Talk:Differential of a function

Title

I'd call it something like Differential (analysis) to deliniate the area in which it is a valid definition. It isn't what people mean by it in topology for instance and it isn't the infinitessmal version. Dmcq (talk) 23:32, 15 August 2009 (UTC)

Dmcq, there is no book that will tell you a differential has an "infinitesimal version" as in "the differential is an infinitesimal". It is more likely that you will find, if lucky, in some book that the differential is "the ratio of two infinitesimals" multiplyied by a finit increment. Read that carefully.

And then you could find some equation like this one:

$dy={\frac {\mathrm {d} y}{\mathrm {d} x}}\Delta x\,$

• Where ${\frac {\mathrm {d} y}{\mathrm {d} x}}$ is in Leibniz's notation and means "the derivative of y with respect to x."
• Notice $dy\neq \mathrm {d} y\,$. $dy$ is the differential, while $\mathrm {d} y$ is the infinitesimal. We need to differentiate (contradistinguish the variables) because of the notation choosed to represent the derivative.

Considering this other equality:

$\mathrm {d} y={\frac {\mathrm {d} y}{\mathrm {d} x}}\mathrm {d} x\,$

I see it is true, whether you take it as Leibniz's notation or not.

The last equality in Leibniz's notation, means that dy (in both sides of the equation) is an infinitesimal. The left side of the equation is NOT the differential dy, IT IS the infinitesimal dy.Usuwiki (talk) 00:04, 16 August 2009 (UTC)

You better point that convention out in the article clearly. It doesn't seem to follow it so far, in fact you seem to have the opposite and used the upright d for a differential. The notation has been round long before two lots of d's and d traditionally has meant an infinitessmal amount and it has been referred to as a differential. You can deal with it however you like but you'll need to explain what you mean in the article rather than just complain about people misunderstanding it. Dmcq (talk) 00:53, 16 August 2009 (UTC)

Exactly. You can use whatever d you want as long as you are clear about what notation you are using, if it's Leibniz's one, then you are talking about infinitesimals, if not, then, most likely, you are talking about differentials.Usuwiki (talk) 02:47, 16 August 2009 (UTC)

Why this article?

I am not sure I understand how this article should be different from Differential (infinitesimal)#Differentials as linear maps. If anything, that section is more correct and more in-depth. Expanding a section of an article into a full-fledged article should be done to actually give a more complete treatment of a subject only skimmed there, which is not the case here, as the existing section is larger and clearer than this article. If I had not known something about the subject, I'd be quite baffled. For instance, what does "is defined as the result of a product of two values, one of wich is obtained [...]" mean? Is this a definition? Should it help intuition? It is almost as saying "the area of a rectangle is defined as the product of two things, one of which is obtained by measuring the height of the rectangle"... [[::User:Goochelaar|Goochelaar]] ([[::User talk:Goochelaar|talk]]) 14:47, 16 August 2009 (UTC)

First, the section "differentials" as linear maps explains something different from what is explained here. Here is explained what a differential is. In the other article it seem that infinitesimals are explained.

See the equation here for the one-dimensional case and compare it with the one in the talk page here. They are different, fundamentally different, the one in the talk page expresses a differential, the other in the linear maps's section doesn't.

Second, you are right about the "product of two things" part. Sorry about that. Lets try to fix it.Usuwiki (talk) 16:22, 16 August 2009 (UTC)

Thanks for your answer, but I am still not sure I see your point. It would help if in the first sentence of the article (thanks for somewhat fixing it) there were such a sentence as, say, "In the field of mathematics called calculus, the differential is..." what? a map? a number? a set? From what follows, I don't understand how your definition is different from saying that a differential is a linear map as in the other article, given as you say that it is the product of the derivative of the function at the point, i.e. a number, and the independent variable $\Delta x$. This looks exactly to be a linear function of the variable $\Delta x$, as said, with a slightly different notation, in the other article. Anyhow, thanks for your contributions, [[::User:Goochelaar|Goochelaar]] ([[::User talk:Goochelaar|talk]]) 16:56, 16 August 2009 (UTC)

