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

Revision as of 14:56, 19 June 2008

677 bytes removed ,  14:56, 19 June 2008
→‎Expanded on an Algebraic Basis: finish table
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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%"
−
|+ '''Differential Forms Expanded on a Logical Basis'''
+
|+ '''Differential Forms Expanded on an Algebraic Basis'''
 
|- style="background:ghostwhite; height:36px"
 
|- style="background:ghostwhite; height:36px"
 
|  
 
|  
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
    (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
+
(y) & \operatorname{d}x & + &
−
(x)     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
(x) & \operatorname{d}y & + &
−
((x, y)) &  \operatorname{d}x\ \operatorname{d}y \\
+
      \operatorname{d}x\ \operatorname{d}y \\
−
    y   & \operatorname{d}x\ (\operatorname{d}y) & + &
+
y & \operatorname{d}x & + &
−
(x)     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
(x) & \operatorname{d}y & + &
−
(x, y)  &  \operatorname{d}x\ \operatorname{d}y \\
+
      \operatorname{d}x\ \operatorname{d}y \\
−
    (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
+
(y) & \operatorname{d}x & + &
−
  x     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
x & \operatorname{d}y & + &
−
(x, y)  &  \operatorname{d}x\ \operatorname{d}y \\
+
      \operatorname{d}x\ \operatorname{d}y \\
−
    y   & \operatorname{d}x\ (\operatorname{d}y) & + &
+
y & \operatorname{d}x & + &
−
  x     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
x & \operatorname{d}y & + &
−
((x, 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) & \partial x & + & (x) & \partial y \\
+
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
−
  y  & \partial x & + & (x) & \partial y \\
+
  y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
−
(y) & \partial x & + &  x  & \partial y \\
+
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
−
  y  & \partial x & + &  x  & \partial y \\
+
  y  & \operatorname{d}x & + &  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}x\  \operatorname{d}y  \\
+
\operatorname{d}x \\
−
\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}
−
\partial x \\
+
\operatorname{d}x \\
−
\partial x \\
+
\operatorname{d}x \\
 
\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 \\
−
\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}
−
\partial x & + & \partial y \\
+
\operatorname{d}x & + & \operatorname{d}y \\
−
\partial x & + & \partial 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}y \\
−
\operatorname{d}x\  \operatorname{d}y \\
+
\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}
−
\partial y \\
+
\operatorname{d}y \\
−
\partial y \\
+
\operatorname{d}y \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
    y   & \operatorname{d}x\ (\operatorname{d}y) & + &
+
y & \operatorname{d}x & + &
−
  x     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
x & \operatorname{d}y & + &
−
((x, y)) &  \operatorname{d}x\ \operatorname{d}y \\
+
      \operatorname{d}x\ \operatorname{d}y \\
−
    (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
+
(y) & \operatorname{d}x & + &
−
  x     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
x & \operatorname{d}y & + &
−
(x, y)  &  \operatorname{d}x\ \operatorname{d}y \\
+
      \operatorname{d}x\ \operatorname{d}y \\
−
    y   & \operatorname{d}x\ (\operatorname{d}y) & + &
+
y & \operatorname{d}x & + &
−
(x)     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
(x) & \operatorname{d}y & + &
−
(x, y)  &  \operatorname{d}x\ \operatorname{d}y \\
+
      \operatorname{d}x\ \operatorname{d}y \\
−
    (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
+
(y) & \operatorname{d}x & + &
−
(x)     & (\operatorname{d}x)\ \operatorname{d}y & + &
+
(x) & \operatorname{d}y & + &
−
((x, 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  & \partial x & + &  x  & \partial y \\
+
  y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
−
(y) & \partial x & + &  x  & \partial y \\
+
(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
−
  y  & \partial x & + & (x) & \partial y \\
+
  y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
−
(y) & \partial x & + & (x) & \partial y \\
+
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
Jon Awbrey
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