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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 16:18, 27 June 2008
, 16:18, 27 June 2008→Appendix 2 @ PlanetMath : TeX Format: update
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\PMlinkescapephrase{action}
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\PMlinkescapephrase{Actions}
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\PMlinkescapephrase{Algebraic}
\PMlinkescapephrase{Algebraic}
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The actions of the \PMlinkname{difference operator}{FiniteDifference} $\operatorname{D}$ and the \PMlinkname{tangent operator}{TangentMap} $\operatorname{d}$ on each of the 16 propositional forms on two variables are shown in the Tables below.
Table A7 expands the resulting differential forms over a so-called ``logical basis":
\begin{center}
$\{ (\operatorname{d}x)(\operatorname{d}y),\ \operatorname{d}x\,(\operatorname{d}y),\ (\operatorname{d}x)\,\operatorname{d}y,\ \operatorname{d}x\,\operatorname{d}y \}.$
\end{center}
This is a set of singular propositions indicating mutually exclusive and exhaustive ``cells" or coordinate points of the universe of discourse. For this reason, it may also be referred to as a cell-basis, point-basis, or singular basis. In this setting it is frequently convenient to use the following abbreviations:
\begin{center}
$\partial x = \operatorname{d}x\,(\operatorname{d}y)$ and $\partial y = (\operatorname{d}x)\,\operatorname{d}y.$
\end{center}
Table A8 expands the resulting differential forms over a so-called ``algebraic basis":
\begin{center}
$\{ 1,\ \operatorname{d}x,\ \operatorname{d}y,\ \operatorname{d}x\,\operatorname{d}y \}.$
\end{center}
\tableofcontents
\tableofcontents
\subsection{Differential Forms Expanded on a Logical Basis}
\subsection{Table A7. Differential Forms Expanded on a Logical Basis}
\begin{center}\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{Table A7. Differential Forms Expanded on a Logical Basis}} \\
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\end{tabular}\end{center}
\end{tabular}\end{center}
\subsection{Differential Forms Expanded on an Algebraic Basis}
\subsection{Table A8. Differential Forms Expanded on an Algebraic Basis}
\begin{center}\begin{tabular}{|c|c|c|c|}
\begin{center}\begin{tabular}{|c|c|c|c|}
\multicolumn{4}{c}{\textbf{Differential Forms Expanded on an Algebraic Basis}} \\
\multicolumn{4}{c}{\textbf{Table A8. Differential Forms Expanded on an Algebraic Basis}} \\
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\hline
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