differential propositional calculus : appendix 3

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differential propositional calculus : appendix 3

Contents:

0.1 Taylor Series Expansion
0.2 Partial Differentials and Relative Differentials

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
	$f$	$\frac{\partial f}{\partial x}$	$\frac{\partial f}{\partial y}$	$\begin{matrix}\operatorname{d}f=\\ \partial_{x}f\cdot\operatorname{d}x\ +\ \partial_{y}f\cdot\operatorname{d}y% \end{matrix}$	$\frac{\partial x}{\partial y}\big\|f$	$\frac{\partial y}{\partial x}\big\|f$
$f_{0}$	$(~{})$	$0$	$0$	$0$	$0$	$0$
$\begin{matrix}f_{{1}}\\ f_{{2}}\\ f_{{4}}\\ f_{{8}}\\ \end{matrix}$	$\begin{matrix}(x)(y)\\ (x)~{}y\\ x~{}(y)\\ x~{}~{}y\\ \end{matrix}$	$\begin{matrix}(y)\\ y\\ (y)\\ y\\ \end{matrix}$	$\begin{matrix}(x)\\ (x)\\ x\\ x\\ \end{matrix}$	$\begin{matrix}(y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&x&\operatorname{d}y\\ \end{matrix}$	$\begin{matrix}\\ \\ \\ \\ \end{matrix}$	$\begin{matrix}\\ \\ \\ \\ \end{matrix}$
$\begin{matrix}f_{{3}}\\ f_{{12}}\\ \end{matrix}$	$\begin{matrix}(x)\\ x\\ \end{matrix}$	$\begin{matrix}1\\ 1\\ \end{matrix}$	$\begin{matrix}0\\ 0\\ \end{matrix}$	$\begin{matrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{matrix}$	$\begin{matrix}\\ \\ \end{matrix}$	$\begin{matrix}\\ \\ \end{matrix}$
$\begin{matrix}f_{{6}}\\ f_{{9}}\\ \end{matrix}$	$\begin{matrix}(x,~{}y)\\ ((x,~{}y))\\ \end{matrix}$	$\begin{matrix}1\\ 1\\ \end{matrix}$	$\begin{matrix}1\\ 1\\ \end{matrix}$	$\begin{matrix}\operatorname{d}x+\operatorname{d}y\\ \operatorname{d}x+\operatorname{d}y\\ \end{matrix}$	$\begin{matrix}\\ \\ \end{matrix}$	$\begin{matrix}\\ \\ \end{matrix}$
$\begin{matrix}f_{{5}}\\ f_{{10}}\\ \end{matrix}$	$\begin{matrix}(y)\\ y\\ \end{matrix}$	$\begin{matrix}0\\ 0\\ \end{matrix}$	$\begin{matrix}1\\ 1\\ \end{matrix}$	$\begin{matrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{matrix}$	$\begin{matrix}\\ \\ \end{matrix}$	$\begin{matrix}\\ \\ \end{matrix}$
$\begin{matrix}f_{{7}}\\ f_{{11}}\\ f_{{13}}\\ f_{{14}}\\ \end{matrix}$	$\begin{matrix}(x~{}~{}y)\\ (x~{}(y))\\ ((x)~{}y)\\ ((x)(y))\\ \end{matrix}$	$\begin{matrix}y\\ (y)\\ y\\ (y)\\ \end{matrix}$	$\begin{matrix}x\\ x\\ (x)\\ (x)\\ \end{matrix}$	$\begin{matrix}y&\operatorname{d}x&+&x&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ \end{matrix}$	$\begin{matrix}\\ \\ \\ \\ \end{matrix}$	$\begin{matrix}\\ \\ \\ \\ \end{matrix}$
$f_{{15}}$	$((~{}))$	$0$	$0$	$0$	$0$	$0$

Major Section:

Reference

Type of Math Object:

Application

Parent:

differential propositional calculus

Mathematics Subject Classification

53A40 Other special differential geometries
39A12 Discrete version of topics in analysis
34G99 None of the above, but in MSC2010 section 34Gxx
03B44 Temporal logic
03B42 Logics of knowledge and belief (including belief change)
03B05 Classical propositional logic

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