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)

This article is complete nonsense. A rigorous, precise and useful definition of the differential of a function can be found in any book on Differential Geometry. I propose that this page is deleted. Tomgg (talk) 05:37, 24 August 2011 (UTC)

Well then some eminent mathematicians have wasted their time writing books as referenced at the start of the definition section. Have you had a look at the disambiguation page Differential (mathematics) where you can see other meanings including the pushforward in differenfial geometry? Dmcq (talk) 09:00, 24 August 2011 (UTC)

I doubt one of these "eminent" mathematicians (Boyer is a historian) would have written the first equation on the page; it's a perfect example of a circular definition. This page is a source of ambiguity which serves the sole purpose of embracing a qualitative idea housed only in the minds of undergraduates and bad physicists. This page needs a rewrite, which stresses that these ideas (the development of the differential) are incredibly inchoate, or needs to be taken down. Tomgg (talk) 13:12, 27 August 2011 (UTC)

It's not circular. The derivative is defined independently of differentials. Sławomir Biały (talk) 13:35, 27 August 2011 (UTC)

Well Boyer was just referenced in the history section and isn't a source for the definition. Kline who was one of the ones I was referencing to did also write Mathematical thought from ancient to modern times but I don't think that makes his work irrelevant! Dmcq (talk) 14:53, 27 August 2011 (UTC)

"dy" is a differential. So is "dx". It is indeed circular. Tomgg (talk) 22:26, 27 August 2011 (UTC)

As explained in the article, dy and dx are new real variables. dy is a function of x and dx. Sławomir Biały (talk) 22:59, 27 August 2011 (UTC)

"dx" and "dy" are not variables; they're functionals. The equation is not even quasi rigorous. Once I work out how to put forward a page deletion request, this page will be there immediately. Tomgg (talk) 01:02, 28 August 2011 (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)

Forgive me for inserting this comment out of temporal order, but I just wanted to second the claim that the books by Apostol and Spivak are excellent. WardenWalk (talk) 09:43, 19 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)

I added a note to linear approximation, and I now suggest that the article here be made a redirect to that article, as suggested by Plclark.WardenWalk (talk) 09:41, 19 August 2009 (UTC)

I still think that a better redirect target is differential (infinitesimal). Although, per Plclark, linear approximation is also relevant, differential (infinitesimal) is much more directly so. Given the subsequent thread, I think that further discussion should be undertaken to foster a consensus about where to redirect the article. Sławomir Biały (talk) 14:50, 19 August 2009 (UTC)

It is a 1-form

The differential dy of a function y is a 1-form. This blindingly obvious fact should be covered, shouldn't it? Geometry guy 23:09, 17 August 2009 (UTC)

I think what should be covered is this: It is not a precisely defined concept, but admits a number of reasonable interpretations including (perhaps especially) non-rigorous heuristic interpretations (and the 1-form interpretation is one of the reasonable interpretations, but Leibniz wouldn't recognize it, so it shouldn't be near the beginning of the list). The notion that a differential of a function y = ƒ(x) is dy = ƒ ′(x) Δx was invented by people who are unjustifiably squeamish about the fact that ideas that are not precisely defined concepts exist. And that notion is seen to be absurd when you think about integrals. Michael Hardy (talk) 23:20, 17 August 2009 (UTC)

(I wrote most of Differential (infinitesimal), with some care to source it, and it seems positively regarded here. I hope this article, if not made into a redirect, will benefit from a similar approach.) Geometry guy 00:01, 18 August 2009 (UTC)

I just had a look on google books for a more modern definition of differential by putting in differential form meaning and the first one that had anything relevant, Differential geometry: Cartan's generalization of Klein's Erlangen program by RW Sharpe on page 53, had the dy as a linear map and Δx as an infinitessmal. He used the format

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

it says dy is the general allowed infinitessmal change in y, and that it can be applied to any specific infinitessmal change Δx. Different yet again and Δx doesn't really need to be infinitessmal but overall better in the notation department, dy isn't treated as the result of a function of two variables. Perhaps this will put the bit about infinitessmals to rest, it is a bit much saying in effect show me the evidence but dismiss anything derived from Leibniz notation as that doesn't count, and anything saying it is infinitessmal is Leibniz notation!

