Talk:PlanetPhysics/Differential Propositional Calculus Appendix 3

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: differential propositional calculus : appendix 3 %%% Primary Category Code: 02. %%% Filename: DifferentialPropositionalCalculusAppendix3.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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\subsection{Taylor Series Expansion}

\begin{center}\begin{tabular}{|c|c|c||c|c|c|c|} \multicolumn{7}{c}{\textbf{\htmladdnormallink{Taylor series}{http://planetphysics.us/encyclopedia/TaylorFormula.html} Expansion $\operatorname{D}f = \operatorname{d}f + \operatorname{d}^2 f$}} \\ \hline & $\begin{matrix} \operatorname{d}f = \\ \partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}^2 f = \\ \partial_{xy} f \cdot \operatorname{d}x\, \operatorname{d}y \\ \end{matrix}$ & $\operatorname{d}f|_{x\ y}$ & $\operatorname{d}f|_{x\ (y)}$ & $\operatorname{d}f|_{(x)\ y}$ & $\operatorname{d}f|_{(x)(y)}$ \\ \hline $f_0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline $\begin{matrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{matrix}$ & $\begin{matrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & x & \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} 0 \\ \operatorname{d}x \\ \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ 0 \\ \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ 0 \\ \operatorname{d}x \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \operatorname{d}x \\ 0 \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{3} \\ f_{12} \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ & $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{6} \\ f_{9} \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{5} \\ f_{10} \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{matrix}$ & $\begin{matrix} y & \operatorname{d}x & + & x & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \operatorname{d}x \\ 0 \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ 0 \\ \operatorname{d}x \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ 0 \\ \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} 0 \\ \operatorname{d}x \\ \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ \\ \hline $f_{15}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline \end{tabular}\end{center}

\subsection{Partial Differentials and Relative Differentials}

\begin{center}\begin{tabular}{|c|c|c|c|c|c|c|} \multicolumn{7}{c}{\textbf{Partial Differentials and Relative Differentials}} \\ \hline & $f$ & $\frac{\partial f}{\partial x}$ & $\frac{\partial f}{\partial y}$ & $\begin{matrix} \operatorname{d}f = \\ \partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y \end{matrix}$ & $\frac{\partial x}{\partial y} \big| f$ & $\frac{\partial y}{\partial x} \big| f$ \\ \hline $f_0$ & $(~)$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline $\begin{matrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{matrix}$ & $\begin{matrix} (x)(y) \\ (x)~y \\ x~(y) \\ x~~y \\ \end{matrix}$ & $\begin{matrix} (y) \\ y \\ (y) \\ y \\ \end{matrix}$ & $\begin{matrix} (x) \\ (x) \\ x \\ x \\ \end{matrix}$ & $\begin{matrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & x & \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{3} \\ f_{12} \\ \end{matrix}$ & $\begin{matrix} (x) \\ x \\ \end{matrix}$ & $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ & $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{6} \\ f_{9} \\ \end{matrix}$ & $\begin{matrix} (x,~y) \\ ((x,~y)) \\ \end{matrix}$ & $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ & $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{5} \\ f_{10} \\ \end{matrix}$ & $\begin{matrix} (y) \\ y \\ \end{matrix}$ & $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ & $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ & $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ \\ \hline $\begin{matrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{matrix}$ & $\begin{matrix} (x~~y) \\ (x~(y)) \\ ((x)~y) \\ ((x)(y)) \\ \end{matrix}$ & $\begin{matrix} y \\ (y) \\ y \\ (y) \\ \end{matrix}$ & $\begin{matrix} x \\ x \\ (x) \\ (x) \\ \end{matrix}$ & $\begin{matrix} y & \operatorname{d}x & + & x & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ & $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ \\ \hline $f_{15}$ & $((~))$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline \end{tabular}\end{center}

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