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

Revision as of 05:06, 22 June 2008

829 bytes added ,  05:06, 22 June 2008
→‎Partial Differentials and Relative Differentials: add 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 an Algebraic Basis'''
+
|+ '''Partial Differentials and Relative Differentials'''
−
|- style="background:ghostwhite; height:36px"
+
|- style="background:ghostwhite; height:48px"
 
|  
 
|  
 
| <math>f\!</math>
 
| <math>f\!</math>
−
| <math>\operatorname{D}f</math>
+
| <math>\frac{\partial f}{\partial x}</math>
−
| <math>\operatorname{d}f</math>
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| <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"
 
|- style="height:36px"
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| <math>f_{0}\!</math>
+
| <math>f_0\!</math>
 
| <math>(~)\!</math>
 
| <math>(~)\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
 +
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
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|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
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(y) & \operatorname{d}x & + &
+
(y) \\
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(x) & \operatorname{d}y & + &
+
y \\
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      \operatorname{d}x\ \operatorname{d}y \\
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(y) \\
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  y  & \operatorname{d}x & + &
+
  y  \\
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(x) & \operatorname{d}y & + &
+
\end{smallmatrix}</math>
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      \operatorname{d}x\ \operatorname{d}y \\
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|}
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(y) & \operatorname{d}x & + &
+
|
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x  & \operatorname{d}y & + &
+
{| align="center"
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      \operatorname{d}x\ \operatorname{d}y \\
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|
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  y & \operatorname{d}x & + &
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<math>\begin{smallmatrix}
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  x  & \operatorname{d}y & + &
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(x) \\
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      \operatorname{d}x\ \operatorname{d}y \\
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(x) \\
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  x \\
 +
  x  \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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(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>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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~ \\
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~ \\
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~ \\
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~ \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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~ \\
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~ \\
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~ \\
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~ \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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(x) \\
 
(x) \\
 
  x  \\
 
  x  \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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1 \\
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1 \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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|
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<math>\begin{smallmatrix}
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0 \\
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0 \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
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\operatorname{d}x \\
+
~ \\
−
\operatorname{d}x \\
+
~ \\
 +
\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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~ \\
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~ \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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  (x, & y)  \\
 
  (x, & y)  \\
 
((x, & y)) \\
 
((x, & y)) \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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1 \\
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1 \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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1 \\
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1 \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
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\operatorname{d}x & + & \operatorname{d}y \\
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~ \\
−
\operatorname{d}x & + & \operatorname{d}y \\
+
~ \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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~ \\
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~ \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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(y) \\
 
(y) \\
 
  y  \\
 
  y  \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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0 \\
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0 \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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1 \\
 +
1 \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
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\operatorname{d}y \\
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~ \\
−
\operatorname{d}y \\
+
~ \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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~ \\
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~ \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
−
  y  & \operatorname{d}x & + &
+
  y  \\
−
x  & \operatorname{d}y & + &
+
(y) \\
−
      \operatorname{d}x\ \operatorname{d}y \\
+
y \\
−
(y) & \operatorname{d}x & + &
+
(y) \\
−
x  & \operatorname{d}y & + &
+
\end{smallmatrix}</math>
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      \operatorname{d}x\ \operatorname{d}y \\
+
|}
−
  y & \operatorname{d}x & + &
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|
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(x) & \operatorname{d}y & + &
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{| align="center"
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      \operatorname{d}x\ \operatorname{d}y \\
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|
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(y) & \operatorname{d}x & + &
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<math>\begin{smallmatrix}
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(x) & \operatorname{d}y & + &
+
  x \\
−
      \operatorname{d}x\ \operatorname{d}y \\
+
x \\
 +
(x) \\
 +
(x) \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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  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>
 +
|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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~ \\
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~ \\
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~ \\
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~ \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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~ \\
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~ \\
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~ \\
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~ \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
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| <math>f_{15}\!</math>
 
| <math>f_{15}\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
|}
 
|}
 
<br>
 
<br>
Jon Awbrey
12,080

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