PlanetPhysics/Differential Propositional Calculus Appendix 3

Taylor Series Expansion
\begin{tabular}{|c|c|c||c|c|c|c|} \multicolumn{7}{\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}

Partial Differentials and Relative Differentials
\begin{tabular}{|c|c|c|c|c|c|c|} \multicolumn{7}{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 $$\begin{matrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{matrix}$$ & $$\begin{matrix} (x)(y) \\ (x)~y \\ x~(y) \\ xy \\ \end{matrix}$$ & $$\begin{matrix} (y) \\ y \\ (y) \\ y \\ \end{matrix}$$ & $$\begin{matrix} (x) \\ (x) \\ x \\ x \\ \end{matrix}$$ & $$\begin{matrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x  & \operatorname{d}y \\ y & \operatorname{d}x & + &  x  & \operatorname{d}y \\ \end{matrix}$$ & $$\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$$ & $$\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$$ \\ \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 $$\begin{matrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{matrix}$$ & $$\begin{matrix} (xy) \\ (x~(y)) \\ ((x)~y) \\ ((x)(y)) \\ \end{matrix}$$ & $$\begin{matrix} y \\ (y) \\ y \\ (y) \\ \end{matrix}$$ & $$\begin{matrix} x \\ x \\ (x) \\ (x) \\ \end{matrix}$$ & $$\begin{matrix} y & \operatorname{d}x & + &  x  & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x  & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ \end{matrix}$$ & $$\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$$ & $$\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$$ \\ \hline $$f_{15}$$ & $$((~))$$ & $$0$$ & $$0$$ & $$0$$ & $$0$$ & $$0$$ \\ \hline \end{tabular}