Changes

Line 5,430: Line 5,430:     
{| 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"
 
|  
 
|  
Line 5,466: Line 5,466:  
|
 
|
 
<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>
 
|}
 
|}
Line 5,484: Line 5,484:  
|
 
|
 
<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>
 
|}
 
|}
Line 5,511: Line 5,511:  
|
 
|
 
<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>
 
|}
 
|}
Line 5,521: Line 5,519:  
|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
\partial x \\
+
\operatorname{d}x \\
\partial x \\
+
\operatorname{d}x \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
Line 5,546: Line 5,544:  
|
 
|
 
<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>
 
|}
 
|}
Line 5,556: Line 5,552:  
|
 
|
 
<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>
 
|}
 
|}
Line 5,581: Line 5,577:  
|
 
|
 
<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>
 
|}
 
|}
Line 5,591: Line 5,585:  
|
 
|
 
<math>\begin{smallmatrix}
 
<math>\begin{smallmatrix}
\partial y \\
+
\operatorname{d}y \\
\partial y \\
+
\operatorname{d}y \\
 
\end{smallmatrix}</math>
 
\end{smallmatrix}</math>
 
|}
 
|}
Line 5,620: Line 5,614:  
|
 
|
 
<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>
 
|}
 
|}
Line 5,638: Line 5,632:  
|
 
|
 
<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>
 
|}
 
|}
12,080

edits