Changes

We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.

We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.

{| align="center" cellpadding="6" style="text-align:center" width="90%"

{| align="center" cellpadding="6" width="90%"

| <math>x ~\operatorname{and}~ y \quad \xrightarrow{~\operatorname{Diff}~} \quad dx ~\operatorname{or}~ dy</math>

| align="center" |

<math>x ~\operatorname{and}~ y \quad \xrightarrow{~\operatorname{Diff}~} \quad dx ~\operatorname{or}~ dy</math>

|}

|}

{| align="center" cellpadding="6" style="text-align:center" width="90%"

{| align="center" cellpadding="6" width="90%"

|

| align="center" |

<pre>

<pre>

o---------------------------------------o

o---------------------------------------o

<math>\operatorname{d}x</math> when crossing the border between <math>x\!</math> and <math>\texttt{(} x \texttt{)}</math> and as <math>\operatorname{d}y</math> when crossing the border between <math>y\!</math> and <math>\texttt{(} y \texttt{)},</math> in either direction, in either case.

<math>\operatorname{d}x</math> when crossing the border between <math>x\!</math> and <math>\texttt{(} x \texttt{)}</math> and as <math>\operatorname{d}y</math> when crossing the border between <math>y\!</math> and <math>\texttt{(} y \texttt{)},</math> in either direction, in either case.

{| align="center" cellpadding="6" style="text-align:center" width="90%"

{| align="center" cellpadding="6" width="90%"

|

| align="center" |

<pre>

<pre>

o---------------------------------------o

o---------------------------------------o

|}

|}

Propositions can be formed on differential variables, or any combination of ordinary logical varibles and differential logical variables, in the same ways that propositions can be formed on the ordinary logical variables alone.  For instance, the proposition <math>\texttt{(} \operatorname{d}x \texttt{(} \operatorname{d}y \texttt{))}</math> may be read to say that <math>\operatorname{d}x \Rightarrow \operatorname{d}y,</math> in other words, there is "no change in <math>x\!</math> without a change in <math>y\!</math>".

Propositions can be formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in all the same ways that propositions can be formed on ordinary logical variables alone.  For instance, the proposition <math>\texttt{(} \operatorname{d}x \texttt{(} \operatorname{d}y \texttt{))}</math> may be read to say that <math>\operatorname{d}x \Rightarrow \operatorname{d}y,</math> in other words, there is "no change in <math>x\!</math> without a change in <math>y\!</math>".

 

formula Ef(x, y, dx, dy) = f(x + dx, y + dy).

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

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>

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

Applying the enlargement operator <math>\operatorname{E}</math> to the present example, <math>f(x, y) = xy,\!</math> we may compute the result as follows:

o---------------------------------------o

o---------------------------------------o

|                                      |

|                                      |

| Ef =      (x, dx) (y, dy)            |

| Ef =      (x, dx) (y, dy)            |

o---------------------------------------o

o---------------------------------------o

Given the proposition f(x, y) in U = X x Y,

Given the proposition f(x, y) in U = X x Y,

the (first order) 'difference' of f is the

the (first order) 'difference' of f is the