Overall I think something like this has to be put in as it satisfies the notability criteria and quite a few people will be taught like this whatever about people's thoughts on the teching. Dmcq (talk) 12:09, 18 August 2009 (UTC)

Two notions of differential

There are two mathematically precise notions of differential of a function y=f that are relevant to this discussion.

• One is that it is a 1-form dy, i.e., an object whose value at any x is a cotangent vector. It is usually this notion that one is using when one writes dy = f'(x) dx.
• The other is that dy is the derivative Df of a map f between 1-dimensional manifolds, i.e., an object whose value at any x is a linear map from the tangent space of R at x to the tangent space of R at f(x). That linear map can be identified with a $1\times 1$ matrix whose entry is f'(x). Sharpe is talking about this one when he writes $dy(\Delta x)=f'(x)\Delta x$.

Anyway, it would be good to include a down-to-earth (i.e., understandable to calculus students) explanation of what is meant by equations like dy = g(x) dx. I think what should be said is something like

In elementary calculus, the differential dy is not defined. When an "equation" dy = f'(x) dx is written in elementary calculus, it means only that the derivative of y with respect to x equals f'(x).

Nevertheless, for intuition it often helps to imagine that dx and dy represent actual small numbers. If one replaces dx in the "equation" by an actual difference $\epsilon$ of x-values, then the resulting right hand side $f'(x)\epsilon$ represents not the actual difference in y-values, $f(x+\epsilon )-f(x)$, but the corresponding difference $F(x+\epsilon )-F(x)$ where F is the linear approximation whose graph is the tangent line at (x,f(x)). So if one imagines that dx represents $\epsilon$, then one should also imagine that dy represents $F(x+\epsilon )-F(x)$.

The main point I am trying to make is that it is OK to give intuition as long as our article is not pretending that it is giving a mathematically precise definition of differential.

Also, I don't think a proliferation of different articles on differentials is what we want. It would be best to have a redirect (Plclark has the right idea - he is worth listening to), and to modify one of the existing pages as necessary.

WardenWalk (talk) 00:06, 19 August 2009 (UTC)

We should not say that the expression is not defined. It is defined, and its definition is important. What we should say is that the definition used at that point is not logically rigorous. You can't expect liberal-arts students with no interest in becoming mathematicians to the meanings of "defined" and "undefined" from a logically rigorous point of view. Telling them whether or not something is "defined" from that point of view is a waste of time. One should probably mention such things, but one should not rely on them to know just what such things mean. If you start by saying it's not defined, you're acting as if you assume they can understand that. Michael Hardy (talk) 01:50, 19 August 2009 (UTC)

You've convinced me that saying "it is not defined" in confusing, even with the qualifier "in elementary calculus". I certainly agree with you that it is defined in more advanced mathematics courses. I would avoid saying "we are giving a definition but it is not a logically rigorous definition", since this makes it sound as if there are two kinds of definition, ones that are rigorous and ones that are not, whereas I think it is more honest to avoid calling the nonrigorous ones definitions at all.

Actually (even though this partially goes against what I was saying earlier about it not being defined in elementary calculus), if you say that dy is a two-variable function in indeterminates called x and $\Delta x$, and its value is $f'(x)\,\Delta x$, then that is a perfectly rigorous definition of a function (even if it is a hack that does not fit well with the elegant definition given in differential geometry).

