Talk:Differential (infinitesimal)

Question
Is this page about derivative or diferential? I don't see any diferentials??? Stijak 12:05, 18 September 2006 (UTC)

Definitions
at the top of this article, it says that a differential is infinitely small, without defining what `infinitely small' means. Later on, the definition from Apostal states that a differential is a real variable; in fact, the `t' in the definition of the differential is the differential itself. Hence the differential can be any real number; it doesn't have to be small.

It's not particularly useful if it's not small, but it's not required to be small.

Differential calculus vs Differential (calculus)
This page is linked from Calculus as 'Differential Calculus' so we expect a succinct exposition of same, not an explanation of the historically important but fairly minor term 'differential' in calculus. I think the whole page needs a rewrite, making use of some of the good existing material. I am considering this. Expitheta 20:33, 7 January 2007 (UTC)


 * For the time being, I've relinked Differential calculus to Derivative, which is a much more complete article. However, a separate article on differentiation and differential calculus would still be very welcome. As you rightly point out, this is not it! Geometry guy 11:06, 22 February 2007 (UTC)

What is this article about?
I've tagged this article for a clean-up or rewrite. It jumps so much from one vague idea to another that it is not even clear what it is supposed to be about. I think there is some scope to turn this into an article about differentials like dx and their role in:
 * the traditional approach to calculus via infinitesimally small quantities
 * the history of calculus
 * notation such as dy/dx, and $$ \int f(x) \mathrm{d} x$$
 * motivating contemporary ideas to make all this rigorous, such as
 * the differential as a linear map and differential 1-forms
 * approaches in algebraic and differential geometry using ringed spaces such as schemes
 * nonstandard analysis and invertible infinitesimals
 * smooth infinitesimal analysis, synthetic differential geometry and nilpotent infinitesimals

I don't think any of these need to be discussed in detail, as they all have their own articles, but they could be tied together here.

Comments? Geometry guy 23:55, 21 February 2007 (UTC)


 * Looks excellent (though I don't really have the knowledge to comment on the last point). I was pleasantly surprised with our article on infinitesimals, so it might not be too hard to pull it off. -- Jitse Niesen (talk) 01:50, 25 February 2007 (UTC)


 * Okay, I've started. More to follow... Geometry guy 21:27, 25 February 2007 (UTC)


 * I've now sketched out this approach. I'll leave the clean-up tag in place for a while to encourage other people to clarify, expand and correct what I have done. Geometry guy 19:46, 28 February 2007 (UTC)

d versus d
Any special reason why "d" for differential is not italized anywhere in this article? iNic 00:31, 22 March 2007 (UTC)


 * That's quite common, though probably not as common as italics. See Wikipedia talk:WikiProject Mathematics/Archive 4 and other discussions throughout Wikipedia. -- Jitse Niesen (talk) 04:03, 22 March 2007 (UTC)

Aha, wow! Didn't know it was an UK/US-thing. (I can't remind myself of ever seeing the plain text d anywhere in math literature, but probably I have.) Thanks! iNic 05:02, 22 March 2007 (UTC)
 * Just sampling from some books I've got lying around:
 * Upright: Fraleigh & Beauregard, Martin & Shaw, Wilmshurst
 * Cursive: Griffiths, Ibach & Lüth, Garrod, Callen, Barger & Olsson, Zeilik & Gregory, Russel & Norvig, Foster & Nightingale, Pedrotti², Sagan, Stewart
 * Everything upright, inc. variables: Regtien
 * All in all, even though the sample size is too small of course, it looks like cursive is most popular, although I think the other styles are at least as readable. 82.139.87.238 (talk) 15:29, 21 September 2009 (UTC)

Differentials as linear maps
The whole part on differentials as linear maps is a bit too confusing. I think a rewrite would be most welcome. I found a very understandable version of this type of treatment in Spivak's Calculus on Manifolds (starting on page 15). A rewrite in that level would be much clearer, I think. Anyone with the proper knowledge would like to do that?

Goldencako 17:24, 7 September 2008 (UTC)

The precise definition of a differential.
Ricently I made a change in this article, but was reverted. The ideas where taken from the book Calculus: Early Transcendentals (Sixth Edition - James Stewart - McMaster University), see page 250. Here is what I wrote:

In calculus, a differential is traditionally a change in a variable (the independent one, or the respect variable for wich the diferentiantion ocurrs). If x is a variable, then a change in the value of x is often denoted Δx (or δx when this change is considered to be small). The differential dx, if x is an independent variable, is the differential of the identity function $${f(x) = x } \,$$. The precise definition of the differential of a function is as follows $${\mathrm d f(x) = f'(x) \Delta x } \,$$


