differential propositional calculus : appendix 3

0.1 Taylor Series Expansion

Taylor Series Expansion $\mathrm{D}\mathbf{}f\mathrm{=}\mathrm{d}\mathbf{}f\mathrm{+}{\mathrm{d}}^{\mathrm{2}}\mathbf{}f$
	$\begin{array}{c}\hfill \mathrm{d}f=\hfill \\ \hfill {\partial }_{x}f\cdot \mathrm{d}x+{\partial }_{y}f\cdot \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill {\mathrm{d}}^{2}f=\hfill \\ \hfill {\partial }_{xy}f\cdot \mathrm{d}x\mathrm{d}y\hfill \end{array}$	${\mathrm{d}f|}_{xy}$	${\mathrm{d}f|}_{x(y)}$	${\mathrm{d}f|}_{(x)y}$	${\mathrm{d}f|}_{(x)(y)}$
$f_0$	$0$	$0$	$0$	$0$	$0$	$0$
$\begin{array}{c}\hfill f_1\hfill \\ \hfill f_2\hfill \\ \hfill f_4\hfill \\ \hfill f_8\hfill \end{array}$		$\begin{array}{c}\hfill \mathrm{d}x\mathrm{d}y\hfill \\ \hfill \mathrm{d}x\mathrm{d}y\hfill \\ \hfill \mathrm{d}x\mathrm{d}y\hfill \\ \hfill \mathrm{d}x\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill 0\hfill \\ \hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill 0\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill 0\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}x\hfill \\ \hfill 0\hfill \end{array}$
$\begin{array}{c}\hfill f_3\hfill \\ \hfill f_{12}\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$
$\begin{array}{c}\hfill f_6\hfill \\ \hfill f_9\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$
$\begin{array}{c}\hfill f_5\hfill \\ \hfill f_{10}\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$
$\begin{array}{c}\hfill f_7\hfill \\ \hfill f_{11}\hfill \\ \hfill f_{13}\hfill \\ \hfill f_{14}\hfill \end{array}$		$\begin{array}{c}\hfill \mathrm{d}x\mathrm{d}y\hfill \\ \hfill \mathrm{d}x\mathrm{d}y\hfill \\ \hfill \mathrm{d}x\mathrm{d}y\hfill \\ \hfill \mathrm{d}x\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}x\hfill \\ \hfill 0\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill 0\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill 0\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill 0\hfill \\ \hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$
$f_{15}$	$0$	$0$	$0$	$0$	$0$	$0$

0.2 Partial Differentials and Relative Differentials

Partial Differentials and Relative Differentials
	$f$	$\frac{\partial f}{\partial x}$	$\frac{\partial f}{\partial y}$	$\begin{array}{c}\hfill \mathrm{d}f=\hfill \\ \hfill {\partial }_{x}f\cdot \mathrm{d}x+{\partial }_{y}f\cdot \mathrm{d}y\hfill \end{array}$	$\frac{\partial x}{\partial y}|f$	$\frac{\partial y}{\partial x}|f$
$f_0$	$()$	$0$	$0$	$0$	$0$	$0$
$\begin{array}{c}\hfill f_1\hfill \\ \hfill f_2\hfill \\ \hfill f_4\hfill \\ \hfill f_8\hfill \end{array}$	$\begin{array}{c}\hfill (x)(y)\hfill \\ \hfill (x)y\hfill \\ \hfill x(y)\hfill \\ \hfill xy\hfill \end{array}$	$\begin{array}{c}\hfill (y)\hfill \\ \hfill y\hfill \\ \hfill (y)\hfill \\ \hfill y\hfill \end{array}$	$\begin{array}{c}\hfill (x)\hfill \\ \hfill (x)\hfill \\ \hfill x\hfill \\ \hfill x\hfill \end{array}$		$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}$
$\begin{array}{c}\hfill f_3\hfill \\ \hfill f_{12}\hfill \end{array}$	$\begin{array}{c}\hfill (x)\hfill \\ \hfill x\hfill \end{array}$	$\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}$	$\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \end{array}$
$\begin{array}{c}\hfill f_6\hfill \\ \hfill f_9\hfill \end{array}$	$\begin{array}{c}\hfill (x,y)\hfill \\ \hfill ((x,y))\hfill \end{array}$	$\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}$	$\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x+\mathrm{d}y\hfill \\ \hfill \mathrm{d}x+\mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \end{array}$
$\begin{array}{c}\hfill f_5\hfill \\ \hfill f_{10}\hfill \end{array}$	$\begin{array}{c}\hfill (y)\hfill \\ \hfill y\hfill \end{array}$	$\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}$	$\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \end{array}$
$\begin{array}{c}\hfill f_7\hfill \\ \hfill f_{11}\hfill \\ \hfill f_{13}\hfill \\ \hfill f_{14}\hfill \end{array}$	$\begin{array}{c}\hfill (xy)\hfill \\ \hfill (x(y))\hfill \\ \hfill ((x)y)\hfill \\ \hfill ((x)(y))\hfill \end{array}$	$\begin{array}{c}\hfill y\hfill \\ \hfill (y)\hfill \\ \hfill y\hfill \\ \hfill (y)\hfill \end{array}$	$\begin{array}{c}\hfill x\hfill \\ \hfill x\hfill \\ \hfill (x)\hfill \\ \hfill (x)\hfill \end{array}$		$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}$	$\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}$
$f_{15}$	$(())$	$0$	$0$	$0$	$0$	$0$

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