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

Revision as of 03:15, 22 June 2008

3,006 bytes added ,  03:15, 22 June 2008
→‎Appendix 3 @ PlanetMath : TeX Format: update
Line 1,533: Line 1,533:  
\hline
 
\hline
 
$f_{15}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
 
$f_{15}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
 +
\hline
 +
\end{tabular}\end{center}
 +
 +
\subsection{Partial Differentials and Relative Differentials}
 +
 +
\begin{center}\begin{tabular}{|c|c|c|c|c|c|c|}
 +
\multicolumn{7}{c}{\textbf{Partial Differentials and Relative Differentials}} \\
 +
\hline
 +
&
 +
$f$
 +
&
 +
$\frac{\partial f}{\partial x}$
 +
&
 +
$\frac{\partial f}{\partial y}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}f = \\
 +
\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y
 +
\end{matrix}$
 +
&
 +
$\frac{\partial x}{\partial y} \big| f$
 +
&
 +
$\frac{\partial y}{\partial x} \big| f$ \\
 +
\hline
 +
$f_0$ & $(~)$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{1} \\
 +
f_{2} \\
 +
f_{4} \\
 +
f_{8} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(x)(y) \\
 +
(x)~y  \\
 +
x~(y) \\
 +
x~~y  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(y) \\
 +
y  \\
 +
(y) \\
 +
y  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(x) \\
 +
(x) \\
 +
x  \\
 +
x  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{3}  \\
 +
f_{12} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(x) \\
 +
x  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
1 \\
 +
1 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
0 \\
 +
0 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{6} \\
 +
f_{9} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(x,~y)  \\
 +
((x,~y)) \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
1 \\
 +
1 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
1 \\
 +
1 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{5}  \\
 +
f_{10} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(y) \\
 +
y  \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
0 \\
 +
0 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
1 \\
 +
1 \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
\end{matrix}$ \\
 +
\hline
 +
$\begin{matrix}
 +
f_{7} \\
 +
f_{11} \\
 +
f_{13} \\
 +
f_{14} \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
(x~~y)  \\
 +
(x~(y)) \\
 +
((x)~y)  \\
 +
((x)(y)) \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
y  \\
 +
(y) \\
 +
y  \\
 +
(y) \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
x  \\
 +
x  \\
 +
(x) \\
 +
(x) \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
 +
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{matrix}$
 +
&
 +
$\begin{matrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{matrix}$ \\
 +
\hline
 +
$f_{15}$ & $((~))$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
 
\hline
 
\hline
 
\end{tabular}\end{center}
 
\end{tabular}\end{center}
Jon Awbrey
12,080

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