Changes

We can form propositions from these differential variables in the same way that we would any other logical variables, for instance, interpreting the proposition <math>(\!| \operatorname{d}x\ (\!| \operatorname{d}y |\!) |\!)</math> to say "<math>\operatorname{d}x \Rightarrow \operatorname{d}y</math>&nbsp;", in other words, however you wish to take it, whether indicatively or injunctively, as saying something to the effect that there is "no change in <math>x\!</math> without a change in <math>y\!</math>&nbsp;".

We can form propositions from these differential variables in the same way that we would any other logical variables, for instance, interpreting the proposition <math>(\!| \operatorname{d}x\ (\!| \operatorname{d}y |\!) |\!)</math> to say "<math>\operatorname{d}x \Rightarrow \operatorname{d}y</math>&nbsp;", in other words, however you wish to take it, whether indicatively or injunctively, as saying something to the effect that there is "no change in <math>x\!</math> without a change in <math>y\!</math>&nbsp;".

Given the proposition ''f''(''x'', ''y'') in ''U'' = ''X''&nbsp; &times;&nbsp;''Y'', the (''first order'') ''enlargement'' of ''f'' is the proposition ''Ef'' in ''EU'' that is defined by the formula ''Ef''(''x'', ''y'', ''dx'', ''dy'') = ''f''(''x'' + ''dx'', ''y'' + ''dy'').

Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y\!, </math> the (''first order'') ''enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f</math> in <math>\operatorname{E}U</math> that is defined by the formula <math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) = f(x + \operatorname{d}x, y + \operatorname{d}y).</math>

In the example ''f''(''x'', ''y'') = ''xy'', we obtain:

In the example <math>f(x, y) = xy,\!</math> we obtain:

: ''Ef''(''x'', ''y'', ''dx'', ''dy'') = (''x'' + ''dx'')(''y'' + ''dy'').

: <p><math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) = (x + \operatorname{d}x)(y + \operatorname{d}y).</math></p>

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