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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 05:20, 17 September 2013

11,044 bytes added ,  05:20, 17 September 2013
update
Line 9,275: Line 9,275:  
| style="border-right:none" | <math>\texttt{((~))}\!</math>
 
| style="border-right:none" | <math>\texttt{((~))}\!</math>
 
| style="border-left:4px double black" | <math>0\!</math>
 
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A9. Differential = Pointwise Linear Approximation to the Difference====
 +
 +
====Table A10. Taylor Series Expansion====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" |
 +
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{D}f
 +
\\
 +
= & \mathrm{d}f & + & \mathrm{d}^2\!f
 +
\\
 +
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}\!</math>
 +
| <math>\mathrm{d}f|_{x \, y}</math>
 +
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
 +
|-
 +
| style="border-right:none" | <math>f_0\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}\!</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>f_{15}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A11. Partial Differentials and Relative Differentials====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math>
 +
|- style="background:ghostwhite; height:50px"
 +
| &nbsp;
 +
| <math>f\!</math>
 +
| <math>\frac{\partial f}{\partial x}</math>
 +
| <math>\frac{\partial f}{\partial y}</math>
 +
|
 +
<p><math>\operatorname{d}f =</math></p>
 +
<p><math>\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y</math></p>
 +
| <math>\left. \frac{\partial x}{\partial y} \right| f</math>
 +
| <math>\left. \frac{\partial y}{\partial x} \right| f</math>
 +
|- style="height:36px"
 +
| <math>f_0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
f_{1} \\
 +
f_{2} \\
 +
f_{4} \\
 +
f_{8} \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(x) & (y) \\
 +
(x) &  y  \\
 +
x  & (y) \\
 +
x  &  y  \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(y) \\
 +
y  \\
 +
(y) \\
 +
y  \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(x) \\
 +
(x) \\
 +
x  \\
 +
x  \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(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{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
f_{3}  \\
 +
f_{12} \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(x) \\
 +
x  \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
1 \\
 +
1 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
0 \\
 +
0 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
f_{6} \\
 +
f_{9} \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(x, & y)  \\
 +
((x, & y)) \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
1 \\
 +
1 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
1 \\
 +
1 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}x & + & \operatorname{d}y \\
 +
\operatorname{d}x & + & \operatorname{d}y \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
f_{5}  \\
 +
f_{10} \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(y) \\
 +
y  \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
0 \\
 +
0 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
1 \\
 +
1 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
f_{7}  \\
 +
f_{11} \\
 +
f_{13} \\
 +
f_{14} \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
(x  &  y)  \\
 +
(x  & (y)) \\
 +
((x) &  y)  \\
 +
((x) & (y)) \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
y  \\
 +
(y) \\
 +
y  \\
 +
(y) \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
x  \\
 +
x  \\
 +
(x) \\
 +
(x) \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
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{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
~ \\
 +
~ \\
 +
~ \\
 +
~ \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|- style="height:36px"
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
|}
 
|}
Jon Awbrey
12,080

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