This article is worthless garbage

Saying that a differential is defined as

$dy=f'(x)\,\Delta x\,$

is a bit of nonsense that modern textbook writers have adopted out of squeamishness about infinitesimals, stemming from the fact that you can't present infinitesimals to freshmen in a logically rigorous way. Insisting on logical rigor is clearly a mistake—typical freshmen can't be expected to appreciate that. The absurdity of that convention becomes apparent as soon as you think about expressions like

$\int _{0}^{1}f(x)\,dx.$

Modern calculus textbooks are just copies of each other. Their authors don't know or care about the subject. They care about standardized testing. Michael Hardy (talk) 18:39, 16 August 2009 (UTC)

...following up on my last paragraph, it occurs to me that among the exceptions to the zeroxing method of writing calculus textbooks, written by thoughtful authors who care about it, two obvious ones come to mind: those by Apostol and Spivak. Both are written for students who want to think like mathematicians. Maybe there is no honest calculus textbook for liberal-arts students. Michael Hardy (talk) 18:44, 16 August 2009 (UTC)

I've brought up this issue at Wikipedia talk:WikiProject Mathematics. Michael Hardy (talk) 04:47, 17 August 2009 (UTC)

I think there is a bit of a culture clash here. As far as I can make out, and I could very easily be wrong, this has come from an analysis/numerical viewpoint and may have started in Russia investigating linear differential operators including both Δx and dx and suchlike, and they'd want them in the same terms and comparable. I'd guess more people here see differentials as being more part of studying manifolds and start with a topological outlook and aren't so interested in finite differences. You got them both using linear maps and the same symbols so it grates. Dmcq (talk) 06:36, 17 August 2009 (UTC)

Upon reflection, I think the present article is trying to get at the material of linear approximation, however with a somewhat awkward setup. My biggest problem is with the notation $df(x)=f'(x)\Delta x$.

On the left hand side, we have (apparently) a function of one variable, x, whereas on the right hand side we have functions of x and $\Delta x$. This notation suggests that the dependence on x is somehow more important than the dependence on $\Delta x$, which is completely absurd. Still, this approach is not that different from what one finds in many calculus textbooks, in which "the differential approximation" is

$dx\approx \Delta x,dy\approx \Delta y$

and "therefore" (because the quotient of quantities which are approximately equal must be approximately equal?!?)

${\frac {\Delta y}{\Delta x}}\approx {\frac {dy}{dx}}=f'(x)$.

This is a lot of mystical language to "justify" approximating a differentiable function near a point $x_{0}$ by its tangent line at $x_{0}$. What's even worse, the justification is never given quantitatively (except possibly hundreds of pages later when Taylor's theorem with remainder is covered): there is a conflation of the fact that the error goes to zero as $\Delta x\rightarrow 0$ (by definition of the derivative!) with the hope that if $\Delta x$ is "reasonably small" then dy will be "reasonably close" to $\Delta y$, which, if it means anything at all, is certainly false without additional hypotheses. End of rant.

Oh, about the article? Perhaps it would make sense to delete what is present and redirect to linear approximation. Plclark (talk) 08:24, 17 August 2009 (UTC)

Perhaps redirecting to differential (infinitesimal) would be more useful to a prospective reader? Sławomir Biały (talk) 16:26, 17 August 2009 (UTC)

Both linear approximation and differential (infinitesimal) are competently written, relevant articles. In terms of levels of sophistication, the latter article is rather high, whereas the former seems to be at about the same level as the intended audience of Differential of a function. Plclark (talk) 20:26, 17 August 2009 (UTC)

It seem that I'm the only one defending this possition. And it seems that I'm defending a textbook. Let whatever you decide, happen.Usuwiki (talk) 22:16, 17 August 2009 (UTC)