I'm not yet sure what we should say in the article, but in any case it should be honest. WardenWalk (talk) 07:44, 19 August 2009 (UTC)

I'm very unkeen on the business of saying dy is a function of Δx. I'd prefer to get rid of the dx = Δx in this article and replace it with dx(Δx) = Δx. We then have df(x)(Δx)=f'(x)Δx and df(x)=f'(x)dx but we don't get that horrible df(x)=f'(x)Δx. Saying dx=Δx is then just an abbreviation, and not a nice one at that. Dmcq (talk) 08:16, 19 August 2009 (UTC)

Then again it looked like they actually wanted to be able to write Δy-dy as the error term when doing numeric stuff. Which really doesn't fit in except with how this article puts it. Dmcq (talk) 08:48, 19 August 2009 (UTC)

I too don't like the hack of saying that dy is a function of (x and) Δx, or equivalently that df(x) is a function of Δx, and I am not advocating this even though I acknowledge that it is well-defined. But when you write df(x)(Δx)=f'(x)Δx, you seem to be defining df(x) as a function of Δx. If not, then what kind of object are you defining df(x) as, if not a function? And if it is a function to you, what are you thinking of as its domain and codomain? WardenWalk (talk) 08:50, 19 August 2009 (UTC)

To me d is an operator df(x) is a linear map, here used as a function. It is applied to the vector Δx to produce a number. Hope I'm not too confused. Dmcq (talk) 09:58, 19 August 2009 (UTC)

That's basically OK, except that if Δx is a vector (presumably a tangent vector), then at each x, f'(x)Δx is a scalar multiple of that vector and hence is a tangent vector of the same type, not a number. This is the second notion of differential (in my two-item list at the top of this section). On the other hand, I gather that what the current article is trying to do is to get away with having a variant of the first notion, a variant in which 1-forms are interpreted as actual numbers, which conflicts with the actual definition of 1-forms.

Anyway, as mentioned above, I added a note to linear approximation, and I hope (along with Plclark) that this article can be replaced by a redirect to that page. WardenWalk (talk) 13:38, 19 August 2009 (UTC)

Sweeping revision

I have just radically revised the whole article.

I deleted the "Disputed" tag I added earlier.

You'll notice the definition of total differential and partial differential. One of the various great virtues of the Leibniz notation is that it makes ideas like this so simple. Is there any easier heuristic argument for the chain rule for partial derivatives than that?

(And at this time, chain rule for partial derivatives is a red link! Should we remedy that?)

Also, I've proposed a merger with differential (calculus).

We should consider adding to the article the more advanced and otherwise different viewpoints, including 1-forms. Michael Hardy (talk) 16:26, 20 August 2009 (UTC)

Differential (calculus) is really a disambiguation page at the moment, so if it's going to be merged with anything, it should be merged with differential, not this page.

I object to the article's statement df = (dy/dx)Δx. There is no sense in which this is true. One can write Δf or dx instead, and then the statement will make sense; but what's there presently doesn't.

If this article is not to be merged, then I think it should focus on approximation: Δf, not df. This is the way in which differentials are usually introduced in calculus, and it's much more elementary than talking precisely about infinitesimal information. Ozob (talk) 22:48, 20 August 2009 (UTC)

I think you have just gotten confused by the notation, and perhaps justifiably so. In the expression df = (dy/dx)Δx that you are reacting to, the (dy/dx) retains its usual definition as the derivative of the function y = ƒ(x). That is to say, I had intended it to be read in exactly the same way as the very next formula,

$df(x,\Delta x)=f'(x)\,\Delta x.$

I will just get rid of the first formula for the sake of clarity. Secondly, the notation df in the article does not refer to an infinitesimal. Rather the first paragraph may convey a false impression that there are infinitesimals floating about. Anyway, I think I am basically in agreement with you about what the style of treatment here should be. As you have already said, there is a notable concept that is (or was) the intended focus of this article. Moreover, for folks like Michael Hardy who think the infinitesimal approach is the way to go, we have Differential (infinitesimal). So I think this article could fill an important niche. Any chance I elicit more detailed feedback? Sławomir Biały (talk) 03:39, 21 August 2009 (UTC)

Cauchy is being misrepresented

Cauchy is being misused to justify this definition of differential. Cauchy never made the definition written in this Wikipedia article, as far as I can tell from reading the cited 1823 text. First of all, Cauchy is still talking about "infinitely small quantities". For him, Δx is a an infinitely small quantity (see p. 30). He talks about limits, but does not have a definition of limit that any modern mathematician would consider to be a definition - that came much later in the 19th century. Modern textbooks generally do not follow Cauchy's treatment.

