Changes

These days, <math>\operatorname{E}</math> is more often called the ''shift operator''.

These days, <math>\operatorname{E}</math> is more often called the ''shift operator''.

In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse.  We mount up from the space <math>U = X \times Y</math> to its ''differential extension'', <math>\operatorname{E}U = U \times \operatorname{d}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> with <math>\operatorname{d}X = \{ (\!|\operatorname{d}x|\!), \operatorname{d}x \}</math> and <math>\operatorname{d}Y = \{ (\!|\operatorname{d}y|\!), \operatorname{d}y \}.</math> The interpretations of these new symbols can be diverse, but the easiest for now is just to say that <math>\operatorname{d}x</math> means "change <math>x\!</math>" and <math>\operatorname{d}y</math> means "change <math>y\!</math>". To draw the differential extension <math>\operatorname{E}U</math> of our present universe <math>U = X \times Y</math> as a venn diagram, it would take us four logical dimensions <math>X, Y, \operatorname{d}X, \operatorname{d}Y,</math> but we can project a suggestion of what it's about on the universe <math>X \times Y</math> by drawing arrows that cross designated borders, labeling the arrows as <math>\operatorname{d}x</math> when crossing the border between <math>x\!</math> and <math>(\!|x|\!)</math> and as <math>\operatorname{d}y</math> when crossing the border between <math>y\!</math> and <math>(\!|y|\!),</math> in either direction, in either case.

In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse.  We mount up from the space <math>U = X \times Y</math> to its ''differential extension'':

 

: <p><math>\operatorname{E}U = U \times \operatorname{d}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y</math></p>

 

: <p><math>\operatorname{d}X = \{ (\!|\operatorname{d}x|\!), \operatorname{d}x \}</math></p>

 

: <p><math>\operatorname{d}Y = \{ (\!|\operatorname{d}y|\!), \operatorname{d}y \}</math></p>

 

The interpretations of these new symbols can be diverse, but the easiest for now is just to say that <math>\operatorname{d}x</math> means "change <math>x\!</math>&nbsp;" and <math>\operatorname{d}y</math> means "change <math>y\!</math>&nbsp;".

 

To draw the differential extension <math>\operatorname{E}U</math> of our present universe <math>U = X \times Y</math> as a venn diagram, it would take us four logical dimensions <math>X, Y, \operatorname{d}X, \operatorname{d}Y,</math> but we can project a suggestion of what it's about on the universe <math>X \times Y</math> by drawing arrows that cross designated borders, labeling the arrows as <math>\operatorname{d}x</math> when crossing the border between <math>x\!</math> and <math>(\!|x|\!)</math> and as <math>\operatorname{d}y</math> when crossing the border between <math>y\!</math> and <math>(\!|y|\!),</math> in either direction, in either case.

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We can form propositions from these differential variables in the same way that we would any other logical variables, for instance, interpreting the proposition (''dx'' (''dy'')) to say "''dx''&nbsp;&rArr;&nbsp;''dy''", 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 x without a change in y".

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 ''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'').