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

Revision as of 19:42, 20 June 2008

6,132 bytes added ,  19:42, 20 June 2008
→‎Appendices: add [taylor series expansion] table
Line 7,983: Line 7,983:     
==Appendix 3==
 
==Appendix 3==
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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%"
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|+ '''Taylor Series Expansion <math>\operatorname{D}f = \operatorname{d}f + \operatorname{d}^2 f</math>'''
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|- style="background:ghostwhite"
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| &nbsp;
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|
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<p><math>\operatorname{d}f = </math></p>
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<p><math>\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y</math></p>
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|
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<p><math>\operatorname{d}^2 f = </math></p>
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<p><math>\partial_{xy} f \cdot \operatorname{d}x\ \operatorname{d}y</math></p>
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| <math>df|_{x\ y}\!</math>
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| <math>df|_{x\ (y)}\!</math>
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| <math>df|_{(x)\ y}\!</math>
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| <math>df|_{(x)(y)}\!</math>
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|-
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| <math>f_0\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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| <math>0\!</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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|
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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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|
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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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\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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|
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<math>\begin{smallmatrix}
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0 \\
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\operatorname{d}x \\
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\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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\operatorname{d}x \\
 +
0 \\
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\operatorname{d}x + \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}x + \operatorname{d}y \\
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0 \\
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\operatorname{d}x \\
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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}
 +
\operatorname{d}x + \operatorname{d}y \\
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\operatorname{d}y \\
 +
\operatorname{d}x \\
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0 \\
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\end{smallmatrix}</math>
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|}
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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_{3}  \\
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f_{12} \\
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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 \\
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\operatorname{d}x \\
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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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0 \\
 +
0 \\
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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}
 +
\operatorname{d}x \\
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\operatorname{d}x \\
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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 \\
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\operatorname{d}x \\
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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}
 +
\operatorname{d}x \\
 +
\operatorname{d}x \\
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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}
 +
\operatorname{d}x \\
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\operatorname{d}x \\
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\end{smallmatrix}</math>
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|}
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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_{6} \\
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f_{9} \\
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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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|
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{| align="center"
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|
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<math>\begin{smallmatrix}
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0 \\
 +
0 \\
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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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|
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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 \\
 +
\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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\operatorname{d}x + \operatorname{d}y \\
 +
\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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\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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|
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{| align="center"
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|
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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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|
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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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|
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<math>\begin{smallmatrix}
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0 \\
 +
0 \\
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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 \\
 +
\operatorname{d}y \\
 +
\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 \\
 +
\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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|
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<math>\begin{smallmatrix}
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\operatorname{d}y \\
 +
\operatorname{d}y \\
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\end{smallmatrix}</math>
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|}
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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_{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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|
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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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|
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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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\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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|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\operatorname{d}y \\
 +
\operatorname{d}x \\
 +
0 \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}y \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
0 \\
 +
\operatorname{d}x \\
 +
\end{smallmatrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{smallmatrix}
 +
\operatorname{d}x \\
 +
0 \\
 +
\operatorname{d}x + \operatorname{d}y \\
 +
\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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0 \\
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\operatorname{d}x \\
 +
\operatorname{d}y \\
 +
\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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| <math>f_{15}\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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| <math>0\!</math>
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|}
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<br>
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==Appendix 4==
    
===Detail of Calculation for the Difference Map===
 
===Detail of Calculation for the Difference Map===
Jon Awbrey
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