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Revision as of 04:01, 24 September 2013
, 04:01, 24 September 2013update
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−<math>\begin{array}{*{4}{l}}
|-
−|
−<math>\begin{array}{*{9}{l}}
|-
−|
−<math>\begin{array}{*{9}{l}}
u \cdot v \cdot \qquad 0+|-+|+<math>\begin{array}{*{9}{l}}
|-+|
−<math>\begin{array}{*{9}{l}}
|-
−|
−<math>\begin{array}{*{9}{l}}
<br>
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}+u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}+|-
−|
−<math>\begin{array}{*{9}{l}}
|-
−|
−<math>\begin{array}{*{9}{l}}
& + & \mathrm{d}f_{9}
& + & \mathrm{d}f_{14}
|+ 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]
−\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>
|}
|}
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|
|
<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>
−\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]
−\mathrm{D}f_{8}
\mathrm{D}f_{8}
−& = &
+& = && 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]
−\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>
|}
|}
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& + & 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]
−\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{)}
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& + & \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]
−\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>
|}
|}
−=====Computation of d''f''<sub>8</sub>=====
=====Computation of d''f''<sub>8</sub>=====
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<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 \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 \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>
|}
|}
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<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]
−\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]
−\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}
−\\[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}
−\\[20pt]
\\[20pt]
\mathrm{D}f_{14}
\mathrm{D}f_{14}
