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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 04:01, 24 September 2013

992 bytes removed ,  04:01, 24 September 2013
update
Line 10,342: Line 10,342:  
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math>
 
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math>
 
|
 
|
−
<math>\begin{array}{*{9}{l}}
+
<math>\begin{array}{*{10}{l}}
−
\boldsymbol\varepsilon f_{8} & = & f_{8}(u, v)
+
\boldsymbol\varepsilon f_{8}
 +
& = && f_{8}(u, v)
 
\\[4pt]
 
\\[4pt]
−
& = & u \cdot v
+
& = && uv
 
\\[4pt]
 
\\[4pt]
−
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}
+
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
−
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }
+
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
−
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}
+
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
−
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }
+
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v
−
\end{array}\!</math>
+
\\[20pt]
−
|-
  −
|
  −
<math>\begin{array}{*{4}{l}}
   
\boldsymbol\varepsilon f_{8}
 
\boldsymbol\varepsilon f_{8}
−
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}
+
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 
\\[4pt]
 
\\[4pt]
−
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}
+
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 
\\[4pt]
 
\\[4pt]
−
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}
+
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 
\\[4pt]
 
\\[4pt]
−
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}
+
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 
\end{array}\!</math>
 
\end{array}\!</math>
 
|}
 
|}
Line 10,376: Line 10,374:  
|
 
|
 
<math>\begin{array}{*{9}{l}}
 
<math>\begin{array}{*{9}{l}}
−
\mathrm{E}f_{8} & = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
+
\mathrm{E}f_{8}
 +
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
 
\\[4pt]
 
\\[4pt]
−
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
+
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
 
\\[4pt]
 
\\[4pt]
−
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot f_{8}(1\!+\!\mathrm{d}u, 1\!+\!\mathrm{d}v)
+
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
−
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot f_{8}(1\!+\!\mathrm{d}u, \mathrm{d}v)
+
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)
−
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot f_{8}(\mathrm{d}u, 1\!+\!\mathrm{d}v)
+
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})
−
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
 
\\[4pt]
 
\\[4pt]
−
& = &
+
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
−
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
+
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
−
& + &
+
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
−
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + &
+
\\[20pt]
−
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})
  −
& + &
  −
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
  −
\end{array}\!</math>
  −
|-
  −
|
  −
<math>\begin{array}{*{9}{l}}
   
\mathrm{E}f_{8}
 
\mathrm{E}f_{8}
−
& = &
+
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
−
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
   
\\[4pt]
 
\\[4pt]
−
&&& + &
+
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
−
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }
   
\\[4pt]
 
\\[4pt]
−
&&&&& + &
+
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
−
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}
   
\\[4pt]
 
\\[4pt]
−
&&&&&&& + &
+
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
−
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }
+
\end{array}\!</math>
−
\end{array}</math>
   
|}
 
|}
    
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math>
 
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math>
 
|
 
|
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\\[6pt]
 
\\[6pt]
 
\mathrm{E}f_{8}
 
\mathrm{E}f_{8}
−
& = &
+
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
−
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
+
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
−
& + &
+
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
−
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + &
  −
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}
  −
& + &
  −
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }
   
\end{array}\!</math>
 
\end{array}\!</math>
 
|}
 
|}
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|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math>
 
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math>
 
|
 
|
−
<math>\begin{array}{*{9}{l}}
+
<math>\begin{array}{*{10}{l}}
 
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
−
& = & \mathrm{E}f_{8}
+
& = && \mathrm{E}f_{8}
−
& + & \boldsymbol\varepsilon f_{8}
+
& + & \boldsymbol\varepsilon f_{8}
−
\\[6pt]
+
\\[4pt]
−
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
+
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
−
& + & f_{8}(u, v)
+
& + & f_{8}(u, v)
−
\\[6pt]
+
\\[4pt]
−
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
+
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
−
& + & u \cdot v
+
& + & uv
−
\end{array}\!</math>
+
\\[20pt]
−
|-
  −
|
  −
<math>\begin{array}{*{9}{l}}
   
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
−
& = &
+
& = && 0
−
u \cdot v \cdot \qquad 0
+
& + &  0
−
\\[6pt]
+
& + &  0
−
& + &
+
& + &  0
−
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
+
\\[4pt]
−
& + &
+
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}
−
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
+
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
−
\\[6pt]
+
& + &  0
−
& + &
+
& + &  0
−
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
+
\\[4pt]
−
&&& + &
+
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
−
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
+
& + & 0
−
\\[6pt]
+
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
−
& + &
+
& + &  0
−
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}
+
\\[4pt]
−
&&&&& + &
+
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
−
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}
+
& + & 0
−
\end{array}</math>
+
& + & 0
−
|-
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}
−
|
+
\\[20pt]
−
<math>\begin{array}{*{9}{l}}
   
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
−
& = &
+
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
−
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
+
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
−
& + &
+
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
−
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}
−
& + &
+
\end{array}\!</math>
−
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
  −
& + &
  −
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}
  −
\end{array}</math>
   
|}
 
|}
   Line 10,508: Line 10,484:  
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
 
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
 
\\[6pt]
 
\\[6pt]
−
& = & u \cdot v
+
& = & uv
−
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
+
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
 
\\[6pt]
 
\\[6pt]
 
& = & 0
 
& = & 0
 
& + & u \cdot \mathrm{d}v
 
& + & u \cdot \mathrm{d}v
 
& + & v \cdot \mathrm{d}u
 
& + & v \cdot \mathrm{d}u
−
& + & \mathrm{d}u \cdot \mathrm{d}v
+
& + & \mathrm{d}u ~ \mathrm{d}v
 
