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

Revision as of 03:32, 22 June 2008

4,080 bytes added ,  03:32, 22 June 2008
→‎Work Area: add section [partial differentials and relative differentials] + model table
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<br>
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==Partial Differentials and Relative Differentials==
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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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|+ '''Differential Forms Expanded on an Algebraic Basis'''
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|- style="background:ghostwhite; height:36px"
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| &nbsp;
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| <math>f\!</math>
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| <math>\operatorname{D}f</math>
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| <math>\operatorname{d}f</math>
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|- style="height:36px"
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| <math>f_{0}\!</math>
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| <math>(~)\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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|-
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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f_{1} \\
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f_{2} \\
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f_{4} \\
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f_{8} \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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<math>\begin{smallmatrix}
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(x) & (y) \\
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(x) &  y  \\
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x  & (y) \\
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x  &  y  \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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|
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<math>\begin{smallmatrix}
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(y) & \operatorname{d}x & + &
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(x) & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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y  & \operatorname{d}x & + &
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(x) & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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(y) & \operatorname{d}x & + &
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x  & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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y  & \operatorname{d}x & + &
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x  & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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\end{smallmatrix}</math>
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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 & + & (x) & \operatorname{d}y \\
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y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
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(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
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y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
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\end{smallmatrix}</math>
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|}
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|-
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{| align="center"
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<math>\begin{smallmatrix}
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f_{3}  \\
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f_{12} \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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<math>\begin{smallmatrix}
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(x) \\
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x  \\
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{| align="center"
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<math>\begin{smallmatrix}
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\operatorname{d}x \\
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\operatorname{d}x \\
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<math>\begin{smallmatrix}
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\operatorname{d}x \\
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|-
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{| align="center"
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<math>\begin{smallmatrix}
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f_{6} \\
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f_{9} \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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<math>\begin{smallmatrix}
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(x, & y)  \\
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((x, & y)) \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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\operatorname{d}x & + & \operatorname{d}y \\
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\operatorname{d}x & + & \operatorname{d}y \\
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\end{smallmatrix}</math>
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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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\operatorname{d}x & + & \operatorname{d}y \\
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\end{smallmatrix}</math>
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|}
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|-
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{| align="center"
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<math>\begin{smallmatrix}
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f_{5}  \\
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f_{10} \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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<math>\begin{smallmatrix}
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(y) \\
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y  \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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<math>\begin{smallmatrix}
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\operatorname{d}y \\
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\operatorname{d}y \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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\operatorname{d}y \\
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\operatorname{d}y \\
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\end{smallmatrix}</math>
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|}
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|-
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{| align="center"
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<math>\begin{smallmatrix}
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f_{7}  \\
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f_{11} \\
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f_{13} \\
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f_{14} \\
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\end{smallmatrix}</math>
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|}
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{| align="center"
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<math>\begin{smallmatrix}
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(x  &  y)  \\
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(x  & (y)) \\
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((x) &  y)  \\
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((x) & (y)) \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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y  & \operatorname{d}x & + &
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x  & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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(y) & \operatorname{d}x & + &
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x  & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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y  & \operatorname{d}x & + &
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(x) & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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(y) & \operatorname{d}x & + &
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(x) & \operatorname{d}y & + &
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      \operatorname{d}x\ \operatorname{d}y \\
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\end{smallmatrix}</math>
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|}
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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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y  & \operatorname{d}x & + &  x  & \operatorname{d}y \\
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(y) & \operatorname{d}x & + &  x  & \operatorname{d}y \\
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y  & \operatorname{d}x & + & (x) & \operatorname{d}y \\
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(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
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\end{smallmatrix}</math>
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|}
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|- style="height:36px"
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| <math>f_{15}\!</math>
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| <math>((~))\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
 
|}
 
|}
 
<br>
 
<br>
Jon Awbrey
12,122

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