 * Is it wrong?
 * I think the book is right and that is the definition of a differential.Usuwiki (talk) 00:23, 15 August 2009 (UTC)
 * Yes that is wrong. The book is right. You haven't read the book properly. The delta symbol is for a finite amount and it does not appear in the book in the way you have put it. A differential stands for an infinitessmal amount and the expression is derived as alimit. Dmcq (talk) 01:00, 15 August 2009 (UTC)
 * Sorry, the delta symbol appears in the book. Look inside Figure 5 for this equality:
 * $$\mathrm d x = \Delta x \,$$
 * Now replace the $$\mathrm d x \,$$ that appears in the definition with $$\Delta x \,$$ and that's it.
 * Also you say that a differential stands for an infinitesimal amount although the book says "$$\mathrm d x $$ can be given the value of any real number." I think you haven't read the book properly.Usuwiki (talk) 01:54, 15 August 2009 (UTC)
 * Ok. I have done so. And here is some things I note.
 * There is never the word infinitesimal in the reference I gave. In the article there is.
 * The book says that the numerical value of $$dy$$ (can be) determined. In the article it says that this value is infinitesimally small.
 * Last, the article says that This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimally small while the book says that the differential is the numerical value wich is is a dependent variable; it depends on the values of $$x \,$$ and $$dx$$, and the geometric meaning of the diferential is the verticall increase of the tangent from the point where you take the diferential. And it gives a graphic like this one:
 * [[file:Sintido geométrico del diferencial de una función en un punto.svg]]
 * There is nothing infinitesimally small in the concept of the diferential (other tan what you can get to define the derivative)Usuwiki (talk) 01:18, 15 August 2009 (UTC)
 * One last thing, if you never give this first definition:
 * $${\mathrm d f(x) = f'(x) \Delta x } \,$$
 * What sense makes this second one?
 * $$\mathrm d y = \frac{\mathrm d y}{\mathrm d x} \mathrm d x,$$
 * Now, if you have the first definition then you get for the second one this logic:
 * $$\mathrm d y = \frac{\mathrm d y}{\mathrm d x} \mathrm d x = \frac{f'(x) \Delta x}{\mathrm D_x[x] \Delta x} \mathrm D_x[x] \Delta x = \frac{f'(x) \Delta x}{1.\Delta x} 1.\Delta x = f'(x) \Delta x $$
 * Wich makes sense. Isn't it?
 * Notice that $$\mathrm d x = \Delta x$$ because of the first definition, and that is in the book too.Usuwiki (talk) 01:36, 15 August 2009 (UTC)
 * Notice that $$\mathrm d x = \Delta x$$ because of the first definition, and that is in the book too.Usuwiki (talk) 01:36, 15 August 2009 (UTC)


 * The book is badly wrong in sticking in that business with equating d's and deltas. And it shouldn't really talk in terms of numeric values for dx or dy, see differential form for how differentials are turned into something useful. You really should be asking questions on maths reference desk if you are having problems with the book rather than just changing an article and having people have to explain their reverts. You stuck in the delta instead of the d yourself rather than following the book so you obviously were thinking about the problem. Dmcq (talk) 01:54, 15 August 2009 (UTC)
 * I'm only saying that I have reference on a book about the topic. Not you neither the article have a reference on what you are saying. —Preceding unsigned comment added by Usuwiki (talk • contribs) 02:03, 15 August 2009 (UTC)
 * No what you wrote is wrong and I could easily have got rid of you by quoting the no original research policy about your derivation rather than try explaining the diagram is badly confusing. Also doing a derivation yourself and then saying it came from a book is very wrong - and then you said it was the precise definition! I've pointed out where you can get help with things like this if you're trying to learn it, you'll find people better than me at maths and at explaining things there. The article has plenty of references for you to look up a better author. What you wrote did not make sense. Dmcq (talk) 07:10, 15 August 2009 (UTC)
 * This article has a discussion about the interpretation of dx at Differential (infinitesimal). Treating themas actual changes in x or y like a delta in Finite differences makes a mess of things. I'm not sure what the author was thinking of with that diagram, I guess he was trying to illustrate a tangent space with actual numbers but the dx=delta x is just saying if the differential was this size rather than a definition of dx. If you have a tutor you should ask them about it. Personally I wouldn't introduce differentials except in the context of tangent spaces and differential forms but there has been some people thinking it easier to treat infinitessmals directly in elementary calculus. It isn't it just leads to errors when people try and differentiate twice and do it wrong. Dmcq (talk) 12:02, 15 August 2009 (UTC)


