Changes

==Note 15==

==Note 15==

<pre>

<br>

| 'Tis a derivative from me to mine,

 

|  And only that I stand for.

{| cellpadding="2" cellspacing="2" width="100%"

|

| Winter's Tale, 3.2.43-44

'Tis a derivative from me to mine,<br>

And only that I stand for.

|-

| &nbsp;

| &mdash; ''Winter's Tale'', 3.2.43&ndash;44

 

We've talked about differentials long enough

We've talked about differentials long enough that I think it's past time we met with some.

that I think it's past time we met with some.

When the term is being used with its more exact sense,

When the term is being used with its more exact sense, a ''differential'' is a locally linear approximation to a function, in the context of this logical discussion, then, a locally linear approximation to a proposition.

a "differential" is a locally linear approximation to

a function, in the context of this logical discussion,

then, a locally linear approximation to a proposition.

I think that it would be best to just go ahead and

I think that it would be best to just go ahead and exhibit the simplest form of a differential <math>\operatorname{d}F\!</math> for the current example of a logical transformation <math>F,\!</math> after which the majority of the easiest questions will have been answered in visually intuitive terms.

exhibit the simplest form of a differential dF for

the current example of a logical transformation F,

after which the majority of the easiest questions

will've been answered in visually intuitive terms.

For F = <f, g> we have dF = <df, dg>, and so we can proceed

For <math>F = (f, g)\!</math> we have <math>\operatorname{d}F = (\operatorname{d}f, \operatorname{d}g),</math> and so we can proceed componentwise, patching the pieces back together at the end.

componentwise, patching the pieces back together at the end.

We have prepared the ground already by computing these terms:

We have prepared the ground already by computing these terms: