Changes

<math>\operatorname{D}</math> is called a "difference operator" because it tells us about the relative change in the value of <math>f\!</math> along the shift from <math>x\!</math> to <math>x + y.\!</math>

<math>\operatorname{D}</math> is called a "difference operator" because it tells us about the relative change in the value of <math>f\!</math> along the shift from <math>x\!</math> to <math>x + y.\!</math>

In practice, one of the variables, <math>x\!</math> or <math>y\!</math>, is often considered to be "less variable" than the other one, being fixed in the context of a concrete discussion. Thus, we might find any one of the following idioms:

In practice, one of the variables, x or y, is often

considered to be "less variable" than the other one,

being fixed in the context of a concrete discussion.

Thus, we might find any one of the following idioms:

1.  Df : X x X -> Y,

: <math>\operatorname{D}f : X \times X \to Y,</math>

    Df(c, x) = f(c + x) - f(c).

: <math>\operatorname{D}f(c, x) = f(c + x) - f(c).</math>

Here, c is held constant and Df(c, x) is regarded

Here, <math>c\!</math> is held constant and <math>\operatorname{D}f(c, x)</math> is regarded mainly as a function of the second variable <math>x\!</math>, giving the relative change in <math>f\!</math> at various distances <math>x\!</math> from the center <math>c\!</math>.

mainly as a function of the second variable x,

giving the relative change in f at various

distances x from the center c.

2.  Df : X x X -> Y,

: <math>\operatorname{D}f : X \times X \to Y,</math>

    Df(x, h) = f(x + h) - f(x).

: <math>\operatorname{D}f(x, h) = f(x + h) - f(x).</math>

Here, h is either a constant (usually 1), in discrete contexts,

Here, <math>h\!</math> is either a constant (usually 1), in discrete contexts, or a variably "small" amount (near to 0) over which a limit is being taken, as in continuous contexts.  <math>\operatorname{D}f(x, h)</math> is regarded mainly as a function of the first variable <math>x\!</math>, in effect, giving the differences in the value of <math>f\!</math> between <math>x\!</math> and a neighbor that is a distance of <math>h\!</math> away, all the while that <math>x\!</math> itself ranges over its various possible locations.

or a variably "small" amount (near to 0) over which a limit is

being taken, as in continuous contexts.  Df(x, h) is regarded

mainly as a function of the first variable x, in effect, giving

the differences in the value of f between x and a neighbor that

is a distance of h away, all the while that x itself ranges over

its various possible locations.

3.  Df : X x X -> Y,

: <math>\operatorname{D}f : X \times X \to Y,</math>

    Df(x, dx) = f(x + dx) - f(x).

: <math>\operatorname{D}f(x, \operatorname{d}x) = f(x + \operatorname{d}x) - f(x).</math>

This is yet another variant of the previous form,

This is yet another variant of the previous form, with <math>\operatorname{d}x</math> denoting small changes contemplated in <math>x\!</math>.

with dx denoting small changes contemplated in x.

   

   

That's the basic idea.  The next order of business is to develop

That's the basic idea.  The next order of business is to develop the logical side of the analogy a bit more fully, and to take up the elaboration of some moderately simple applications of these ideas to a selection of relatively concrete examples.

the logical side of the analogy a bit more fully, and to take up

the elaboration of some moderately simple applications of these

ideas to a selection of relatively concrete examples.

===Note 2===

===Note 2===