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Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
Just to be clear about what's being indicated here,
it's a visual way of specifying the following data:
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{array}{rcccccc}
\operatorname{d}(pq)
& = & p & \cdot & q & \cdot &
\texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)}
\\[4pt]
& + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q
\\[4pt]
& + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p
\\[4pt]
& + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & 0
\end{array}</math>
|}
To understand the extended interpretations, that is, the conjunctions of basic and differential features that are being indicated here, it may help to note the following equivalences:
To understand the extended interpretations, that is,
the conjunctions of basic and differential features
that are being indicated here, it may help to note
the following equivalences:
<pre>
(dp, dq) = dp + dq = dp(dq) + (dp)dq
(dp, dq) = dp + dq = dp(dq) + (dp)dq
dq = dp dq + (dp)dq
dq = dp dq + (dp)dq
</pre>
Capping the series that analyzes the proposition pq
Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, Figure 26-2 shows the remainder map <math>\operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.
in terms of succeeding orders of linear propositions,
Figure 26-2 shows the remainder map r[pq] : EX -> B,
that happens to be linear in pairs of variables.
{| align="center" cellspacing="10" style="text-align:center; width:90%"
{| align="center" cellspacing="10" style="text-align:center; width:90%"
Reading the arrows off the map produces the following data:
Reading the arrows off the map produces the following data:
<pre>
r[pq]
r[pq]
(p)(q) . dp dq
(p)(q) . dp dq
</pre>
In short, r[pq] is a constant field,
In short, <math>\operatorname{r}(pq)</math> is a constant field, having the value <math>\operatorname{d}p~\operatorname{d}q</math> at each cell.
having the value dp dq at each cell.
A more detailed presentation of Differential Logic can be found here:
A more detailed presentation of Differential Logic can be found here:
DLOG D. http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
* DLOG D. http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
* DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
* DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
==Document History==
==Document History==
