Line 8,324: |
Line 8,324: |
| | <math>f_{15}\!</math> | | | <math>f_{15}\!</math> |
| | <math>0\!</math> | | | <math>0\!</math> |
| + | | <math>0\!</math> |
| + | | <math>0\!</math> |
| + | | <math>0\!</math> |
| + | | <math>0\!</math> |
| + | | <math>0\!</math> |
| + | |} |
| + | <br> |
| + | |
| + | ===Partial Differentials and Relative Differentials=== |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Partial Differentials and Relative Differentials''' |
| + | |- style="background:ghostwhite; height:48px" |
| + | | |
| + | | <math>f\!</math> |
| + | | <math>\frac{\partial f}{\partial x}</math> |
| + | | <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" |
| + | | <math>f_0\!</math> |
| + | | <math>(~)\!</math> |
| + | | <math>0\!</math> |
| + | | <math>0\!</math> |
| + | | <math>0\!</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) \\ |
| + | y \\ |
| + | (y) \\ |
| + | y \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | (x) \\ |
| + | (x) \\ |
| + | x \\ |
| + | x \\ |
| + | \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} |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | \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} |
| + | 1 \\ |
| + | 1 \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | 0 \\ |
| + | 0 \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | \operatorname{d}x \\ |
| + | \operatorname{d}x \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | ~ \\ |
| + | ~ \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | ~ \\ |
| + | ~ \\ |
| + | \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} |
| + | 1 \\ |
| + | 1 \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | 1 \\ |
| + | 1 \\ |
| + | \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} |
| + | ~ \\ |
| + | ~ \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | ~ \\ |
| + | ~ \\ |
| + | \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} |
| + | 0 \\ |
| + | 0 \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | 1 \\ |
| + | 1 \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | \operatorname{d}y \\ |
| + | \operatorname{d}y \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | ~ \\ |
| + | ~ \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | ~ \\ |
| + | ~ \\ |
| + | \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 \\ |
| + | (y) \\ |
| + | y \\ |
| + | (y) \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | x \\ |
| + | x \\ |
| + | (x) \\ |
| + | (x) \\ |
| + | \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} |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | ~ \\ |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | |- style="height:36px" |
| + | | <math>f_{15}\!</math> |
| + | | <math>((~))\!</math> |
| | <math>0\!</math> | | | <math>0\!</math> |
| | <math>0\!</math> | | | <math>0\!</math> |