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| <pre> | | <pre> |
| + | \PMlinkescapephrase{action} |
| + | \PMlinkescapephrase{Action} |
| + | \PMlinkescapephrase{actions} |
| + | \PMlinkescapephrase{Actions} |
| \PMlinkescapephrase{algebraic} | | \PMlinkescapephrase{algebraic} |
| \PMlinkescapephrase{Algebraic} | | \PMlinkescapephrase{Algebraic} |
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| \PMlinkescapephrase{expanded} | | \PMlinkescapephrase{expanded} |
| \PMlinkescapephrase{expanded} | | \PMlinkescapephrase{expanded} |
| + | \PMlinkescapephrase{expanded} |
| + | \PMlinkescapephrase{expanded} |
| + | |
| + | 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} |
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| \tableofcontents | | \tableofcontents |
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− | \subsection{Differential Forms Expanded on a Logical Basis} | + | \subsection{Table A7. Differential Forms Expanded on a Logical Basis} |
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| \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}} \\ |
| \hline | | \hline |
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| \end{tabular}\end{center} | | \end{tabular}\end{center} |
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− | \subsection{Differential Forms Expanded on an Algebraic Basis} | + | \subsection{Table A8. Differential Forms Expanded on an Algebraic Basis} |
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| \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}} \\ |
| \hline | | \hline |
| & | | & |