Talk:Derivation (differential algebra)

Untitled

I'm not sure, but shouldn't this article be merged into Derivative? Jon Harald Søby \ ^{no na} 07:05, 4 November 2005 (UTC)[reply]

No - it is too abstract and the concept is important enough in its own right. Charles Matthews 09:58, 4 November 2005 (UTC)[reply]

A graded (anti-)derivation is a sum of of homogenous derivations, cf. for example, Bourbaki, Algebra, Book 3, or J. Braconnier, Eléments d'algèbre différentielle graduée, Dept. Math., Univ. Lyon 1, 1982, ISBN 0076-1656. AlainD 11:18, 25 January 2007 (UTC)[reply]

Jessica writes On the other hand, I think derivation should be concatenated with differential algebra. I'm going to make the change, although I am open to discussion with anyone who wants to change it back. Shellgirl 15:19, 5 April 2007 (UTC)[reply]

If you use epsilon = +1, you kill the grading. You can do that, and that is the usual notion of derivation of a non-graded space. But for a graded space, I think epsilon should always be -1, and then there are still two notions defined as follows: graded derivation has exponent |a||D|, graded antiderivation has exponent |a||D| + 1. —Preceding unsigned comment added by 137.146.194.173 (talk) 13:57, 15 June 2009 (UTC)[reply]

Fox derivative

Should fox derivative be linked?198.189.194.129 (talk) 19:09, 10 April 2012 (UTC)[reply]

No. The Fox derivative has some vague superficial similarities, but it is really a different thing. 67.198.37.16 (talk) 20:36, 24 May 2024 (UTC)[reply]

Derivations at a point

In many differential geometry texts -- such as Sharpe's Differential Geometry, Gallot's Riemannian Geometry, as well as in "Natural Operations in Differential Geometry", which is referenced in this article, and in the article "Tangent Space", under "See also" -- derivations on the Algebra C_p^∞(M) of smooth germs at p are defined not as a an endomorphism, but as a function onto the underlying field, D:C_p^∞(M) -> R, satisfying the slightly modified Leibniz Rule

D(fg) = D(f)g(p) + f(p)D(g)

Admittedly, in the last text cited above, "Natural Operations in Differential Geometry", "Derivation of an algebra" is later defined as in this article. Nonetheless, I find that the concept of "derivation at a point" is widespread enough for its complete omission in this article to be quite confusing (it certainly confused me). — Preceding unsigned comment added by Jslam (talk • contribs) 15:55, 16 April 2013 (UTC)[reply]

I added a single sentence to the lede, to try to say this in a very informal way, but perhaps I was too simplistic. I can't find any WP page to link to, that would provide a clear demo of derivations "at a point". Yes, this article could benefit from a simpler, less formal lede. 67.198.37.16 (talk) 20:55, 24 May 2024 (UTC)[reply]

I'll revert this added sentence. Fisrtly, it is too vague for being useful for anybody. Also it is confusing, as the standard derivative is defined for the derivation at a point, and the sentence suggests that the derivation at a point is a concept specific to manifolds. Also, there is a confusion between "manifold" (used in the edit) and "germ" (used above). Finally, I am not sure whether Jslam's complaint deserves to be implemented. Indeed, if their assertion is correct, it must appear in Germ (mathematics). This assertion (that the derivation of a germ is a number) is doubtful, since it does not allow to define higher order derivatives and Taylor series of a germs. In any case, since constant functions are germs, the above Jslam's definition is an endomorphism of the algebra of germs. D.Lazard (talk) 08:24, 25 May 2024 (UTC)[reply]

Scope and move

The change to the introduction ("Abstract algebra" is too narrow to classify this topic) followed by the move from Derivation (abstract algebra) to Derivation (differential algebra) (in passing what does "clarifying the disambiguation qualificative" mean?) seem to me to unduly limit the scope of the subject. Derivations occur in areas which have nothing to do with differential equations, such as the formal treatment of differentials on algebraic varieties. I suggest reverting both changes. Deltahedron (talk) 20:03, 15 February 2014 (UTC)[reply]

About the move: Normally, the part between parentheses of an article title is intended to disambiguate a title a title which otherwise would be ambiguous. The previous title Derivation (abstract algebra) was misleading and ambiguous because, as you pointed out, derivations are used in many mathematics areas that have nothing to do with abstract algebra. For example, most people interested in differential equations, formal integration or differential fields are not supposed to know that the definition there are looking for belongs to abstract algebra. Myself, when looking at this title, before reading the article, I was thinking that it was about the computation of the derived series. As a differential algebra is simply an algebra equipped with a derivation and derivations are always defined on algebras, Derivation (differential algebra) is certainly the clearest and less ambiguous title. As the other mathematical article about derivations is Derivation (calculus) (redirect to Derivative), when typing "derivation" in the search window, the reader immediately see both articles titles and the new title makes the choice evident.

About the first sentence, I agree that 1/ this is not the place to define "Differential algebra" as an area of mathematics 2/ the given definition is too narrow and may be controversial. Therefore, I'll simply begin the article with "In mathematics". IMO, the definition of the mathematics area called "differential algebra" should added to the article Differential algebra, which, presently, is devoted only to the structure. D.Lazard (talk) 22:43, 15 February 2014 (UTC)[reply]

The problem is, of course, that derivations occur in lots of places besides differential algebra. There are plenty of things in number theory which are derivations, to which concepts of differentiability have got nothing to do with anything. Examples include various systems of orthogonal polynomials, where you are not taking any derivatives of the polynomials, but rather, are observing that polynomials of different orders happen to be related to one another with a relation that just so happens to have the structure of a derivation. These in turn tend to span some Hilbert space, so now you get derivations on Hilbert spaces, and there's nothing to "differentiate", nothing "smooth", for miles around. 67.198.37.16 (talk) 21:03, 24 May 2024 (UTC)[reply]

seemingly stupid question

I'm familiar with the concept, though not, it appears, enough with the English language. In other people, this may be vice versa. => It would be helpful information if the article said whether the verb for the thing, as in "derivXXX x^3 and you get 3x^2", is derive or derivate.--2001:A61:260D:6E01:412F:2D0F:6B21:83F7 (talk) 00:54, 31 January 2018 (UTC)[reply]

There's no verb for this. Linguists have a joke that you can always "verbize your nouns", but for the noun "derivative", this is not generally done. 67.198.37.16 (talk) 21:07, 24 May 2024 (UTC)[reply]