Changes

Figure&nbsp;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&nbsp;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>

|  X                                                                |

|           o-------------------o  o-------------------o            |

|   |                      /|          |\                      |    |

|      \              o<--------------------->o              /      |

|          \                    / \                     /          |

Figure 26-1.  Differential or Tangent d[pq] : EX -> B

</pre>

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<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>