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.