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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 04:46, 21 June 2008

4,390 bytes added ,  04:46, 21 June 2008
→‎Appendix 3 @ PlanetMath : TeX Format: update
Line 1,304: Line 1,304:     
<pre>
 
<pre>
 +
\tableofcontents
    +
\subsection{Taylor Series Expansion}
 +
 +
\begin{center}\begin{tabular}{|c|c|c||c|c|c|c|}
 +
\multicolumn{7}{c}{\textbf{Taylor Series Expansion $\operatorname{D}f = \operatorname{d}f + \operatorname{d}^2 f$}} \\
 +
\hline
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}f = \\
 +
\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}^2 f = \\
 +
\partial_{xy} f \cdot \operatorname{d}x\, \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\operatorname{d}f|_{x\ y}$  &
 +
$\operatorname{d}f|_{x\ (y)}$ &
 +
$\operatorname{d}f|_{(x)\ y}$ &
 +
$\operatorname{d}f|_{(x)(y)}$ \\
 +
\hline
 +
$f_0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{1} \\
 +
f_{2} \\
 +
f_{4} \\
 +
f_{8} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
0 \\
 +
\operatorname{d}x \\
 +
\operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
0 \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
0 \\
 +
\operatorname{d}x \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\operatorname{d}x \\
 +
0 \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{3}  \\
 +
f_{12} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
0 \\
 +
0 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{6} \\
 +
f_{9} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
0 \\
 +
0 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{5}  \\
 +
f_{10} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
0 \\
 +
0 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{7} \\
 +
f_{11} \\
 +
f_{13} \\
 +
f_{14} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\operatorname{d}x\ \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\operatorname{d}x \\
 +
0 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
0 \\
 +
\operatorname{d}x \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
0 \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
0 \\
 +
\operatorname{d}x \\
 +
\operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$ \\
 +
\hline
 +
$f_{15}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
 +
\hline
 +
\end{tabular}\end{center}
 
</pre>
 
</pre>
  
Jon Awbrey
12,080

edits

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