I still feel that most of the content in this article is more carefully presented in already existing articles. It would be better to try to improve those articles than to duplicate them by adding content here. WardenWalk (talk) 23:08, 20 August 2009 (UTC)

I think you are misreading the Cauchy source. He does not say that Δx is infinitely small, but that the error is infinitely small, by which he clearly means (in the context of the statement you are quoting) that the error is what we would today call $o(\Delta x)$ as Δx → 0. Moreover, there is a reliable secondary source (Kline, 1972) attributing the use of finite increments to Cauchy.

It is also instructive to compare with (Goursat, 1904), where the terminology "infinitely small" is also applied, but ultimately only positive real increments are actually considered. Instead, "infinitely small" only refers to the limiting process under which the variable tends to zero. This is explained in the now-cited Boyer reference. I do, by the way, agree with you that Cauchy's notion of a limit differs substantially from the modern notion. But that has no bearing on how Cauchy defined the differential (which is the subject of this article): for Cauchy, neither dy nor dx represented infinitesimal quantities in Leibniz's sense. That aspect of things, as well as Cauchy's overall handling of differentials, is very much in accord with the modern treatments cited in the text. The secondary historical sources also bear this point out. Sławomir Biały (talk) 01:24, 21 August 2009 (UTC)

Also, I find the first statement that Cauchy is being "misused" to be a quite curious way of putting things, as if it is to suggest that some conspiracy in presenting things in this way. Let me just add to the above that I have, to the best of my ability, attempted to represent what reliable sources have to say about the matter. This includes very well-regarded sources, not just by Cauchy, but Edouard Goursat's analysis, Richard Courant's influential calculus textbook, as well as that of Morris Kline. Other similarly highly-regarded sources, reflecting other points of view, would be appreciated. However, I am actually surprised by the substantial agreement that I have found among the various calculus and mathematical analysis sources when researching this topic. Sławomir Biały (talk) 00:10, 21 August 2009 (UTC)

In the 1st sentence, on p.27, Cauchy says that i always represents an infinitely small quantity. On p.30, Δx is taken to be i. (P.S. Sorry, I didn't mean to question your motives; I'm just pointing out what Cauchy actually says.) WardenWalk (talk) 07:20, 21 August 2009 (UTC)

I have expounded on Cauchy's precise contribution, and how he broke with the traditional infinitesimals of Leibniz by introducing the limit concept. Significant for this article is, of course, the point that dx and dy are not "fixed infinitesimals", nor "smaller than any given positive quantity", but merely new real-valued variables representing an increment in the function. This is, of course, in marked contrast with how Leibniz regarded the differentials. This material is well-sourced and not contentious. I ask that you now remove the {{disputed}} tag, since I find it very disruptive. Sławomir Biały (talk) 12:40, 21 August 2009 (UTC)

Hi, thank you for making changes. It is looking better. It's still not clear to me whether Cauchy was really thinking of dx as an increment in x values. Also, even if dx is no longer an infinitesimal, Cauchy still uses infinitesimals in his explanation of what the derivative is, even if he also talks about limits. He seems halfway between Leibniz and Weierstrass. At times, it almost seems as if h is playing the role of a tangent vector in his text. I'll remove the disputed tag; there are parts of the article that look fine; but there are still things that I find dubious, so I will leave some comments about this in the hope that others will express their opinions here. Thank you, WardenWalk (talk) 13:26, 21 August 2009 (UTC)

So, we have now dispensed with the objection of dx being infinitesimal. Now, it is true that Cauchy uses terms like "infinitely small" in his explanation of the derivative. The important conceptual advance here over the approach of Leibniz is that "infinitely small" does not actually refer to a fixed infinitesimal quantity, but rather explicitly to the limit; see his own explanation of the locution "infinitely small" on p. 12. Thus Cauchy is careful to refer to the derivative as the quotient