 * I have just had a look at the Springer encyclopaedia of maths for differential and it agrees with you symbolism. I am very surprised. I will refer this to maths reference desk and perhaps someone there will have a look and decide what's right. Dmcq (talk) 14:33, 15 August 2009 (UTC)
 * Have put a query at WP:RD/MA Dmcq (talk) 14:51, 15 August 2009 (UTC)
 * Good. I think here is our problem:
 * taken from http://www.geocities.com/pkving4math2tor2/2_the_der/2_06_the_diffl.htm (see Differentials Vs Infinitesimals)
 * It says: "there are 2 interpretations for the set of the quantities dx and dy: differentials and infinitesimals. Calculus with dx and dy interpreted as differentials is called standard analysis. Calculus with dx and dy interpreted as infinitesimals and the set of real numbers extended to include them is called non-standard analysis."
 * So, this article may be a part of the non-standard analysis taking dx and dy as infinitesimals and confiusing them with differentials. Or something. Lets wait to see what is decided.Usuwiki (talk) 14:57, 15 August 2009 (UTC)


 * The title of the article is differential (infinitessmal) but no, even if non-standard analysis is the modern way of putting the old idea it shouldn't go so forward I think.. I do think that a lot more introduction would be needed before using Δx as in the Springer article rather than just dump in it in the start without saying anything else about it. It isn't required for an intuitive notion and as the article says there are a number of ways of making the idea more mathematical. And it doesn't come into some of the other meanings. I'm not sure your approach and Springer's fits any of the ones there currently, even the linear map one which is closest, perhaps yet another meaning of differential should be added for analysis. Dmcq (talk) 15:49, 15 August 2009 (UTC)

Saying that it's
 * $$ dy = f'(x)\,\Delta x \, $$

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&mdash;typical freshmen can't be expected to appreciate that. The absurdity of that convention becomes apparent as soon as you think about expressions like
 * $$ \int_0^1 f(x) \, dx. $$

Michael Hardy (talk) 19:23, 15 August 2009 (UTC)

The calculus textbook writing industry is an abomination. They all just copy each other, and serious mathematicians don't want to work on this stuff because it's all about teaching the sort of freshmen who don't want to understand the subject but just want to know what to write on the test. That is cheating, and that's what the enterprise is about. Michael Hardy (talk) 19:26, 15 August 2009 (UTC)


 * Agree it is an abomination but it does seem to be notable. I don't believe the leader should be changed but somehow this way of dealing with things will have to be accomodated. It already has a number of ways of dealing with differentials. It isn't an infinitessmal but neither are they in the linear map view. It is unfortunate that Differential (calculus) is so overloaded. Dmcq (talk) 21:59, 15 August 2009 (UTC)

Here I created the article Differential of a function. Is a rough traduction from the spanish version I made. I repeat Differential $$\neq$$ Infinitesimal.Usuwiki (talk) 22:17, 15 August 2009 (UTC)


 * And differential often does mean infinitessmal if you willl look up some english maths dictionaries. yet another differential topic is fine I gues. I'd call it differential (analysis) instead of Differential of a function, that is claiming other definitions and uses of the word is invalid which migght be okay if you were writing a textbook but not an encyclopaedia. Dmcq (talk) 23:30, 15 August 2009 (UTC)

Separate the differential from the infinitesimal.
This article should get removed the phrase "a differential is traditionally an infinitesimally small change in a variable"

I propose:


 * "the differential of a function can be defined through infinitesimals (a infinitesimally small change in a variable)"
 * Or this reverted edit, but changed to end up like this:

In calculus, a differential is traditionally a change in a variable (the independent one, or the respect variable for wich the diferentiantion ocurrs). If $$x$$ is a variable, then a change in the value of $$x$$ is often denoted $$\Delta x$$ (or $$\delta x$$ when this change is considered to be small). The differential $$\mathrm dx$$, if $$x$$ is an independent variable, is the differential of the identity function $${f(x) = x } \,$$.

Through infinitesimals the differential can be represented as follows

$$\mathrm d y = \frac{dy}{dx} \mathrm d x \,$$
 * Notice that $$\frac{dy}{dx} \,$$ is in Leibniz's notation meaning the derivative of $$y$$ with respect to $$x$$.

This equation can be confusing because as can be seen there are two differntials in it, the one being defined in the left side is $$\mathrm d y \,$$ (the differential of $$y$$), and the one wich is the multiplying factor on the right side $$\mathrm d x \,$$ (the differential of the fucntion $$f(x) = x$$), but there are two infinitesimals on it also, both, interrelated in Leibniz's fraction $$\frac{dy}{dx} \,$$ at the right side of the equeation. —Preceding unsigned comment added by Usuwiki (talk • contribs) 13:07, 16 August 2009 (UTC)


