & (x\ y)  & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & (x & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  x      & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & (x & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & (x & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &    0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,277: Line 1,329:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  (x & (y)) & \operatorname{d}x & \operatorname{d}y \\
+
   &  (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\
−
+ & ((x) &  y)  & \operatorname{d}x & \operatorname{d}y \\
+
+ & ((x)\ y)  & \operatorname{d}x & \operatorname{d}y \\
 +
= &  (x, y)  & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  (x & y) & \operatorname{d}x & (\operatorname{d}y) \\
+
   &  (x\ y) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & ((x) & y)  & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &      y & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
+
   & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & ((x) &  y) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & ((x)\ y)  & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  (x)     & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) &  y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & ((x) & y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &    0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline
 
\hline
Line 1,299: Line 1,355:  
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  (x &  y)  & \operatorname{d}x & \operatorname{d}y \\
+
   &  (x\ y)  & \operatorname{d}x & \operatorname{d}y \\
−
+ & ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\
+
+ & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\
 +
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   &  (x & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
+
   &  (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\
−
+ & ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
+
+ & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\
 +
= &      (y) & \operatorname{d}x & (\operatorname{d}y) \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) &  y)  & (\operatorname{d}x) & \operatorname{d}y \\
+
   & ((x)\ y)  & (\operatorname{d}x) & \operatorname{d}y \\
−
+ & ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
+
+ & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\
 +
= &  (x)     & (\operatorname{d}x) & \operatorname{d}y \\
 
\end{smallmatrix}$
 
\end{smallmatrix}$
 
&
 
&
 
$\begin{smallmatrix}
 
$\begin{smallmatrix}
−
   & ((x) & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
   & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
−
+ & ((x) & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+
+ & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
 +
= &    0    & (\operatorname{d}x) & (\operatorname{d}y) \\
 
\end{smallmatrix}$ \\[6pt]
 
\end{smallmatrix}$ \\[6pt]
 
\hline\hline
 
\hline\hline
−
$f_{15}$ & $1 + 1$ & $1 + 1$ & $1 + 1$ & $1 + 1$ \\[6pt]
+
$f_{15}$ & $1 + 1 = 0$ & $1 + 1 = 0$ & $1 + 1 = 0$ & $1 + 1 = 0$ \\[6pt]
 
\hline\hline
 
\hline\hline
 
\end{tabular}\end{quote}
 
\end{tabular}\end{quote}
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/67851"

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