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


Contents

Taylor Series Expansion

Taylor series Expansion $\operatorname{D}f = \operatorname{d}f + \operatorname{d}^2 f$
  $\begin{matrix} \operatorname{d}f = \ \partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}^2 f = \ \partial_{xy} f \cdot \operatorname{d}x\, \operatorname{d}y \ \end{matrix}$ $\operatorname{d}f\vert _{x\ y}$ $\operatorname{d}f\vert _{x\ (y)}$ $\operatorname{d}f\vert _{(x)\ y}$ $\operatorname{d}f\vert _{(x)(y)}$
$f_0$ 0 0 0 0 0 0
$\begin{matrix} f_{1} \ f_{2} \ f_{4} \ f_{8} \ \end{matrix}$ $\begin{matrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \ y & \o... ...name{d}y \ y & \operatorname{d}x & + & x & \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}x\ \operatorname{d}y \ \operatorname{d}x\ \ope... ...}x\ \operatorname{d}y \ \operatorname{d}x\ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} 0 \ \operatorname{d}x \ \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}x \ 0 \ \operatorname{d}x + \operatorname{d}y \ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ 0 \ \operatorname{d}x \ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \ \operatorname{d}y \ \operatorname{d}x \ 0 \ \end{matrix}$
$\begin{matrix} f_{3} \ f_{12} \ \end{matrix}$ $\begin{matrix} \operatorname{d}x \ \operatorname{d}x \ \end{matrix}$ $\begin{matrix} 0 \ 0 \ \end{matrix}$ $\begin{matrix} \operatorname{d}x \ \operatorname{d}x \ \end{matrix}$ $\begin{matrix} \operatorname{d}x \ \operatorname{d}x \ \end{matrix}$ $\begin{matrix} \operatorname{d}x \ \operatorname{d}x \ \end{matrix}$ $\begin{matrix} \operatorname{d}x \ \operatorname{d}x \ \end{matrix}$
$\begin{matrix} f_{6} \ f_{9} \ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ \end{matrix}$ $\begin{matrix} 0 \ 0 \ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ \end{matrix}$
$\begin{matrix} f_{5} \ f_{10} \ \end{matrix}$ $\begin{matrix} \operatorname{d}y \ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} 0 \ 0 \ \end{matrix}$ $\begin{matrix} \operatorname{d}y \ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}y \ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}y \ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}y \ \operatorname{d}y \ \end{matrix}$
$\begin{matrix} f_{7} \ f_{11} \ f_{13} \ f_{14} \ \end{matrix}$ $\begin{matrix} y & \operatorname{d}x & + & x & \operatorname{d}y \ (y) & \ope... ...{d}y \ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}x\ \operatorname{d}y \ \operatorname{d}x\ \ope... ...}x\ \operatorname{d}y \ \operatorname{d}x\ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} \operatorname{d}x + \operatorname{d}y \ \operatorname{d}y \ \operatorname{d}x \ 0 \ \end{matrix}$ $\begin{matrix} \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ 0 \ \operatorname{d}x \ \end{matrix}$ $\begin{matrix} \operatorname{d}x \ 0 \ \operatorname{d}x + \operatorname{d}y \ \operatorname{d}y \ \end{matrix}$ $\begin{matrix} 0 \ \operatorname{d}x \ \operatorname{d}y \ \operatorname{d}x + \operatorname{d}y \ \end{matrix}$
$f_{15}$ 0 0 0 0 0 0

Partial Differentials and Relative Differentials

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


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Cross-references: Taylor series

This is version 1 of differential propositional calculus : appendix 3, born on 2009-05-25.
Object id is 780, canonical name is DifferentialPropositionalCalculusAppendix3.
Accessed 359 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
 02.10.Ab (Logic and set theory)
 02.10.Ox (Combinatorics; graph theory)
 02.10.Ud (Linear algebra)
 02.20.-a (Group theory )
 02.30.-f (Function theory, analysis)
 02.40.-k (Geometry, differential geometry, and topology )
 02.40.Yy (Geometric mechanics )
 02.50.Tt (Inference methods)
 02.70.-c (Computational techniques )
 02.70.Bf (Finite-difference methods)
 02.70.Wz (Symbolic computation )
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