Changes

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

Revision as of 03:50, 19 June 2008

985 bytes added , 03:50, 19 June 2008

→‎Differential Forms: copy table model for refitting

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

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|+ '''~~Orbit Table Template~~'''

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|+ '''Differential Forms Expanded on a Logical Basis'''

|- style="background:ghostwhite; height:36px"

|  

| <math>f\!</math>

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| <math>\operatorname{F}f~~|_{xy}</math>~~

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| <math>\operatorname{D}f</math>

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~~| <math>\operatorname{F}f|_{x(y)}</math>~~

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| <math>\operatorname{d}f</math>

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~~| <math>\operatorname{F}f|_{(x)y}~~</math>

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| <math>\operatorname{F}f~~|_{(x)(y)}~~</math>

|- style="height:36px"

| <math>f_{0}\!</math>

| <math>(~)\!</math>

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| <math>~~(~)~~\!</math>

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| <math>0\!</math>

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| <math>~~(~)\!</math>~~

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| <math>0\!</math>

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~~| <math>(~)\!</math>~~

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~~| <math>(~)~~\!</math>

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{| align="center"

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|

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~~| height="36px" |~~ <math>f_{1}\~~!</math>~~

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<math>\begin{smallmatrix}

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f_{1} \\

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~~| height="36px" | <math>~~f_{2}\~~!</math>~~

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f_{2} \\

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f_{4} \\

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~~| height="36px" | <math>~~f_{4}\~~!</math>~~

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f_{8} \\

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\end{smallmatrix}</math>

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~~| height="36px" | <math>~~f_{8}\!</math>

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{| align="center"

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|

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~~| height="36px" |~~ <math>(x)(y)\~~!</math>~~

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<math>\begin{smallmatrix}

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(x) & (y) \\

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~~| height="36px" | <math>~~(x)\ y\~~!</math>~~

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(x) & y \\

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x & (y) \\

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~~| height="36px" | <math>~~x\ (y)\~~!</math>~~

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x & y \\

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\end{smallmatrix}</math>

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~~| height="36px" | <math>~~x\ y\!</math>

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{| align="center"

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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|-

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(y) & \operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

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(x) & (\operatorname{d}x)\ \operatorname{d}y & + &

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((x, y)) & \operatorname{d}x\ \operatorname{d}y \\

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~~| height="36px" |~~

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y & \operatorname{d}x\ (\operatorname{d}y) & + &

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(x) & (\operatorname{d}x)\ \operatorname{d}y & + &

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~~| height="36px" |~~

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(x, y) & \operatorname{d}x\ \operatorname{d}y \\

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(y) & \operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

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x & (\operatorname{d}x)\ \operatorname{d}y & + &

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(x, y) & \operatorname{d}x\ \operatorname{d}y \\

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y & \operatorname{d}x\ (\operatorname{d}y) & + &

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x & (\operatorname{d}x)\ \operatorname{d}y & + &

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((x, y)) & \operatorname{d}x\ \operatorname{d}y \\

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\end{smallmatrix}</math>

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{| align="center"

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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(y) & \partial x & + & (x) & \partial y \\

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~~| height="36px" |~~

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y & \partial x & + & (x) & \partial y \\

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(y) & \partial x & + & x & \partial y \\

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~~| height="36px" |~~

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y & \partial x & + & x & \partial y \\

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\end{smallmatrix}</math>

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~~| height="36px" |~~

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~~| height="36px" |~~

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{| align="center"

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|

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~~| height="36px" |~~ <math>f_{3}\~~!</math>~~

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<math>\begin{smallmatrix}

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f_{3} \\

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~~| height="36px" | <math>~~f_{12}\!</math>

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f_{12} \\

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\end{smallmatrix}</math>

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{| align="center"

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+

|

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~~| height="36px" |~~ <math>(x)\~~!</math>~~

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<math>\begin{smallmatrix}

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(x) \\

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~~| height="36px" | <math>x~~\!</math>

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x \\

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\end{smallmatrix}</math>

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{| align="center"

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|-

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~~| height="36px" |~~

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~~| height="36px" |~~

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|}

|

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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\operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

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\operatorname{d}x\ \operatorname{d}y \\

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\operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

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\operatorname{d}x\ \operatorname{d}y \\

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\end{smallmatrix}</math>

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{| align="center"

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|-

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~~| height="36px" |~~

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~~| height="36px" |~~

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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\partial x \\

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~~| height="36px" |~~

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\partial x \\

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\end{smallmatrix}</math>

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~~| height="36px" |~~

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{| align="center"

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|

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~~| height="36px" |~~ <math>f_{6}\~~!</math>~~

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<math>\begin{smallmatrix}

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f_{6} \\

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~~| height="36px" | <math>~~f_{9}\!</math>

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f_{9} \\

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\end{smallmatrix}</math>

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{| align="center"

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|

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~~| height="36px" |~~ <math>(x,\ y)\~~!</math>~~

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<math>\begin{smallmatrix}

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(x, & y) \\

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~~| height="36px" | <math>~~((x,\ y))\!</math>

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((x, & y)) \\

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\end{smallmatrix}</math>

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{| align="center"

