Line 4,806:
Line 4,806:
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|}
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<br>
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==Taylor Series Expansion==
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+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
|+ '''Differential Forms Expanded on an Algebraic Basis'''
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|- style="background:ghostwhite; height:36px"
+
|
+
| <math>f\!</math>
+
| <math>\operatorname{D}f</math>
+
| <math>\operatorname{d}f</math>
+
|- style="height:36px"
+
| <math>f_{0}\!</math>
+
| <math>(~)\!</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) & \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 \\
+
y & \operatorname{d}x & + &
+
x & \operatorname{d}y & + &
+
\operatorname{d}x\ \operatorname{d}y \\
+
\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}
+
f_{3} \\
+
f_{12} \\
+
\end{smallmatrix}</math>
+
|}
+
|
+
{| align="center"
+
|
+
<math>\begin{smallmatrix}
+
(x) \\
+
x \\
+
\end{smallmatrix}</math>
+
|}
+
|
+
{| align="center"
+
|
+
<math>\begin{smallmatrix}
+
\operatorname{d}x \\
+
\operatorname{d}x \\
+
\end{smallmatrix}</math>
+
|}
+
|
+
{| align="center"
+
|
+
<math>\begin{smallmatrix}
+
\operatorname{d}x \\
+
\operatorname{d}x \\
+
\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}
+
\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}x & + & \operatorname{d}y \\
+
\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}
+
\operatorname{d}y \\
+
\operatorname{d}y \\
+
\end{smallmatrix}</math>
+
|}
+
|
+
{| align="center"
+
|
+
<math>\begin{smallmatrix}
+
\operatorname{d}y \\
+
\operatorname{d}y \\
+
\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 & \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 \\
+
(y) & \operatorname{d}x & + &
+
(x) & \operatorname{d}y & + &
+
\operatorname{d}x\ \operatorname{d}y \\
+
\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>
+
|}
+
|- style="height:36px"
+
| <math>f_{15}\!</math>
+
| <math>((~))\!</math>
+
| <math>0\!</math>
+
| <math>0\!</math>
|}
|}
<br>
<br>