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

Revision as of 20:30, 19 June 2008

3,347 bytes added ,  20:30, 19 June 2008
update
Line 975: Line 975:     
<pre>
 
<pre>
 +
\PMlinkescapephrase{algebraic}
 +
\PMlinkescapephrase{Algebraic}
 
\PMlinkescapephrase{basis}
 
\PMlinkescapephrase{basis}
 
\PMlinkescapephrase{Basis}
 
\PMlinkescapephrase{Basis}
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\PMlinkescapephrase{mode}
 
\PMlinkescapephrase{mode}
 
\PMlinkescapephrase{Mode}
 
\PMlinkescapephrase{Mode}
−
  −
\textbf{Note.}  The following Tables are best viewed in the Page Image mode.
      
\tableofcontents
 
\tableofcontents
Line 990: Line 990:  
\subsection{Differential Forms Expanded on a Logical Basis}
 
\subsection{Differential Forms Expanded on a Logical Basis}
   −
\begin{tabular}{|c|c|c|c|}
+
\begin{center}\begin{tabular}{|c|c|c|c|}
 
\multicolumn{4}{c}{\textbf{Differential Forms Expanded on a Logical Basis}} \\
 
\multicolumn{4}{c}{\textbf{Differential Forms Expanded on a Logical Basis}} \\
 
\hline
 
\hline
Line 1,003: Line 1,003:  
$0$    \\
 
$0$    \\
 
\hline
 
\hline
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
f_{1} \\
 
f_{1} \\
 
f_{2} \\
 
f_{2} \\
 
f_{4} \\
 
f_{4} \\
 
f_{8} \\
 
f_{8} \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
(x) & (y) \\
 
(x) & (y) \\
 
(x) &  y  \\
 
(x) &  y  \\
 
  x  & (y) \\
 
  x  & (y) \\
 
  x  &  y  \\
 
  x  &  y  \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
     (y)  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 
     (y)  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 
  (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
 
  (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
Line 1,030: Line 1,030:  
   x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
 
   x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
 
((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
 
((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
(y) & \partial x & + & (x) & \partial y \\
 
(y) & \partial x & + & (x) & \partial y \\
 
  y  & \partial x & + & (x) & \partial y \\
 
  y  & \partial x & + & (x) & \partial y \\
 
(y) & \partial x & + &  x  & \partial y \\
 
(y) & \partial x & + &  x  & \partial y \\
 
  y  & \partial x & + &  x  & \partial y \\
 
  y  & \partial x & + &  x  & \partial y \\
−
\end{matrix}$ \\
+
\end{smallmatrix}$ \\
 
\hline
 
\hline
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
f_{3}  \\
 
f_{3}  \\
 
f_{12} \\
 
f_{12} \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
(x) \\
 
(x) \\
 
  x  \\
 
  x  \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
\operatorname{d}x\ (\operatorname{d}y) & + &
 
\operatorname{d}x\ (\operatorname{d}y) & + &
 
\operatorname{d}x\  \operatorname{d}y  \\
 
\operatorname{d}x\  \operatorname{d}y  \\
 
\operatorname{d}x\ (\operatorname{d}y) & + &
 
\operatorname{d}x\ (\operatorname{d}y) & + &
 
\operatorname{d}x\  \operatorname{d}y  \\
 
\operatorname{d}x\  \operatorname{d}y  \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
\partial x \\
 
\partial x \\
 
\partial x \\
 
\partial x \\
−
\end{matrix}$ \\
+
\end{smallmatrix}$ \\
 
\hline
 
\hline
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
f_{6} \\
 
f_{6} \\
 
f_{9} \\
 
f_{9} \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
  (x, & y)  \\
 
  (x, & y)  \\
 
((x, & y)) \\
 
((x, & y)) \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
  \operatorname{d}x\ (\operatorname{d}y) & + &
 
  \operatorname{d}x\ (\operatorname{d}y) & + &
 
(\operatorname{d}x)\ \operatorname{d}y  \\
 
(\operatorname{d}x)\ \operatorname{d}y  \\
 
  \operatorname{d}x\ (\operatorname{d}y) & + &
 
  \operatorname{d}x\ (\operatorname{d}y) & + &
 
(\operatorname{d}x)\ \operatorname{d}y  \\
 
(\operatorname{d}x)\ \operatorname{d}y  \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
\partial x & + & \partial y \\
 
\partial x & + & \partial y \\
 
\partial x & + & \partial y \\
 
\partial x & + & \partial y \\
−
\end{matrix}$ \\
+
\end{smallmatrix}$ \\
 
\hline
 
\hline
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
f_{5}  \\
 
f_{5}  \\
 
f_{10} \\
 
f_{10} \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
(y) \\
 
(y) \\
 
  y  \\
 
  y  \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
(\operatorname{d}x)\ \operatorname{d}y & + &
 
(\operatorname{d}x)\ \operatorname{d}y & + &
 
  \operatorname{d}x\  \operatorname{d}y \\
 
  \operatorname{d}x\  \operatorname{d}y \\
 
(\operatorname{d}x)\ \operatorname{d}y & + &
 
(\operatorname{d}x)\ \operatorname{d}y & + &
 
  \operatorname{d}x\  \operatorname{d}y \\
 
  \operatorname{d}x\  \operatorname{d}y \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
\partial y \\
 
