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MyWikiBiz, Author Your Legacy — Tuesday November 26, 2024
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<pre>
 
<pre>
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\PMlinkescapephrase{action}
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\PMlinkescapephrase{Action}
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\PMlinkescapephrase{actions}
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\PMlinkescapephrase{Actions}
 
\PMlinkescapephrase{algebraic}
 
\PMlinkescapephrase{algebraic}
 
\PMlinkescapephrase{Algebraic}
 
\PMlinkescapephrase{Algebraic}
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\PMlinkescapephrase{expanded}
 
\PMlinkescapephrase{expanded}
 
\PMlinkescapephrase{expanded}
 
\PMlinkescapephrase{expanded}
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\PMlinkescapephrase{expanded}
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\PMlinkescapephrase{expanded}
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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.
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Table A7 expands the resulting differential forms over a so-called ``logical basis":
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\begin{center}
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$\{ (\operatorname{d}x)(\operatorname{d}y),\ \operatorname{d}x\,(\operatorname{d}y),\ (\operatorname{d}x)\,\operatorname{d}y,\ \operatorname{d}x\,\operatorname{d}y \}.$
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\end{center}
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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:
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\begin{center}
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$\partial x = \operatorname{d}x\,(\operatorname{d}y)$ and $\partial y = (\operatorname{d}x)\,\operatorname{d}y.$
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\end{center}
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Table A8 expands the resulting differential forms over a so-called ``algebraic basis":
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\begin{center}
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$\{ 1,\ \operatorname{d}x,\ \operatorname{d}y,\ \operatorname{d}x\,\operatorname{d}y \}.$
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\end{center}
    
\tableofcontents
 
\tableofcontents
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\subsection{Differential Forms Expanded on a Logical Basis}
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\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}} \\
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\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}
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\subsection{Differential Forms Expanded on an Algebraic Basis}
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\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}} \\
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\multicolumn{4}{c}{\textbf{Table A8.  Differential Forms Expanded on an Algebraic Basis}} \\
 
\hline
 
\hline
 
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