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

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<pre>
 
<pre>
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\PMlinkescapephrase{basis}
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\PMlinkescapephrase{Basis}
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\PMlinkescapephrase{expanded}
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\PMlinkescapephrase{expanded}
 
\PMlinkescapephrase{image}
 
\PMlinkescapephrase{image}
 
\PMlinkescapephrase{Image}
 
\PMlinkescapephrase{Image}
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\tableofcontents
 
\tableofcontents
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\subsection{Table A7.  Detail of Calculation for the Difference Map}
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\subsection{Differential Forms Expanded on a Logical Basis}
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\begin{tabular}{|c|c|c|c|}
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\multicolumn{4}{c}{\textbf{Differential Forms Expanded on a Logical Basis}} \\
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\hline
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&
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$f$ &
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$\operatorname{D}f$ &
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$\operatorname{d}f$ \\
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\hline
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$f_{0}$ &
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$(~)$  &
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$0$    &
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$0$    \\
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\hline
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$\begin{matrix}
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f_{1} \\
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f_{2} \\
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f_{4} \\
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f_{8} \\
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\end{matrix}$
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&
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$\begin{matrix}
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(x) & (y) \\
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(x) &  y  \\
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x  & (y) \\
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x  &  y  \\
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\end{matrix}$
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&
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$\begin{matrix}
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    (y)  &  \operatorname{d}x\ (\operatorname{d}y) & + &
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(x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
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((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
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    y  &  \operatorname{d}x\ (\operatorname{d}y) & + &
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(x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
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(x, y)  &  \operatorname{d}x\  \operatorname{d}y  \\
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    (y)  &  \operatorname{d}x\ (\operatorname{d}y) & + &
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  x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
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(x, y)  &  \operatorname{d}x\  \operatorname{d}y  \\
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    y  &  \operatorname{d}x\ (\operatorname{d}y) & + &
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  x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
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((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
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\end{matrix}$
 +
&
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$\begin{matrix}
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(y) & \partial x & + & (x) & \partial y \\
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y  & \partial x & + & (x) & \partial y \\
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(y) & \partial x & + &  x  & \partial y \\
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y  & \partial x & + &  x  & \partial y \\
 +
\end{matrix}$ \\
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\hline
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$\begin{matrix}
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f_{3}  \\
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f_{12} \\
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\end{matrix}$
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&
 +
$\begin{matrix}
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(x) \\
 +
x  \\
 +
\end{matrix}$
 +
&
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$\begin{matrix}
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\operatorname{d}x\ (\operatorname{d}y) & + &
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\operatorname{d}x\  \operatorname{d}y  \\
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\operatorname{d}x\ (\operatorname{d}y) & + &
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\operatorname{d}x\  \operatorname{d}y  \\
 +
\end{matrix}$
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&
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$\begin{matrix}
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\partial x \\
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\partial x \\
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\end{matrix}$ \\
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\hline
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$\begin{matrix}
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f_{6} \\
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f_{9} \\
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\end{matrix}$
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&
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$\begin{matrix}
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(x, & y)  \\
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((x, & y)) \\
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\end{matrix}$
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&
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$\begin{matrix}
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\operatorname{d}x\ (\operatorname{d}y) & + &
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(\operatorname{d}x)\ \operatorname{d}y  \\
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\operatorname{d}x\ (\operatorname{d}y) & + &
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(\operatorname{d}x)\ \operatorname{d}y  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
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\partial x & + & \partial y \\
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\partial x & + & \partial y \\
 +
\end{matrix}$ \\
 +
\hline
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$\begin{matrix}
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f_{5}  \\
 +
f_{10} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
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(y) \\
 +
y  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
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(\operatorname{d}x)\ \operatorname{d}y & + &
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\operatorname{d}x\  \operatorname{d}y \\
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(\operatorname{d}x)\ \operatorname{d}y & + &
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\operatorname{d}x\  \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
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\partial y \\
 +
\partial y \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{7}  \\
 +
f_{11} \\
 +
f_{13} \\
 +
f_{14} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(x  &  y)  \\
 +
(x  & (y)) \\
 +
((x) &  y)  \\
 +
((x) & (y)) \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
    y  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 +
  x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
 +
((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
 +
    (y)  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 +
  x      & (\operatorname{d}x)\ \operatorname{d}y  & + &
 +
(x, y)  &  \operatorname{d}x\  \operatorname{d}y  \\
 +
    y  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 +
(x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
 +
(x, y)  &  \operatorname{d}x\  \operatorname{d}y  \\
 +
    (y)  &  \operatorname{d}x\ (\operatorname{d}y) & + &
 +
(x)    & (\operatorname{d}x)\ \operatorname{d}y  & + &
 +
((x, y)) &  \operatorname{d}x\  \operatorname{d}y  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
y  & \partial x & + &  x  & \partial y \\
 +
(y) & \partial x & + &  x  & \partial y \\
 +
y  & \partial x & + & (x) & \partial y \\
 +
(y) & \partial x & + & (x) & \partial y \\
 +
\end{matrix}$ \\
 +
\hline
 +
$f_{15}$ &
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$((~))$  &
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$0$      &
 +
$0$      \\
 +
\hline
 +
\end{tabular}
 +
</pre>
 +
 
 +
=Appendix 3 @ PlanetMath : TeX Format=
 +
 
 +
<pre>
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\PMlinkescapephrase{image}
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\PMlinkescapephrase{Image}
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\PMlinkescapephrase{mode}
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\PMlinkescapephrase{Mode}
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\textbf{Note.} The following Tables are best viewed in the Page Image mode.
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\tableofcontents
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\subsection{Detail of Calculation for the Difference Map}
    
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
 
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
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\multicolumn{5}{c}{\textbf{Table A7.  Detail of Calculation for} $\operatorname{D}f = \operatorname{E}f + f$} \\[6pt]
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\multicolumn{5}{c}{\textbf{Detail of Calculation for} $\operatorname{D}f = \operatorname{E}f + f$} \\[6pt]
 
\hline\hline
 
\hline\hline
 
&
 
&
Jon Awbrey
12,080

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