Changes

: <p><math>\operatorname{E}f(x_1, \ldots x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k) = f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k).</math></p>

: <p><math>\operatorname{E}f(x_1, \ldots x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k) = f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k).</math></p>

It should be noted that the so-called ''differential variables'' ''dx''_''j'' are really just the same kind of boolean variables as the other ''x''_''j''. It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.

It should be noted that the so-called ''differential variables'' <math>\operatorname{d}x_j</math> are really just the same kind of boolean variables as the other <math>x_j.\!</math>  It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.

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

For 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>

Given that this expression uses nothing more than the boolean ring operations of addition (+) and multiplication (&middot;), it is permissible to multiply things out in the usual manner to arrive at the result:

Given that this expression uses nothing more than the boolean ring operations of addition <math>(+)\!</math> and multiplication <math>(\cdot),</math> it is permissible to multiply things out in the usual manner to arrive at the result:

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

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

To understand what this means in logical terms, for instance, as expressed in a boolean expansion or a ''disjunctive normal form'' (DNF), it is perhaps a little better to go back and analyze the expression the same way that we did for ''Df''.  Thus, let us compute the value of the enlarged proposition ''Ef'' at each of the points in the universe of discourse ''U'' = ''X''&nbsp;&times;&nbsp;''Y''.

To understand what this means in logical terms, for instance, as expressed in a boolean expansion or a ''disjunctive normal form'' (DNF), it is perhaps a little better to go back and analyze the expression the same way that we did for ''Df''.  Thus, let us compute the value of the enlarged proposition ''Ef'' at each of the points in the universe of discourse ''U'' = ''X''&nbsp;&times;&nbsp;''Y''.