Line 4,068:
Line 4,068:
==Note 25==
==Note 25==
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<pre>
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Continuing with the example <math>pq : X \to \mathbb{B},</math> Figure 25-1 shows the enlargement or shift map <math>\operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
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Staying with the example pq : X -> B, Figure 25-1 shows
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the enlargement or shift map E[pq] : EX -> B in the same
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style of differential field picture that we drew for the
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tacit extension !e![pq] : EX -> B.
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o---------------------------------------------------------------------o
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{| align="center" cellspacing="10" style="text-align:center"
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| |
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| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
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| X |
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|-
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| o-------------------o o-------------------o |
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| <math>\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>
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| / \ / \ |
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|-
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| / P o Q \ |
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|
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| / / \ \ |
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<math>\begin{array}{rcccccc}
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| / / \ \ |
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\operatorname{E}(pq)
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| / / \ \ |
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& = &
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| / / \ \ |
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p
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| / / \ \ |
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& \cdot &
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| o o (dp) (dq) o o |
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q
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| | | o-->--o | | |
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& \cdot &
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| | | \ / | | |
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\texttt{(} \operatorname{d}p \texttt{)}
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| | (dp) dq | \ / | dp (dq) | |
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\texttt{(} \operatorname{d}q \texttt{)}
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| | o----------------->o<-----------------o | |
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\\[4pt]
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| | | ^ | | |
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& + &
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| | | | | | |
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p
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| | | | | | |
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& \cdot &
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| o o | o o |
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\texttt{(} q \texttt{)}
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| \ \ | / / |
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& \cdot &
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| \ \ | / / |
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\texttt{(} \operatorname{d}p \texttt{)}
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| \ \ | / / |
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\texttt{~} \operatorname{d}q \texttt{~}
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| \ \ | / / |
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\\[4pt]
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| \ \|/ / |
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& + &
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| \ | / |
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\texttt{(} p \texttt{)}
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| \ /|\ / |
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& \cdot &
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| o-------------------o | o-------------------o |
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q
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| | |
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& \cdot &
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| dp | dq |
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\texttt{~} \operatorname{d}p \texttt{~}
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| | |
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\texttt{(} \operatorname{d}q \texttt{)}
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| | |
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\\[4pt]
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| o |
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& + &
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| |
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\texttt{(} p \texttt{)}
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o---------------------------------------------------------------------o
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& \cdot &
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Figure 25-1. Enlargement E[pq] : EX -> B
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\texttt{(} q \texttt{)}
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& \cdot &
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\texttt{~} \operatorname{d}p \texttt{~}
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\texttt{~} \operatorname{d}q \texttt{~}
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\end{array}</math>
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|}
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A very important conceptual transition has just occurred here,
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A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields <math>\varepsilon f</math> and <math>\operatorname{E}f,</math> both of the type <math>\operatorname{E}X \to \mathbb{B},</math> is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.
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almost tacitly, as it were. Generally speaking, having a set
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of mathematical objects of compatible types, in this case the
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two differential fields !e!f and Ef, both of the type EX -> B,
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is very useful, because it allows us to consider these fields
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as integral mathematical objects that can be operated on and
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combined in the ways that we usually associate with algebras.
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In this case one notices that the tacit extension !e!f and the
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In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other. The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true. The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure 25-2.
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enlargement Ef are in a certain sense dual to each other, with
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!e!f indicating all of the arrows out of the region where f is
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true, and with Ef indicating all of the arrows into the region
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where f is true. The only arc that they have in common is the
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no-change loop (dp)(dq) at pq. If we add the two sets of arcs
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mod 2, then the common loop drops out, leaving the 6 arrows of
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D[pq] = !e![pq] + E[pq] that are illustrated in Figure 25-2.
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o---------------------------------------------------------------------o
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{| align="center" cellspacing="10" style="text-align:center"
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| |
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| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
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| X |
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|-
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| o-------------------o o-------------------o |
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| <math>\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>
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| / \ / \ |
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|-
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| / P o Q \ |
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|
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| / / \ \ |
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<math>\begin{array}{rcccccc}
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| / / \ \ |
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\operatorname{D}(pq)
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| / / \ \ |
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& = &
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| / / \ \ |
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p
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| / / \ \ |
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& \cdot &
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| o o o o |
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q
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| | | | | |
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& \cdot &
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| | | | | |
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\texttt{(}
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| | (dp) dq | | dp (dq) | |
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\texttt{(} \operatorname{d}p \texttt{)}
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| | o<---------------->o<---------------->o | |
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\texttt{(} \operatorname{d}q \texttt{)}
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| | | ^ | | |
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\texttt{)}
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| | | | | | |
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\\[4pt]
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| | | | | | |
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& + &
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| o o | o o |
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p
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| \ \ | / / |
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& \cdot &
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| \ \ | / / |
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\texttt{(} q \texttt{)}
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| \ \ | / / |
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& \cdot &
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| \ \ | / / |
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\texttt{~}
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| \ \|/ / |
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\texttt{(} \operatorname{d}p \texttt{)}
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| \ | / |
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\texttt{~} \operatorname{d}q \texttt{~}
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| \ /|\ / |
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\texttt{~}
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| o-------------------o | o-------------------o |
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\\[4pt]
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| | |
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& + &
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| dp | dq |
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\texttt{(} p \texttt{)}
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| | |
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& \cdot &
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| v |
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q
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| o |
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& \cdot &
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| |
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\texttt{~}
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o---------------------------------------------------------------------o
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\texttt{~} \operatorname{d}p \texttt{~}
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Figure 25-2. Difference Map D[pq] : EX -> B
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\texttt{(} \operatorname{d}q \texttt{)}
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\texttt{~}
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The differential features of D[pq] may be collected cell by cell of
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\\[4pt]
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the underlying universe X% = [p, q] to give the following expansion:
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& + &
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\texttt{(} p \texttt{)}
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D[pq]
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& \cdot &
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\texttt{(}q \texttt{)}
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=
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& \cdot &
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\texttt{~}
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p q . ((dp)(dq))
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\texttt{~} \operatorname{d}p \texttt{~}
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\texttt{~} \operatorname{d}q \texttt{~}
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\texttt{~}
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\end{array}</math>
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p (q) . (dp) dq
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|}
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(p) q . dp (dq)
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(p)(q) . dp dq
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</pre>
==Note 26==
==Note 26==