# differential propositional calculus : appendix 3

## 0.1 Taylor Series Expansion

Taylor Series Expansion $\operatorname{D}f=\operatorname{d}f+\operatorname{d}^{2}f$
$\begin{matrix}\operatorname{d}f=\\ \partial_{x}f\cdot\operatorname{d}x\ +\ \partial_{y}f\cdot\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}^{2}f=\\ \partial_{xy}f\cdot\operatorname{d}x\,\operatorname{d}y\\ \end{matrix}$ $\operatorname{d}f|_{x\ y}$ $\operatorname{d}f|_{x\ (y)}$ $\operatorname{d}f|_{(x)\ y}$ $\operatorname{d}f|_{(x)(y)}$
$f_{0}$ $0$ $0$ $0$ $0$ $0$ $0$
$\begin{matrix}f_{1}\\ f_{2}\\ f_{4}\\ f_{8}\\ \end{matrix}$ $\begin{matrix}(y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&x&\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\ \operatorname{d}y\\ \operatorname{d}x\ \operatorname{d}y\\ \operatorname{d}x\ \operatorname{d}y\\ \operatorname{d}x\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}0\\ \operatorname{d}x\\ \operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ 0\\ \operatorname{d}x+\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ 0\\ \operatorname{d}x\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}y\\ \operatorname{d}x\\ 0\\ \end{matrix}$
$\begin{matrix}f_{3}\\ f_{12}\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{matrix}$ $\begin{matrix}0\\ 0\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{matrix}$
$\begin{matrix}f_{6}\\ f_{9}\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}0\\ 0\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$
$\begin{matrix}f_{5}\\ f_{10}\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}0\\ 0\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$
$\begin{matrix}f_{7}\\ f_{11}\\ f_{13}\\ f_{14}\\ \end{matrix}$ $\begin{matrix}y&\operatorname{d}x&+&x&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\ \operatorname{d}y\\ \operatorname{d}x\ \operatorname{d}y\\ \operatorname{d}x\ \operatorname{d}y\\ \operatorname{d}x\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}y\\ \operatorname{d}x\\ 0\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ 0\\ \operatorname{d}x\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ 0\\ \operatorname{d}x+\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}0\\ \operatorname{d}x\\ \operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$
$f_{15}$ $0$ $0$ $0$ $0$ $0$ $0$

## 0.2 Partial Differentials and Relative Differentials

Partial Differentials and Relative Differentials
$f$ $\frac{\partial f}{\partial x}$ $\frac{\partial f}{\partial y}$ $\begin{matrix}\operatorname{d}f=\\ \partial_{x}f\cdot\operatorname{d}x\ +\ \partial_{y}f\cdot\operatorname{d}y% \end{matrix}$ $\frac{\partial x}{\partial y}\big{|}f$ $\frac{\partial y}{\partial x}\big{|}f$
$f_{0}$ $(~{})$ $0$ $0$ $0$ $0$ $0$
$\begin{matrix}f_{1}\\ f_{2}\\ f_{4}\\ f_{8}\\ \end{matrix}$ $\begin{matrix}(x)(y)\\ (x)~{}y\\ x~{}(y)\\ x~{}~{}y\\ \end{matrix}$ $\begin{matrix}(y)\\ y\\ (y)\\ y\\ \end{matrix}$ $\begin{matrix}(x)\\ (x)\\ x\\ x\\ \end{matrix}$ $\begin{matrix}(y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&x&\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\\ \\ \\ \\ \end{matrix}$ $\begin{matrix}\\ \\ \\ \\ \end{matrix}$
$\begin{matrix}f_{3}\\ f_{12}\\ \end{matrix}$ $\begin{matrix}(x)\\ x\\ \end{matrix}$ $\begin{matrix}1\\ 1\\ \end{matrix}$ $\begin{matrix}0\\ 0\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{matrix}$ $\begin{matrix}\\ \\ \end{matrix}$ $\begin{matrix}\\ \\ \end{matrix}$
$\begin{matrix}f_{6}\\ f_{9}\\ \end{matrix}$ $\begin{matrix}(x,~{}y)\\ ((x,~{}y))\\ \end{matrix}$ $\begin{matrix}1\\ 1\\ \end{matrix}$ $\begin{matrix}1\\ 1\\ \end{matrix}$ $\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\\ \\ \end{matrix}$ $\begin{matrix}\\ \\ \end{matrix}$
$\begin{matrix}f_{5}\\ f_{10}\\ \end{matrix}$ $\begin{matrix}(y)\\ y\\ \end{matrix}$ $\begin{matrix}0\\ 0\\ \end{matrix}$ $\begin{matrix}1\\ 1\\ \end{matrix}$ $\begin{matrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\\ \\ \end{matrix}$ $\begin{matrix}\\ \\ \end{matrix}$
$\begin{matrix}f_{7}\\ f_{11}\\ f_{13}\\ f_{14}\\ \end{matrix}$ $\begin{matrix}(x~{}~{}y)\\ (x~{}(y))\\ ((x)~{}y)\\ ((x)(y))\\ \end{matrix}$ $\begin{matrix}y\\ (y)\\ y\\ (y)\\ \end{matrix}$ $\begin{matrix}x\\ x\\ (x)\\ (x)\\ \end{matrix}$ $\begin{matrix}y&\operatorname{d}x&+&x&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ \end{matrix}$ $\begin{matrix}\\ \\ \\ \\ \end{matrix}$ $\begin{matrix}\\ \\ \\ \\ \end{matrix}$
$f_{15}$ $((~{}))$ $0$ $0$ $0$ $0$ $0$
 Title differential propositional calculus : appendix 3 Canonical name DifferentialPropositionalCalculusAppendix3 Date of creation 2013-03-22 18:09:20 Last modified on 2013-03-22 18:09:20 Owner Jon Awbrey (15246) Last modified by Jon Awbrey (15246) Numerical id 10 Author Jon Awbrey (15246) Entry type Application Classification msc 53A40 Classification msc 39A12 Classification msc 34G99 Classification msc 03B44 Classification msc 03B42 Classification msc 03B05 Related topic DifferentialLogic Related topic MinimalNegationOperator Related topic PropositionalCalculus Related topic ZerothOrderLogic