Changes

To denote lists of propositions and to detail their components, we use notations like:

To denote lists of propositions and to detail their components, we use notations like:

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

{| align="center" cellpadding="10" width="90%"

| <math>\mathbf{a} = (a, b, c), \quad \mathbf{p} = (p, q, r), \quad \mathbf{x} = (x, y, z),</math>

| <math>\mathbf{a} = (a, b, c), \quad \mathbf{p} = (p, q, r), \quad \mathbf{x} = (x, y, z),</math>

|}

|}

or, in more complicated situations:

or, in more complicated situations:

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

{| align="center" cellpadding="10" width="90%"

| <math>x = (x_1, x_2, x_3), \quad y = (y_1, y_2, y_3), \quad z = (z_1, z_2, z_3).\!</math>

| <math>x = (x_1, x_2, x_3), \quad y = (y_1, y_2, y_3), \quad z = (z_1, z_2, z_3).\!</math>

|}

|}

The first operator, <math>\operatorname{E}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{E}f</math>, where <math>\operatorname{E}f : X \times X \to Y</math> is defined by the following equation:

The first operator, <math>\operatorname{E}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{E}f</math>, where <math>\operatorname{E}f : X \times X \to Y</math> is defined by the following equation:

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

| <math>\operatorname{E}f(x, y) ~=~ f(x + y).</math>

<math>\operatorname{E}</math> is called a "shift operator" because it takes us from contemplating the value of <math>f\!</math> at a place <math>x\!</math> to considering the value of <math>f\!</math> at a shift of <math>y\!</math> away.  Thus, <math>\operatorname{E}</math> tells us the absolute effect on <math>f\!</math> that is obtained by changing its argument from <math>x\!</math> by an amount that is equal to <math>y\!</math>.

<math>\operatorname{E}</math> is called a "shift operator" because it takes us from contemplating the value of <math>f\!</math> at a place <math>x\!</math> to considering the value of <math>f\!</math> at a shift of <math>y\!</math> away.  Thus, <math>\operatorname{E}</math> tells us the absolute effect on <math>f\!</math> that is obtained by changing its argument from <math>x\!</math> by an amount that is equal to <math>y\!</math>.

The next operator, <math>\operatorname{D}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{D}f</math>, where <math>\operatorname{D}f : X \times X \to Y</math> is defined by the following equation:

The next operator, <math>\operatorname{D}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{D}f</math>, where <math>\operatorname{D}f : X \times X \to Y</math> is defined by the following equation:

: <math>\operatorname{D}f(x, y) = \operatorname{E}f(x, y) - f(x),</math>

| <math>\operatorname{D}f(x, y) ~=~ \operatorname{E}f(x, y) - f(x),</math>

or, equivalently,

or, equivalently,

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

| <math>\operatorname{D}f(x, y) ~=~ f(x + y) - f(x).</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>

<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, <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:

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

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

 

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

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

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>.

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>.

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

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

 

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

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

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.

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.

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

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

 

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

| <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, with <math>\operatorname{d}x</math> denoting small changes contemplated in <math>x\!</math>.

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