 * The Leibnitz notation is what's traditional and he called them differentials and people find them easiest to start with. This is an encyclopaedia not an analysis textbook and there's more than one view on this subject. Glad to see you got the d's the right way round this time, how about not trying to change this article till you get Differential of a function in a presentable condition? And I still feel it would be better named Differential (analysis). Otherwise it will be necessary sometime to stick a link back to Differential (infinitessmal) into it and explain the difference. Dmcq (talk) 15:53, 16 August 2009 (UTC)
 * Please, this is endless. Tell one book that instead of saying "infinitesimals", it says "differentials". I know Leibniz never used the word infinitesimals, Leibniz used differentials, but that was Leibniz. Technically, what Leibniz called a differential back then, today is called an infinitesimal. —Preceding unsigned comment added by Usuwiki (talk • contribs) 16:11, 16 August 2009 (UTC)
 * I'm perfectly aware usage is changing but Leibniz's usage is still very common when introducing calculus and that's what people here will have mostly learnt. This is an encyclopaedia not a textbook. It may be possible to phrase things better but you most certainly have not. Dmcq (talk) 17:46, 16 August 2009 (UTC)

Differentials as linear maps

 * There is a simple way to make precise sense of differentials by regarding them as linear maps. One way to explain this point of view is to regard the variable x in an expression such as f(x) as a function on the real line, the standard coordinate or identity map which takes a real number p to itself (x(p) = p): then f(x) denotes the composite $$f \circ x$$ of f with x, whose value at p is f(x(p)).

With which of the following does this paragraph mean to identify "the standard coordinate or identity map"? Is it with x or f(x) or "the real line"? The syntax is ambiguous, but if I had to guess, I'd guess it was x, i.e. x is a function whose input is a real number, and the specific function which x stands for here is the identity function from reals to reals.

Are there any constraints the codomain of f, or could it be days of the week, continents, real numbers, other functions, kings of France, human limbs etc.?


 * It may seem fanciful to regard the identity map as an infinitesimal, but it does at least have the property that if $$\varepsilon$$ is very small, then $$\mathrm{d}x_p(\epsilon)$$ is very small. The differential $$\mathrm{d}f_p$$ has the same property, because it is just a multiple of $$\mathrm{d}x_p$$, and this multiple is the derivative $$f'(p)$$ by definition.

Does $$\mathrm{d}x_p(\varepsilon)$$ mean

$$\mathrm{d}(x(p))(\varepsilon)=(\mathrm{d} \circ x(p))(\varepsilon)=\mathrm{d}p(\varepsilon),$$

i.e. when you multiply a small number by an arbitrary number, the result is small. (Not true in general, since it depends on the relative size of p.) Or does it mean

$$\mathrm{d}x(p=\varepsilon)=x(p=\varepsilon)=p=\varepsilon,$$

i.e. tautologically, the smaller p is, the smaller p is.

Let $$f = x(p)^{-100} = p^{-100}$$.

$$f'(p) = -p^{-101}$$.

$$\mathrm{d}f'(p) = -p^{-101} \mathrm{d}x = -p^{-101} \cdot 1 = -p^{-101}$$

The smaller the input, the bigger the output. Would this f or this df be regarded as infinitesimal, since it too is a multiple of dx? If not, is the article (ironically) defining an infinitesimal be any number, regarded as a multiplicative operation x(p) = xp, whose absolute value is greater than or equal to 1? That is, contrary to the intro, an infinitesimal can be any size, as long as it's not very small!? Dependent Variable (talk) 19:21, 21 September 2010 (UTC)

sequences of rationals?
The subsection on nonstandard analysis contains an error. A hyperreal field extending R cannot be developed as equivalence classes of sequences of rationals; this does not quite give enough numbers. Tkuvho (talk) 18:33, 20 April 2011 (UTC)

It's usefulness is intuitive?
I don't think it is. I think you'd need to understand calculus before it becomes "intuitive". Regardless "it's intuitive" violates NPOV.75.129.109.93 (talk) 18:28, 19 April 2014 (UTC)

Missing the point
If we consider the differential as defined by a difference in a function involving an infinitesimal, then the definition in the page follows from a first order Taylor series expansion, truncated after the first derivative, and for well behaved functions the asymptotic truncation should be ok.

$$f(x+h) = f(x) + h \frac{df}{dx}, \; f(x+dx) = f(x) + dx \frac{df}{dx}, \; df := f(x+dx)-f(x) = \frac{df}{dx}\,dx, $$

What about germs of functions
Shouldn't Differential (infinitesimal) mention the exterior algebra defined over germs of functions on a differentiable manifold? Note that there is a reference to that approach in Differential (mathematics). Note that in that approach the differentials are nilpotent but the ring is not commutative. Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:34, 23 February 2015 (UTC)

Merge? Which direction
I don't see the need for both Differential (infinitesimal) andDifferential (mathematics); I would have added a merge template but it is not clear to me which should be merged into which. Comments? Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:42, 23 February 2015 (UTC)

Assessment comment
Substituted at 02:00, 5 May 2016 (UTC)