\partial y \\
 
\partial y \\
 
\partial y \\
−
\end{matrix}$ \\
+
\end{smallmatrix}$ \\
 
\hline
 
\hline
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
f_{7}  \\
 
f_{7}  \\
 
f_{11} \\
 
f_{11} \\
 
f_{13} \\
 
f_{13} \\
 
f_{14} \\
 
f_{14} \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
  (x  &  y)  \\
 
  (x  &  y)  \\
 
  (x  & (y)) \\
 
  (x  & (y)) \\
 
((x) &  y)  \\
 
((x) &  y)  \\
 
((x) & (y)) \\
 
((x) & (y)) \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
     y  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 
     y  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 
   x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
 
   x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
Line 1,132: Line 1,132:  
  (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
 
  (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
 
((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
 
((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
−
\end{matrix}$
+
\end{smallmatrix}$
 
&
 
&
−
$\begin{matrix}
+
$\begin{smallmatrix}
 
  y  & \partial x & + &  x  & \partial y \\
 
  y  & \partial x & + &  x  & \partial y \\
 
(y) & \partial x & + &  x  & \partial y \\
 
(y) & \partial x & + &  x  & \partial y \\
 
  y  & \partial x & + & (x) & \partial y \\
 
  y  & \partial x & + & (x) & \partial y \\
 
(y) & \partial x & + & (x) & \partial y \\
 
(y) & \partial x & + & (x) & \partial y \\
−
\end{matrix}$ \\
+
\end{smallmatrix}$ \\
 +
\hline
 +
$f_{15}$ &
 +
$((~))$  &
 +
$0$      &
 +
$0$      \\
 +
\hline
 +
\end{tabular}\end{center}
 +
 
 +
\subsection{Differential Forms Expanded on an Algebraic Basis}
 +
 
 +
\begin{center}\begin{tabular}{|c|c|c|c|}
 +
\multicolumn{4}{c}{\textbf{Differential Forms Expanded on an Algebraic Basis}} \\
 +
\hline
 +
&
 +
$f$ &
 +
$\operatorname{D}f$ &
 +
$\operatorname{d}f$ \\
 +
\hline
 +
$f_{0}$ &
 +
$(~)$  &
 +
$0$    &
 +
$0$    \\
 +
\hline
 +
$\begin{smallmatrix}
 +
f_{1} \\
 +
f_{2} \\
 +
f_{4} \\
 +
f_{8} \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
(x) & (y) \\
 +
(x) &  y  \\
 +
x  & (y) \\
 +
x  &  y  \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
(y) & \operatorname{d}x & + &
 +
(x) & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
y  & \operatorname{d}x & + &
 +
(x) & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &
 +
x  & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
y  & \operatorname{d}x & + &
 +
x  & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
\end{smallmatrix}$ \\
 +
\hline
 +
$\begin{smallmatrix}
 +
f_{3}  \\
 +
f_{12} \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
(x) \\
 +
x  \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{smallmatrix}$ \\
 +
\hline
 +
$\begin{smallmatrix}
 +
f_{6} \\
 +
f_{9} \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
(x, & y)  \\
 +
((x, & y)) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
\operatorname{d}x & + & \operatorname{d}y \\
 +
\operatorname{d}x & + & \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
\operatorname{d}x & + & \operatorname{d}y \\
 +
\operatorname{d}x & + & \operatorname{d}y \\
 +
\end{smallmatrix}$ \\
 +
\hline
 +
$\begin{smallmatrix}
 +
f_{5}  \\
 +
f_{10} \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
(y) \\
 +
y  \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{smallmatrix}$ \\
 +
\hline
 +
$\begin{smallmatrix}
 +
f_{7}  \\
 +
f_{11} \\
 +
f_{13} \\
 +
f_{14} \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
(x  &  y)  \\
 +
(x  & (y)) \\
 +
((x) &  y)  \\
 +
((x) & (y)) \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
y  & \operatorname{d}x & + &
 +
x  & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &
 +
x  & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
y  & \operatorname{d}x & + &
 +
(x) & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &
 +
(x) & \operatorname{d}y & + &
 +
      \operatorname{d}x\ \operatorname{d}y \\
 +
\end{smallmatrix}$
 +
&
 +
$\begin{smallmatrix}
 +
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
\end{smallmatrix}$ \\
 
\hline
 
\hline
 
$f_{15}$ &
 
$f_{15}$ &
Line 1,146: Line 1,300:  
$0$      \\
 
$0$      \\
 
\hline
 
\hline
−
\end{tabular}
+
\end{tabular}\end{center}
 
</pre>
 
</pre>
   Line 1,156: Line 1,310:  
\PMlinkescapephrase{mode}
 
\PMlinkescapephrase{mode}
 
\PMlinkescapephrase{Mode}
 
\PMlinkescapephrase{Mode}
−
  −
\textbf{Note.}  The following Tables are best viewed in the Page Image mode.
      
\tableofcontents
 
\tableofcontents
Line 1,164: Line 1,316:     
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
 
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
−
\multicolumn{5}{c}{\textbf{Detail of Calculation for} $\operatorname{D}f = \operatorname{E}f + f$} \\[6pt]
+
\multicolumn{5}{c}{\textbf{Detail of Calculation for $\operatorname{D}f = \operatorname{E}f + f$}} \\[6pt]
 
\hline\hline
 
\hline\hline
 
&
 
&
Jon Awbrey
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