Talk:Differential (infinitesimal)

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

message: 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)
replies: text: 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)
thread_title: Differential calculus vs Differential (calculus)

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

text: 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)
thread_title:  What is this article about? 

message: 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)
thread_title:  Differentials as linear maps 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Separate the differential from the infinitesimal. 

message: :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)
thread_title:  Differentials as linear maps 

message: 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)
thread_title:  sequences of rationals? 

message: 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)
thread_title:  It's usefulness is intuitive? 

message: 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)
thread_title:  What about germs of functions 

message: I don't see the need for both Differential (infinitesimal)  andDifferential (mathematics); I would have added a  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)
thread_title:  Merge? Which direction 

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