Talk:PlanetPhysics/Differential Propositional Calculus Appendix 2

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: differential propositional calculus : appendix 2 %%% Primary Category Code: 02. %%% Filename: DifferentialPropositionalCalculusAppendix2.tex %%% Version: 1 %%% Owner: Jon Awbrey %%% Author(s): Jon Awbrey %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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The actions of the difference operator $\operatorname{D}$ and the tangent operator $\operatorname{d}$ on the 16 propositional forms in two variables are shown in the Tables below.

Table A7 expands the resulting differential forms over a \textit{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 set consists of the \htmladdnormallink{singular propositions}{http://planetphysics.us/encyclopedia/PositiveProposition.html} in the first order \htmladdnormallink{differential variables}{http://planetphysics.us/encyclopedia/PositiveProposition.html}, indicating mutually exclusive and exhaustive \textit{cells} of the \htmladdnormallink{tangent universe}{http://planetphysics.us/encyclopedia/PositiveProposition.html} of discourse. Accordingly, this set of \htmladdnormallink{differential propositions}{http://planetphysics.us/encyclopedia/PositiveProposition.html} may also be referred to as the cell-basis, point-basis, or singular \htmladdnormallink{differential basis}{http://planetphysics.us/encyclopedia/PositiveProposition.html}. 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 an \textit{algebraic basis}:

\begin{center} $\{ 1,\ \operatorname{d}x,\ \operatorname{d}y,\ \operatorname{d}x\,\operatorname{d}y \}.$ \end{center}

This set consists of the \htmladdnormallink{positive propositions}{http://planetphysics.us/encyclopedia/PositiveProposition.html} in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.

\tableofcontents

\subsection{Table A7. Differential Forms Expanded on a Logical Basis}

\begin{center}\begin{tabular}{|c|c|c|c|} \multicolumn{4}{c}{\textbf{Table A7. Differential Forms Expanded on a Logical Basis}} \\ \hline & $f$ & $\operatorname{D}f$ & $\operatorname{d}f$ \\ \hline $f_{0}$ & $(~)$ & $0$ & $0$ \\ \hline $\begin{smallmatrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{smallmatrix}$ & $\begin{smallmatrix} (x) & (y) \\ (x) & y  \\ x & (y) \\ x &  y  \\ \end{smallmatrix}$ & $\begin{smallmatrix} (y) &  \operatorname{d}x\ (\operatorname{d}y) & + & (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + & ((x, y)) & \operatorname{d}x\  \operatorname{d}y  \\ y  &  \operatorname{d}x\ (\operatorname{d}y) & + & (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + & (x, y) &  \operatorname{d}x\  \operatorname{d}y  \\ (y) &  \operatorname{d}x\ (\operatorname{d}y) & + & x     & (\operatorname{d}x)\ \operatorname{d}y  & + & (x, y) &  \operatorname{d}x\  \operatorname{d}y  \\ y  &  \operatorname{d}x\ (\operatorname{d}y) & + & x     & (\operatorname{d}x)\ \operatorname{d}y  & + & ((x, y)) & \operatorname{d}x\  \operatorname{d}y  \\ \end{smallmatrix}$ & $\begin{smallmatrix} (y) & \partial x & + & (x) & \partial y \\ y & \partial x & + & (x) & \partial y \\ (y) & \partial x & + & x  & \partial y \\ y & \partial x & + &  x  & \partial y \\ \end{smallmatrix}$ \\ \hline $\begin{smallmatrix} f_{3} \\ f_{12} \\ \end{smallmatrix}$ & $\begin{smallmatrix} (x) \\ x \\ \end{smallmatrix}$ & $\begin{smallmatrix} \operatorname{d}x\ (\operatorname{d}y) & + & \operatorname{d}x\ \operatorname{d}y  \\ \operatorname{d}x\ (\operatorname{d}y) & + & \operatorname{d}x\ \operatorname{d}y  \\ \end{smallmatrix}$ & $\begin{smallmatrix} \partial x \\ \partial x \\ \end{smallmatrix}$ \\ \hline $\begin{smallmatrix} f_{6} \\ f_{9} \\ \end{smallmatrix}$ & $\begin{smallmatrix} (x, & y) \\ ((x, & y)) \\ \end{smallmatrix}$ & $\begin{smallmatrix} \operatorname{d}x\ (\operatorname{d}y) & + & (\operatorname{d}x)\ \operatorname{d}y \\ \operatorname{d}x\ (\operatorname{d}y) & + & (\operatorname{d}x)\ \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} \partial x & + & \partial y \\ \partial x & + & \partial y \\ \end{smallmatrix}$ \\ \hline $\begin{smallmatrix} f_{5} \\ f_{10} \\ \end{smallmatrix}$ & $\begin{smallmatrix} (y) \\ y \\ \end{smallmatrix}$ & $\begin{smallmatrix} (\operatorname{d}x)\ \operatorname{d}y & + & \operatorname{d}x\ \operatorname{d}y \\ (\operatorname{d}x)\ \operatorname{d}y & + & \operatorname{d}x\ \operatorname{d}y \\ \end{smallmatrix}$ & $\begin{smallmatrix} \partial y \\ \partial y \\ \end{smallmatrix}$ \\ \hline $\begin{smallmatrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{smallmatrix}$ & $\begin{smallmatrix} (x &  y)  \\ (x & (y)) \\ ((x) & y)  \\ ((x) & (y)) \\ \end{smallmatrix}$ & $\begin{smallmatrix} y  &  \operatorname{d}x\ (\operatorname{d}y) & + & x     & (\operatorname{d}x)\ \operatorname{d}y  & + & ((x, y)) & \operatorname{d}x\  \operatorname{d}y  \\ (y) &  \operatorname{d}x\ (\operatorname{d}y) & + & x     & (\operatorname{d}x)\ \operatorname{d}y  & + & (x, y) &  \operatorname{d}x\  \operatorname{d}y  \\ y  &  \operatorname{d}x\ (\operatorname{d}y) & + & (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + & (x, y) &  \operatorname{d}x\  \operatorname{d}y  \\ (y) &  \operatorname{d}x\ (\operatorname{d}y) & + & (x)    & (\operatorname{d}x)\ \operatorname{d}y  & + & ((x, y)) & \operatorname{d}x\  \operatorname{d}y  \\ \end{smallmatrix}$ & $\begin{smallmatrix} y & \partial x & + &  x  & \partial y \\ (y) & \partial x & + & x  & \partial y \\ y & \partial x & + & (x) & \partial y \\ (y) & \partial x & + & (x) & \partial y \\ \end{smallmatrix}$ \\ \hline $f_{15}$ & $((~))$ & $0$ & $0$ \\ \hline \end{tabular}\end{center}

\subsection{Table A8. Differential Forms Expanded on an Algebraic Basis}

\begin{center}\begin{tabular}{|c|c|c|c|} \multicolumn{4}{c}{\textbf{Table A8. 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 $f_{15}$ & $((~))$ & $0$ & $0$ \\ \hline \end{tabular}\end{center}

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