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[parent] differential propositional calculus : appendix 3 (Application)


Contents

Taylor Series Expansion

<</SPAN>#60#>Taylor Series Expansion $\operatorname{D}f = \operatorname{d}f + \operatorname{d}^2 f$
  $\begin{matrix} \operatorname{d}f = \\ \partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}^2 f = \\ \partial_{xy} f \cdot \operatorname{d}x\, \operatorname{d}y \\ \end{matrix}$ $\operatorname{d}f|_{x\ y}$ $\operatorname{d}f|_{x\ (y)}$ $\operatorname{d}f|_{(x)\ y}$ $\operatorname{d}f|_{(x)(y)}$
$f_0$ $0$ $0$ $0$ $0$ $0$ $0$
$\begin{matrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{matrix}$ $\begin{matrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & x & \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} 0 \\ \operatorname{d}x \\ \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ 0 \\ \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ 0 \\ \operatorname{d}x \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \operatorname{d}x \\ 0 \\ \end{matrix}$
$\begin{matrix} f_{3} \\ f_{12} \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$
$\begin{matrix} f_{6} \\ f_{9} \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$
$\begin{matrix} f_{5} \\ f_{10} \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$
$\begin{matrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{matrix}$ $\begin{matrix} y & \operatorname{d}x & + & x & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \operatorname{d}x \\ 0 \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ 0 \\ \operatorname{d}x \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ 0 \\ \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} 0 \\ \operatorname{d}x \\ \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$
$f_{15}$ $0$ $0$ $0$ $0$ $0$ $0$

Partial Differentials and Relative Differentials

Partial Differentials and Relative Differentials
  $f$ $\frac{\partial f}{\partial x}$ $\frac{\partial f}{\partial y}$ $\begin{matrix} \operatorname{d}f = \\ \partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y \end{matrix}$ $\frac{\partial x}{\partial y} \big| f$ $\frac{\partial y}{\partial x} \big| f$
$f_0$ $(~)$ $0$ $0$ $0$ $0$ $0$
$\begin{matrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{matrix}$ $\begin{matrix} (x)(y) \\ (x)~y \\ x~(y) \\ x~~y \\ \end{matrix}$ $\begin{matrix} (y) \\ y \\ (y) \\ y \\ \end{matrix}$ $\begin{matrix} (x) \\ (x) \\ x \\ x \\ \end{matrix}$ $\begin{matrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & x & \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$
$\begin{matrix} f_{3} \\ f_{12} \\ \end{matrix}$ $\begin{matrix} (x) \\ x \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x \\ \operatorname{d}x \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$
$\begin{matrix} f_{6} \\ f_{9} \\ \end{matrix}$ $\begin{matrix} (x,~y) \\ ((x,~y)) \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \\ \operatorname{d}x + \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$
$\begin{matrix} f_{5} \\ f_{10} \\ \end{matrix}$ $\begin{matrix} (y) \\ y \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} \operatorname{d}y \\ \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$
$\begin{matrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{matrix}$ $\begin{matrix} (x~~y) \\ (x~(y)) \\ ((x)~y) \\ ((x)(y)) \\ \end{matrix}$ $\begin{matrix} y \\ (y) \\ y \\ (y) \\ \end{matrix}$ $\begin{matrix} x \\ x \\ (x) \\ (x) \\ \end{matrix}$ $\begin{matrix} y & \operatorname{d}x & + & x & \operatorname{d}y \\ (y) & \operatorname{d}x & + & x & \operatorname{d}y \\ y & \operatorname{d}x & + & (x) & \operatorname{d}y \\ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$
$f_{15}$ $((~))$ $0$ $0$ $0$ $0$ $0$



"differential propositional calculus : appendix 3" is owned by Jon Awbrey.
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See Also: differential logic, minimal negation operator, propositional calculus, zeroth order logic


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Cross-references: Taylor series

This is version 7 of differential propositional calculus : appendix 3, born on 2008-06-19, modified 2008-06-21.
Object id is 10712, canonical name is DifferentialPropositionalCalculusAppendix3.
Accessed 1027 times total.

Classification:
AMS MSC: 03B05 (Mathematical logic and foundations :: General logic :: Classical propositional logic)
 03B42 (Mathematical logic and foundations :: General logic :: Logic of knowledge and belief)
 03B44 (Mathematical logic and foundations :: General logic :: Temporal logic)
 34G99 (Ordinary differential equations :: Differential equations in abstract spaces :: Miscellaneous)
 39A12 (Difference and functional equations :: Difference equations :: Discrete version of topics in analysis)
 53A40 (Differential geometry :: Classical differential geometry :: Other special differential geometries)
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