\\[6pt]
 
\\[6pt]
 
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
 
& = & 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 \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
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|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math>
 
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math>
 
|
 
|
−
<math>\begin{array}{*{5}{l}}
+
<math>\begin{array}{c*{9}{l}}
−
\mathrm{D}f_{8} & = & \boldsymbol\varepsilon f_{8} & + & \mathrm{E}f_{8}
+
\mathrm{D}f_{8}
−
\end{array}</math>
+
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}
−
|-
+
\\[20pt]
−
|
  −
<math>\begin{array}{*{9}{l}}
   
\boldsymbol\varepsilon f_{8}
 
\boldsymbol\varepsilon f_{8}
 
& = &  u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
 
& = &  u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
Line 10,546: Line 10,520:  
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v
 
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v
−
\end{array}</math>
+
\\[20pt]
−
|-
  −
|
  −
<math>\begin{array}{*{9}{l}}
   
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
−
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
+
& = & ~~~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
−
& + & ~~ 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 \texttt{,} 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>
 
|}
 
|}
−
  −
<br>
      
=====Computation of d''f''<sub>8</sub>=====
 
=====Computation of d''f''<sub>8</sub>=====
Line 10,564: Line 10,533:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math>
 
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math>
 
|
 
|
 
<math>\begin{array}{c*{8}{l}}
 
<math>\begin{array}{c*{8}{l}}
 
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
−
& = &
+
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
−
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
+
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
−
& + &
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
−
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + &
  −
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}
  −
& + &
  −
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}
   
\\[6pt]
 
\\[6pt]
 
\Downarrow
 
\Downarrow
 
\\[6pt]
 
\\[6pt]
 
\mathrm{d}f_{8}
 
\mathrm{d}f_{8}
−
& = &
+
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
−
& + &
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
−
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
−
& + &
  −
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u
  −
& + &
  −
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
   
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
Line 10,598: Line 10,559:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math>
 
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math>
 
|
 
|
−
<math>\begin{array}{*{5}{l}}
+
<math>\begin{array}{c*{8}{l}}
−
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} & + & \mathrm{d}f_{8}
+
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}
−
\end{array}</math>
+
\\[20pt]
−
|-
  −
|
  −
<math>\begin{array}{*{9}{l}}
   
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
−
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
+
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
−
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 
\\[6pt]
 
\\[6pt]
 
\mathrm{d}f_{8}
 
\mathrm{d}f_{8}
−
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
+
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
−
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
−
\end{array}</math>
+
\\[20pt]
−
|-
  −
|
  −
<math>\begin{array}{*{9}{l}}
   
\mathrm{r}f_{8}
 
\mathrm{r}f_{8}
−
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~~~~~~
+
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
Line 10,641: Line 10,596:  
<math>\begin{array}{c*{8}{l}}
 
<math>\begin{array}{c*{8}{l}}
 
\boldsymbol\varepsilon f_{8}
 
\boldsymbol\varepsilon f_{8}
−
& = & u \!\cdot\! v \cdot 1
+
& = & uv \cdot 1
 
& + & u \texttt{(} v \texttt{)} \cdot 0
 
& + & u \texttt{(} v \texttt{)} \cdot 0
 
& + & \texttt{(} u \texttt{)} v \cdot 0
 
& + & \texttt{(} u \texttt{)} v \cdot 0
Line 10,647: Line 10,602:  
\\[6pt]
 
\\[6pt]
 
\mathrm{E}f_{8}
 
\mathrm{E}f_{8}
−
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
+
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
−
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 
\\[6pt]
 
\\[6pt]
 
\mathrm{D}f_{8}
 
\mathrm{D}f_{8}
−
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
+
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
−
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 
\\[6pt]
 
\\[6pt]
 
\mathrm{d}f_{8}
 
\mathrm{d}f_{8}
−
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
+
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
 
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
Line 10,665: Line 10,620:  
\\[6pt]
 
\\[6pt]
 
\mathrm{r}f_{8}
 
\mathrm{r}f_{8}
−
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
−
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
Line 10,803: Line 10,758:  
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 
\\[4pt]
 
\\[4pt]
−
& = && 0
+
&& + & 0
 
& + &  0
 
& + &  0
 
& + &  0
 
& + &  0
Line 10,813: Line 10,768:  
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
\end{array}</math>
+
\end{array}\!</math>
 
|}
 
|}
   Line 10,823: Line 10,778:  
<math>\begin{array}{*{9}{l}}
 
<math>\begin{array}{*{9}{l}}
 
\mathrm{D}f_{9}
 
\mathrm{D}f_{9}
−
& = & 0 \cdot \mathrm{d}u \cdot \mathrm{d}v
+
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v
 
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
 
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
Line 10,866: Line 10,821:  
|
 
|
 
<math>\begin{array}{c*{8}{l}}
 
<math>\begin{array}{c*{8}{l}}
−
\mathrm{r}f_{9}
+
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}
−
& = & \mathrm{D}f_{9}
  −
& + & \mathrm{d}f_{9}
   
\\[20pt]
 
\\[20pt]
 
\mathrm{D}f_{9}
 
\mathrm{D}f_{9}
Line 11,129: Line 11,082:  
|
 
|
 
<math>\begin{array}{c*{8}{l}}
 
<math>\begin{array}{c*{8}{l}}
−
\mathrm{r}f_{14}
+
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}
−
& = & \mathrm{D}f_{14}
  −
& + & \mathrm{d}f_{14}
   
\\[20pt]
 
\\[20pt]
 
\mathrm{D}f_{14}
 
\mathrm{D}f_{14}
Jon Awbrey
12,080

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