${\frac {f(x+i)-f(x)}{i}}$

when i is "infinitely small", and with the proviso that, as Cauchy makes, the term "infinitely small" is understood as "in the limit as i → 0", this is precisely the modern definition of the derivative of a function of one variable. Boyer, p. 275, without reservation, translates Cauchy's defintion to mean: "The limit of this ratio... as i approaches zero." The reason that the terminology "infinitely small" was adopted appears to be a desire not to break completely with tradition. But the term quite clearly does not refer to an actual infinitesimal quantity: even Weierstrass, for whom the notion of an actual "infinitely small" quantity was anathema, used almost exactly the same terminology in the verbal description of continuity

"Infinitely small variations in the arguments correspond to those of the function."

--Sławomir Biały (talk) 14:10, 21 August 2009 (UTC)

I think you are more or less right, though what still seems to go against this is that Cauchy will speak of a ratio of infinitesimals and then speak of its limit as if the limit were distinct from the quotient itself. I think what (in modern terms) might be closer to his meaning is that one infinitesimal is an indeterminate, and the others are functions of this indeterminate, so what Cauchy calls a ratio of infinitesimals is what we would call a difference quotient today (a ratio of functions of the unspecified increment), and when Cauchy takes a limit, it is truly a limit as the value of the first indeterminate tends to 0 through actual real numbers. --WardenWalk (talk) 18:04, 21 August 2009 (UTC)

It is one thing for Courant to say that Cauchy's approach to differentials is logically satisfactory, and another thing for him to say that it is "the standard approach today". The quotation you give is evidence for the former, but not the latter. WardenWalk (talk) 18:56, 21 August 2009 (UTC)

Dear Sławomir Biały, You wrote

"ultimately, I plan to mention the differential as a linear map, but this is not the place for it, and is not supported by the references here."

But this whole article is about defining dy as a linear function of Δx! You yourself added a lot of this, and the references supporting this! You also wrote

Also, the Stewart calculus book is no longer used.

Actually you can find many college campuses that are using it in Fall 2009. WardenWalk (talk) 19:02, 21 August 2009 (UTC)

The reference to Courant supports the earlier contention that Cauchy originated what might be termed the modern view of the differential of a function in calculus. This statement was, however, moved onto a different sentence (which had already been sourced to the Boyer text), in which it was claimed that the principal insight of Cauchy's approach is that it permits treating dx and dy as additional real variables. I would not have changed it back, except for the fact that Boyer makes a very significant point of this, so the changes made to the history section were, upon reflection, not the best given the sources that I have provided. Finally, I think a section on the differential as a linear functional should be included, but it should be presented in a more systematic way. Also "not used" was a poor locution on my part as "no longer a reference in the article" would not fit in the edit summary field. All of the sources currently used are intended to pass muster with Mr. Hardy, as textbooks written by mathematicians of the highest caliber. Sławomir Biały (talk) 19:05, 21 August 2009 (UTC)

Hi, thank you for the explanations. I don't have Boyer's book in front of me, so I'll trust you on that. Anyway, mainly what I was trying to fix is this:

Currently the article suggests that there is only one modern definition of differential, and that it is identical to Cauchy's except for Weierstrass making precise the definition of limit. In suggesting this, the article completely ignores the definition in advanced calculus of differentials as 1-forms etc. This other definition is related to, but certainly not identical to Cauchy's definition. And in higher mathematics, this other definition is as respected (probably even more respected) as the one in this article. (I could provide references later on if necessary, but I am not with my library at the moment, and in any case, I suspect that you already know several references yourself, given that you seem to be rather knowledgeable about the subject.)

This is what I was trying to get across; if you can think of a suitable way of wording this, then go ahead and update the text yourself. WardenWalk (talk) 20:08, 21 August 2009 (UTC)

I now see that these other definitions are alluded to later in the article. So I guess my point is just that the history and the definition sections at the top need to be written so as to allow for the possibility that these other definitions exist and are used extensively in modern math. WardenWalk (talk) 20:14, 21 August 2009 (UTC)