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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 21:37, 26 August 2013
, 21:37, 26 August 2013update
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{| align="center" border="1" cellpadding="15" cellspacing="0" style="text-align:center; width:60%"
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math>
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0
0
& = & 0 & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))} & = & & 0
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0
\\[8pt]
\\[8pt]
\texttt{(} A \texttt{)}
\texttt{(} A \texttt{)}
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))}
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}
& = & \texttt{(} A \texttt{)} ~\mathrm{d}A~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}
\\[8pt]
\\[8pt]
A
A
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))}
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}
\\[8pt]
\\[8pt]
1
1
& = & 1 & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))} & = & & 1
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1
\end{matrix}</math>
\end{matrix}</math>
|}
|}
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}
\\[4pt]
\\[4pt]
& = & \texttt{(} u, \mathrm{d}u \texttt{)} \cdot \texttt{(} v, \mathrm{d}v \texttt{)}
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[4pt]
\\[4pt]
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}
& = & u \cdot v
& = & u \cdot v
& + & u \cdot \mathrm{d}v
& + & u \cdot \mathrm{d}v
& + & \mathrm{d}u \cdot v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u \cdot \mathrm{d}v
& + & \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\\[6pt]
& + & J_{(u, v)}
& + & J_{(u, v)}
\\[6pt]
\\[6pt]
& = & \texttt{(} u, \mathrm{d}u \texttt{)} \cdot \texttt{(} v, \mathrm{d}v \texttt{)}
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
& + & u \cdot v
& + & u \cdot v
\end{array}</math>
\end{array}</math>
\\[6pt]
\\[6pt]
& = & u \cdot v
& = & u \cdot v
& + & \texttt{(} u, \mathrm{d}u \texttt{)} \cdot \texttt{(} v, \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\\[6pt]
& = & 0
& = & 0
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u, v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v
\end{array}</math>
\end{array}</math>
|}
|}
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~~~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u, v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}\!</math>
\end{array}\!</math>
|}
|}
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math>
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math>
|
|
<math>\begin{array}{cllllllll}
<math>\begin{array}{c*{8}{l}}
\mathrm{D}J
\mathrm{D}J
& = &
& = &