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==Note 3==
==Note 3==
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<pre>
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Given the proposition <math>f(p, q)\!</math> over the space <math>X = P \times Q,</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f</math> over the differential extension <math>\operatorname{E}X</math> that is defined by the
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Given the proposition f<p, q> over the space X = !P! x !Q!,
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the (first order) "enlargement" of f is the proposition Ef
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over the differential extension EX that is defined by the
following formula:
following formula:
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Ef<p, q, dp, dq>
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{| align="center" cellpadding="6" width="90%"
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|
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= f<p + dp, q + dq>
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<math>\begin{matrix}
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\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
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= f<(p, dp), (q, dq)>
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& = &
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f(p + \operatorname{d}p,~ q + \operatorname{d}q)
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& = &
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f( \texttt{(} p, \operatorname{d}p \texttt{)},~ \texttt{(} q, \operatorname{d}q \texttt{)} )
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\end{matrix}</math>
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|}
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<pre>
In the example f<p, q> = pq, the enlargement Ef is given by:
In the example f<p, q> = pq, the enlargement Ef is given by: