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Revision as of 14:30, 28 May 2009
, 14:30, 28 May 2009→Note 2: markup
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We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>xy,\!</math> that is, at the place where <math>x = 1\!</math> and <math>y = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{xy}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
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<math>f ~=~ xy ~=~ x ~\operatorname{and}~ y ~\Rightarrow~ \operatorname{D}f|_{xy} ~=~ \texttt{((} \operatorname{dx} \texttt{)(} \operatorname{d}y \texttt{))} ~=~ \operatorname{d}x ~\operatorname{or}~ \operatorname{d}y.</math>
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<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
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| Df|xy = ((dx)(dy)) |
| Df|xy = ((dx)(dy)) |
o---------------------------------------o
o---------------------------------------o
</pre>
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The picture illustrates the analysis of the inclusive disjunction ((dx)(dy))
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}x \texttt{)(} \operatorname{d}y \texttt{))}</math> into the following exclusive disjunction:
into the exclusive disjunction: dx(dy) + dy(dx) + dx dy, a proposition that
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| align="center" | <math>\operatorname{d}x ~\texttt{(} \operatorname{d}y \texttt{)} ~+~ \operatorname{d}y ~\texttt{(} \operatorname{d}x \texttt{)} ~+~ \operatorname{d}x ~\operatorname{d}y.</math>
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This resulting differential proposition may be interpreted to say "change <math>x\!</math> or change <math>y\!</math> or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.
</pre>
====Note 3====
====Note 3====
