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 ''U'' = ''X''&nbsp; &times;&nbsp;''Y'' to its ''differential extension'',

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.

''EU'' = ''U''&nbsp; &times;&nbsp;''dU'' = ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''dX''&nbsp; &times;&nbsp;''dY'', with ''dX'' = {(''dx''), ''dx''} and ''dY'' = {(''dy''), ''dy''}. .The interpretations of these new symbols can be diverse, but the easiest for now is just to say that ''dx'' means "change x" and ''dy'' means "change y". .To draw the differential extension ''EU'' of our present universe ''U'' = ''X''&nbsp; &times;&nbsp;''Y'' as a venn diagram, it would take us four logical dimensions ''X'', ''Y'', ''dX'', ''dY'', but we can project a suggestion of what it's about on the universe ''X''&nbsp; &times;&nbsp;''Y'' by drawing arrows that cross designated borders, labeling the arrows as ''dx'' when crossing the border between ''x'' and (''x'') and as ''dy'' when crossing the border between ''y'' and (''y''), in either direction, in either case.

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