# 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 ${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\\\end{matrix}}$ & ${\begin{matrix}(x)(y)\\(x)~y\\x~(y)\\x~~y\\\end{matrix}}$ & ${\begin{matrix}(y)\\y\\(y)\\y\\\end{matrix}}$ & ${\begin{matrix}(x)\\(x)\\x\\x\\\end{matrix}}$ & ${\begin{matrix}(y)&\operatorname {d} x&+&(x)&\operatorname {d} y\\y&\operatorname {d} x&+&(x)&\operatorname {d} y\\(y)&\operatorname {d} x&+&x&\operatorname {d} y\\y&\operatorname {d} x&+&x&\operatorname {d} y\\\end{matrix}}$ & ${\begin{matrix}~\\~\\~\\~\\\end{matrix}}$ & ${\begin{matrix}~\\~\\~\\~\\\end{matrix}}$ \\ \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}