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|-

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~~| height="36px" |~~

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~~| height="36px" |~~

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|}

|

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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|-

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\operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

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(\operatorname{d}x)\ \operatorname{d}y \\

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\operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

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(\operatorname{d}x)\ \operatorname{d}y \\

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\end{smallmatrix}</math>

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{| align="center"

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|-

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~~| height="36px" |~~

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~~| height="36px" |~~

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|}

|

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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\partial x & + & \partial y \\

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~~| height="36px" |~~

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\partial x & + & \partial y \\

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\end{smallmatrix}</math>

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~~| height="36px" |~~

|}

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{| align="center"

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|

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~~| height="36px" |~~ <math>f_{5}\~~!</math>~~

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<math>\begin{smallmatrix}

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f_{5} \\

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~~| height="36px" | <math>~~f_{10}\!</math>

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f_{10} \\

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\end{smallmatrix}</math>

|}

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{| align="center"

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|-

+

|

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~~| height="36px" |~~ <math>(y)\~~!</math>~~

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<math>\begin{smallmatrix}

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(y) \\

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~~| height="36px" | <math>y~~\!</math>

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y \\

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\end{smallmatrix}</math>

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{| align="center"

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|-

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~~| height="36px" |~~

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|-

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~~| height="36px" |~~

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|}

|

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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(\operatorname{d}x)\ \operatorname{d}y & + &

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~~| height="36px" |~~

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\operatorname{d}x\ \operatorname{d}y \\

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(\operatorname{d}x)\ \operatorname{d}y & + &

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~~| height="36px" |~~

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\operatorname{d}x\ \operatorname{d}y \\

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\end{smallmatrix}</math>

|}

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{| align="center"

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|-

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~~| height="36px" |~~

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~~| height="36px" |~~

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|}

|

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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\partial y \\

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~~| height="36px" |~~

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\partial y \\

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+

\end{smallmatrix}</math>

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~~| height="36px" |~~

|}

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{| align="center"

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|

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~~| height="36px" |~~ <math>f_{7}\~~!</math>~~

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<math>\begin{smallmatrix}

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f_{7} \\

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~~| height="36px" | <math>~~f_{11}\~~!</math>~~

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f_{11} \\

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f_{13} \\

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~~| height="36px" | <math>~~f_{13}\~~!</math>~~

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f_{14} \\

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\end{smallmatrix}</math>

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~~| height="36px" | <math>~~f_{14}\!</math>

|}

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{| align="center"

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|

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~~| height="36px" |~~ <math>(x\ y)\~~!</math>~~

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<math>\begin{smallmatrix}

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|-

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(x & y) \\

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~~| height="36px" | <math>~~(x\ (y))\~~!</math>~~

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(x & (y)) \\

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|-

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((x) & y) \\

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~~| height="36px" | <math>~~((x)\ y)\~~!</math>~~

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((x) & (y)) \\

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\end{smallmatrix}</math>

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~~| height="36px" | <math>~~((x)(y))\!</math>

|}

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{| align="center"

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~~| height="36px" |~~

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~~| height="36px" |~~

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~~| height="36px" |~~

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~~| height="36px" |~~

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|}

|

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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|-

+

y & \operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

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x & (\operatorname{d}x)\ \operatorname{d}y & + &

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|-

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((x, y)) & \operatorname{d}x\ \operatorname{d}y \\

−

~~| height="36px" |~~

+

(y) & \operatorname{d}x\ (\operatorname{d}y) & + &

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|-

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x & (\operatorname{d}x)\ \operatorname{d}y & + &

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~~| height="36px" |~~

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(x, y) & \operatorname{d}x\ \operatorname{d}y \\

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|-

+

y & \operatorname{d}x\ (\operatorname{d}y) & + &

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~~| height="36px" |~~

+

(x) & (\operatorname{d}x)\ \operatorname{d}y & + &

+

(x, y) & \operatorname{d}x\ \operatorname{d}y \\

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(y) & \operatorname{d}x\ (\operatorname{d}y) & + &

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(x) & (\operatorname{d}x)\ \operatorname{d}y & + &

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((x, y)) & \operatorname{d}x\ \operatorname{d}y \\

+

\end{smallmatrix}</math>

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{| align="center"

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~~| height="36px" |~~

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~~| height="36px" |~~

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~~| height="36px" |~~

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~~| height="36px" |~~

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|}

|

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{~~| align="center"~~

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<math>\begin{smallmatrix}

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|-

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y & \partial x & + & x & \partial y \\

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~~| height="36px" |~~

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(y) & \partial x & + & x & \partial y \\

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|-

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y & \partial x & + & (x) & \partial y \\

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~~| height="36px" |~~

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(y) & \partial x & + & (x) & \partial y \\

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|-

+

\end{smallmatrix}</math>

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~~| height="36px" |~~

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|-

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~~| height="36px" |~~

|}

|- style="height:36px"

| <math>f_{15}\!</math>

| <math>((~))\!</math>

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| ~~ ~~

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| <math>0\!</math>

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| ~~ ~~

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| <math>0\!</math>

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~~|  ~~

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~~|  ~~

|}

<br>

Jon Awbrey

12,080

edits