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Revision as of 13:14, 18 June 2009
, 13:14, 18 June 2009→Note 26: convert graphics
Figure 26-1 shows the differential proposition <math>\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\operatorname{d}f.\!</math>
Figure 26-1 shows the differential proposition <math>\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\operatorname{d}f.\!</math>
{| align="center" cellspacing="10" style="text-align:center; width:90%"
{| align="center" cellspacing="10" style="text-align:center"
|
| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
|-
| <math>\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math>
| o-------------------o o-------------------o |
| | /| |\ | |
Figure 26-1. Differential or Tangent d[pq] : EX -> B
</pre>
|}
|}
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
\operatorname{d}(pq)
\operatorname{d}(pq)
& = & p & \cdot & q & \cdot &
& = &
p & \cdot & q & \cdot &
\texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)}
\texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)}
\\[4pt]
\\[4pt]
& + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q
& + &
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
\operatorname{d}q
\\[4pt]
\\[4pt]
& + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p
& + &
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
\operatorname{d}p
\\[4pt]
\\[4pt]
& + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & 0
& + &
\texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & 0
\end{array}</math>
\end{array}</math>
|}
|}
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
\operatorname{r}(pq)
\operatorname{r}(pq)
& = & p & \cdot & q & \cdot &
& = &
p & \cdot & q & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
\\[4pt]
& + & p & \cdot & \texttt{(} q \texttt{)} & \cdot &
& + &
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
\\[4pt]
& + & \texttt{(} p \texttt{)} & \cdot & q & \cdot &
& + &
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
\\[4pt]
& + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot &
& + &
\texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\operatorname{d}p ~ \operatorname{d}q
\end{array}</math>
\end{array}</math>
