Line 4,806: |
Line 4,806: |
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| + | |} |
| + | <br> |
| + | |
| + | ==Taylor Series Expansion== |
| + | |
| + | {| 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" |
| + | | |
| + | | <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> |