Talk:Differential (mathematics)

Untitled
A differential in mathematics sounds like a derivative. It also implies that it isn't the same thing. More clarification on this needs to be made. Is a differential a derivative? If not, whats the difference? Fresheneesz 22:06, 10 February 2006 (UTC)
 * Perhaps, is a differential what you do when you differentiate explicietly? For example, if dy/dx is the derivative of y with respect to x, is the differential of y dy, and the differential of x, dx? Fresheneesz 09:01, 26 March 2006 (UTC)
 * I know that much is true, but the meaning of a differential by itself is not very clear to me (i.e., not in a derivative or integral). Perhaps it could be improved in this sense. Commander Nemet 05:45, 18 May 2006 (UTC)
 * Actually, I take that back. I know that a differential is the limit of a difference (delta); correct me if I'm wrong.  What I wonder about is what happens when you take the differential of a function, i.e. d(f(x)).  Does this operation even have any meaning? I've seen it done before...Commander Nemet 05:52, 18 May 2006 (UTC)
 * Er, forget it. I remember what it means.  But I wonder if it could be added into the article somehow.  I'm not confident with my formal knowledge of it to do it myself. Commander Nemet 07:05, 25 May 2006 (UTC)
 * ur right d(something) is an operator but the definition isnt the infinitesmall increment. in a function of teh form f(x,y)=C, dy=(dy/dx)dx, it may remind an 8 year old the cancelation rule but thats not wat it means. im still working on partial differentiation. i suggest the removal of infinitesmall quantity its not even rigorously grounded in any area of math -getting stoned is fun :P Smoke Weed 13:01, 8 July 2006 (UTC)
 * oopss just like to make a few thing clear for fresheneesz: yes when u differentiate y=f(x) and the deriavative exist u end up with dy/dx=somthing. multiply both sides by dx and u got ur differential form, although u must separat all the y variable to dy and vice versa for dx to integrate go to differential equations and look there for a cleaer description im too tired and got a stupid history asignment due tommorrow (man i wish i can drop history)
 * A differential IS an infinitely small change in a variable. I'm not sure if d(f(x)) should strictly be considered an operation. it generally represesnts a 'value'. It has the same units as f(x) but if you were to measure it, it would be indistinguishably close to zero. A good example is in a single-variable integral which always ends in a differential. This differential represents the width of the rectangles, of which there are infinitely many. Thus, the rectangles do not have any real thickness, but in concept we must account for that aspect of each one. A derrivative is the ratio of two differentials, typically it is $$\frac{dy}{dx}$$ which is the change in y devided by the change in x (see slope formula) however, since the derivative is taken at a point, not across a secant line, the changes are infintessimally small. A derivative involves differentials, but that is the extent of the connection. They are definatively 'not' synonyms although this is a common misconception. As differentials are one of the fundamental building blocks of calculus (and especially multivariable calculus) i think this definately needs to be expanded into its own article.

48v 20:37, 12 July 2006 (UTC)


 * I intend to pull 'Differential (calculus)' into its own article and refrence it from both of these disambiguation pages in the near future. Once it is created, further comments should probably directed there. 48v 05:40, 13 July 2006 (UTC)

Assessment comment
Substituted at 18:14, 17 July 2016 (UTC)

There are no infinitesimals in the real numbers
The very first sentence in the article says: "In mathematics, differential refers to infinitesimal differences ..."

This is deeply misleading. It gives support to every crank on the Internet who thinks there are infinitesimals in the real numbers. This article is just awful.

189.223.217.159 (talk) 19:30, 14 May 2017 (UTC)

Merger proposal
I propose merging Differential (infinitesimal) into Differential (mathematics). The two articles discuss the same topic, from slightly different perspectives. I believe that there should be a lead of no more than three sentences, followed by Introduction, History and Approaches sections, and that both germs and linear maps should be retained as separate subsections of Approaches. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:05, 24 February 2022 (UTC)
 * I like the sound of that, but it's hard to say for sure without an actual text to evaluate. Maybe you could make a first draft of your proposal in your sandbox? XOR&#39;easter (talk) 21:15, 24 February 2022 (UTC)
 * I've created User:Chatul/Sandbox/Differential (Mathematics) as a first cut; most of the sections need to be filled in from the existing articles. I hope that the outline will be enough to evaluate the suggestion. If there is a consensus to proceed, I invite other editors to clean up my prose or add text to the sandbox. I'd also welcome guidance on the appropriate level of detail. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 01:49, 25 February 2022 (UTC)
 * I think that I have all of the pieces in place, although there may be some rough edges. Does it look good to go? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:24, 2 March 2022 (UTC)

Suggested rewording for germs
In Differential (mathematics), I am considering replacing Define $\mathcal{I}_p$ to be the set of all smooth germs vanishing at p and $\mathcal{I}_p^2$ to be the set of all sums of products of germs in $\mathcal{I}_p$. with Define $\mathcal{I}_p$ to be the set of all smooth germs vanishing at p and $\mathcal{I}_p^2$ to be the product of ideals $\mathcal{I}_p \mathcal{I}_p$.

The ring ideal $$\mathcal{I}_p^2 $$ is also an algebra ideal of $$\mathcal{I}_p$$, but the distinction is probably TMI. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:09, 9 March 2022 (UTC)