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

Revision as of 19:20, 20 June 2008

66 bytes removed ,  19:20, 20 June 2008
→‎Taylor Series Expansion: finish table + cleanup
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
|+ '''Taylor Series Expansion'''
 
|+ '''Taylor Series Expansion'''
−
|- style="background:ghostwhite; height:36px"
+
|- style="background:ghostwhite"
 
|  
 
|  
 
|
 
|
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|
 
|
 
<p><math>\operatorname{d}^2 f = </math></p>
 
<p><math>\operatorname{d}^2 f = </math></p>
−
<p><math>\partial_{xy} f \cdot \operatorname{d}x \cdot \operatorname{d}y</math></p>
+
<p><math>\partial_{xy} f \cdot \operatorname{d}x\ \operatorname{d}y</math></p>
 
| <math>df|_{x\ y}\!</math>
 
| <math>df|_{x\ y}\!</math>
 
| <math>df|_{x\ (y)}\!</math>
 
| <math>df|_{x\ (y)}\!</math>
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\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
−
|}
  −
  −
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
  −
|+ '''Differential Forms Expanded on an Algebraic Basis'''
  −
|- style="background:ghostwhite; height:36px"
  −
| &nbsp;
  −
| <math>f\!</math>
  −
| <math>\operatorname{D}f</math>
  −
| <math>\operatorname{d}f</math>
  −
|- style="height:36px"
  −
| <math>f_{0}\!</math>
  −
| <math>(~)\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
   
|-
 
|-
 
|
 
|
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
f_{1} \\
+
f_{3} \\
−
f_{2} \\
+
f_{12} \\
−
f_{4} \\
  −
f_{8} \\
   
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
(x) & (y) \\
+
\operatorname{d}x \\
−
(x) &  y  \\
+
\operatorname{d}x \\
−
x  & (y) \\
  −
x &  y  \\
   
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
(y) & \operatorname{d}x & + &
+
0 \\
−
(x) & \operatorname{d}y & + &
+
0 \\
−
      \operatorname{d}x\ \operatorname{d}y \\
  −
y  & \operatorname{d}x & + &
  −
(x) & \operatorname{d}y & + &
  −
      \operatorname{d}x\ \operatorname{d}y \\
  −
(y) & \operatorname{d}x & + &
  −
x  & \operatorname{d}y & + &
  −
      \operatorname{d}x\ \operatorname{d}y \\
  −
y  & \operatorname{d}x & + &
  −
x  & \operatorname{d}y & + &
  −
      \operatorname{d}x\ \operatorname{d}y \\
   
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
+
\operatorname{d}x \\
−
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
+
\operatorname{d}x \\
−
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
  −
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
   
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
−
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
f_{3} \\
+
\operatorname{d}x \\
−
f_{12} \\
+
\operatorname{d}x \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
(x) \\
+
\operatorname{d}x \\
−
x \\
+
\operatorname{d}x \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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\operatorname{d}x \\
 
\operatorname{d}x \\
 
\operatorname{d}x \\
 
\operatorname{d}x \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
f_{6} \\
 +
f_{9} \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
\operatorname{d}x \\
+
\operatorname{d}x + \operatorname{d}y \\
−
\operatorname{d}x \\
+
\operatorname{d}x + \operatorname{d}y \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
0 \\
 +
0 \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
−
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
f_{6} \\
+
\operatorname{d}x + \operatorname{d}y \\
−
f_{9} \\
+
\operatorname{d}x + \operatorname{d}y \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
(x, & y)  \\
+
\operatorname{d}x + \operatorname{d}y \\
−
((x, & y)) \\
+
\operatorname{d}x + \operatorname{d}y \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
\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{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
\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{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
(y) \\
+
\operatorname{d}y \\
−
y \\
+
\operatorname{d}y \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
0 \\
 +
0 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}y \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
f_{7} \\
+
f_{7} \\
 
f_{11} \\
 
f_{11} \\
 
f_{13} \\
 
f_{13} \\
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
  (x  & y)  \\
+
  y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
−
  (x  & (y)) \\
+
(y) & \operatorname{d}x & + & x  & \operatorname{d}y \\
−
((x) & y)  \\
+
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
−
((x) & (y)) \\
+
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\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{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\operatorname{d}x \\
 +
0 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
0 \\
 +
\operatorname{d}x \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
y  & \operatorname{d}x & + &
+
\operatorname{d}x \\
−
x  & \operatorname{d}y & + &
+
0 \\
−
      \operatorname{d}x\ \operatorname{d}y \\
+
\operatorname{d}x + \operatorname{d}y \\
−
(y) & \operatorname{d}x & + &
+
\operatorname{d}y \\
−
x  & \operatorname{d}y & + &
  −
      \operatorname{d}x\ \operatorname{d}y \\
  −
y  & \operatorname{d}x & + &
  −
(x) & \operatorname{d}y & + &
  −
      \operatorname{d}x\ \operatorname{d}y \\
  −
(y) & \operatorname{d}x & + &
  −
(x) & \operatorname{d}y & + &
  −
      \operatorname{d}x\ \operatorname{d}y \\
   
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
+
0 \\
−
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
+
\operatorname{d}x \\
−
y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
+
\operatorname{d}y \\
−
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
+
\operatorname{d}x + \operatorname{d}y \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
−
|- style="height:36px"
+
|-
 
| <math>f_{15}\!</math>
 
| <math>f_{15}\!</math>
−
| <math>((~))\!</math>
+
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
|}
 
|}
−
<br>
 
Jon Awbrey
12,080

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