Changes

==Note 3==

==Note 3==

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

Given the proposition f<p, q> over the space X = !P! x !Q!,

the (first order) "enlargement" of f is the proposition Ef

over the differential extension EX that is defined by the

following formula:

following formula:

  Ef<p, q, dp, dq>

 

  =  f<p + dp, q + dq>

<math>\begin{matrix}

 

\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)

  =  f<(p, dp), (q, dq)>

f(p + \operatorname{d}p,~ q + \operatorname{d}q)

f( \texttt{(} p, \operatorname{d}p \texttt{)},~ \texttt{(} q, \operatorname{d}q \texttt{)} )

\end{matrix}</math>

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: