PlanetPhysics/Differential Propositional Calculus Appendix 3

Taylor Series Expansion

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\multicolumn{7}{\htmladdnormallink{Taylor series {http://planetphysics.us/encyclopedia/TaylorFormula.html} Expansion $\mathrm{D}f=\mathrm{d}f+{\mathrm{d}}^{2}f$}} \\ \hline & $\begin{array}{c}\mathrm{d}f=\\ {\partial }_{x}f\cdot \mathrm{d}x+{\partial }_{y}f\cdot \mathrm{d}y\end{array}$ & $\begin{array}{c}{\mathrm{d}}^{2}f=\\ {\partial }_{xy}f\cdot \mathrm{d}x\mathrm{d}y\end{array}$ & $\mathrm{d}f{|}_{xy}$ & $\mathrm{d}f{|}_{x(y)}$ & $\mathrm{d}f{|}_{(x)y}$ & $\mathrm{d}f{|}_{(x)(y)}$ \\ \hline $f_0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline $\begin{array}{c}{f}_1\\ f_2\\ f_4\\ f_8\end{array}$ & $\begin{array}{ccccc}(y)& \mathrm{d}x& +& (x)& \mathrm{d}y\\ y& \mathrm{d}x& +& (x)& \mathrm{d}y\\ (y)& \mathrm{d}x& +& x& \mathrm{d}y\\ y& \mathrm{d}x& +& x& \mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}x\mathrm{d}y\\ \mathrm{d}x\mathrm{d}y\\ \mathrm{d}x\mathrm{d}y\\ \mathrm{d}x\mathrm{d}y\end{array}$ & $\begin{array}{c}0\\ \mathrm{d}x\\ \mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ 0\\ \mathrm{d}x+\mathrm{d}y\\ \mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\\ 0\\ \mathrm{d}x\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}y\\ \mathrm{d}x\\ 0\end{array}$ \\ \hline $\begin{array}{c}{f}_3\\ f_{12}\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ \mathrm{d}x\end{array}$ & $\begin{array}{c}0\\ 0\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ \mathrm{d}x\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ \mathrm{d}x\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ \mathrm{d}x\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ \mathrm{d}x\end{array}$ \\ \hline $\begin{array}{c}{f}_6\\ f_9\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ & $\begin{array}{c}0\\ 0\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ \\ \hline $\begin{array}{c}{f}_5\\ f_{10}\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}y\end{array}$ & $\begin{array}{c}0\\ 0\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}y\end{array}$ \\ \hline $\begin{array}{c}{f}_7\\ f_{11}\\ f_{13}\\ f_{14}\end{array}$ & $\begin{array}{ccccc}y& \mathrm{d}x& +& x& \mathrm{d}y\\ (y)& \mathrm{d}x& +& x& \mathrm{d}y\\ y& \mathrm{d}x& +& (x)& \mathrm{d}y\\ (y)& \mathrm{d}x& +& (x)& \mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}x\mathrm{d}y\\ \mathrm{d}x\mathrm{d}y\\ \mathrm{d}x\mathrm{d}y\\ \mathrm{d}x\mathrm{d}y\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}y\\ \mathrm{d}x\\ 0\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\\ 0\\ \mathrm{d}x\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ 0\\ \mathrm{d}x+\mathrm{d}y\\ \mathrm{d}y\end{array}$ & $\begin{array}{c}0\\ \mathrm{d}x\\ \mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ \\ \hline $f_{15}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline

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Partial Differentials and Relative Differentials

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\multicolumn{7}{Partial Differentials and Relative Differentials } \\ \hline & $f$ & $\frac{\partial f}{\partial x}$ & $\frac{\partial f}{\partial y}$ & $\begin{array}{c}\mathrm{d}f=\\ {\partial }_{x}f\cdot \mathrm{d}x+{\partial }_{y}f\cdot \mathrm{d}y\end{array}$ & $\frac{\partial x}{\partial y}|f$ & $\frac{\partial y}{\partial x}|f$ \\ \hline $f_0$ & $()$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline $\begin{array}{c}{f}_1\\ f_2\\ f_4\\ f_8\end{array}$ & $\begin{array}{c}(x)(y)\\ (x)y\\ x(y)\\ xy\end{array}$ & $\begin{array}{c}(y)\\ y\\ (y)\\ y\end{array}$ & $\begin{array}{c}(x)\\ (x)\\ x\\ x\end{array}$ & $\begin{array}{ccccc}(y)& \mathrm{d}x& +& (x)& \mathrm{d}y\\ y& \mathrm{d}x& +& (x)& \mathrm{d}y\\ (y)& \mathrm{d}x& +& x& \mathrm{d}y\\ y& \mathrm{d}x& +& x& \mathrm{d}y\end{array}$ & $\begin{array}{c}\\ \\ \\ \end{array}$ & $\begin{array}{c}\\ \\ \\ \end{array}$ \\ \hline $\begin{array}{c}{f}_3\\ f_{12}\end{array}$ & $\begin{array}{c}(x)\\ x\end{array}$ & $\begin{array}{c}1\\ 1\end{array}$ & $\begin{array}{c}0\\ 0\end{array}$ & $\begin{array}{c}\mathrm{d}x\\ \mathrm{d}x\end{array}$ & $\begin{array}{c}\\ \end{array}$ & $\begin{array}{c}\\ \end{array}$ \\ \hline $\begin{array}{c}{f}_6\\ f_9\end{array}$ & $\begin{array}{c}(x,y)\\ ((x,y))\end{array}$ & $\begin{array}{c}1\\ 1\end{array}$ & $\begin{array}{c}1\\ 1\end{array}$ & $\begin{array}{c}\mathrm{d}x+\mathrm{d}y\\ \mathrm{d}x+\mathrm{d}y\end{array}$ & $\begin{array}{c}\\ \end{array}$ & $\begin{array}{c}\\ \end{array}$ \\ \hline $\begin{array}{c}{f}_5\\ f_{10}\end{array}$ & $\begin{array}{c}(y)\\ y\end{array}$ & $\begin{array}{c}0\\ 0\end{array}$ & $\begin{array}{c}1\\ 1\end{array}$ & $\begin{array}{c}\mathrm{d}y\\ \mathrm{d}y\end{array}$ & $\begin{array}{c}\\ \end{array}$ & $\begin{array}{c}\\ \end{array}$ \\ \hline $\begin{array}{c}{f}_7\\ f_{11}\\ f_{13}\\ f_{14}\end{array}$ & $\begin{array}{c}(xy)\\ (x(y))\\ ((x)y)\\ ((x)(y))\end{array}$ & $\begin{array}{c}y\\ (y)\\ y\\ (y)\end{array}$ & $\begin{array}{c}x\\ x\\ (x)\\ (x)\end{array}$ & $\begin{array}{ccccc}y& \mathrm{d}x& +& x& \mathrm{d}y\\ (y)& \mathrm{d}x& +& x& \mathrm{d}y\\ y& \mathrm{d}x& +& (x)& \mathrm{d}y\\ (y)& \mathrm{d}x& +& (x)& \mathrm{d}y\end{array}$ & $\begin{array}{c}\\ \\ \\ \end{array}$ & $\begin{array}{c}\\ \\ \\ \end{array}$ \\ \hline $f_{15}$ & $(())$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline

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