differential propositional calculus : appendix 3


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 & \... ...name{d}y \\ y & \operatorname{d}x & + & x & \operatorname{d}y \\ \end{matrix}$ $ \begin{matrix} \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \op... ...}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) & \op... ...{d}y \\ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\ \end{matrix}$ $ \begin{matrix} \operatorname{d}x\ \operatorname{d}y \\ \operatorname{d}x\ \op... ...}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 & \... ...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) & \op... ...{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


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As of this snapshot date, this entry was owned by Jon Awbrey.