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Revision as of 20:30, 19 June 2008
, 20:30, 19 June 2008update
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\textbf{Note.} The following Tables are best viewed in the Page Image mode.
+
+\textbf{Note.} The following Tables are best viewed in the Page Image mode.
<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}
−\tableofcontents
\tableofcontents
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\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
$f_{15}$ &
$f_{15}$ &
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$0$ \\
$0$ \\
\hline
\hline
−\end{tabular}
+\end{tabular}\end{center}
</pre>
</pre>
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\PMlinkescapephrase{mode}
\PMlinkescapephrase{mode}
\PMlinkescapephrase{Mode}
\PMlinkescapephrase{Mode}
−\tableofcontents
\tableofcontents
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\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
&
&
