Talk:Generalizations of the derivative

Untitled
Each entry on this list needs a short description to orient the reader. Dmharvey Talk 30 June 2005 15:52 (UTC)

And since the list is now so long, maybe we should split it into sub-lists, approximately by subject. Dmharvey Talk 30 June 2005 17:18 (UTC)


 * Yeah, I've always meant to do just that, even when this was back on the derivative article. Of course, a few of them, I don't know what they are.  -lethe talk June 30, 2005 17:35 (UTC)

I'm beat
OK, I did some stuff. I didn't write anything about these:


 * Dini derivative
 * Radon-Nikodym derivative, used in measure theory to differentiate one measure with respect to another.
 * Gâteaux derivative
 * Matrix calculus
 * Pincherle derivative
 * Automatic differentiation

Also, sooner or later, we should get rid of those red links. See Dini derivative (perhaps we can just copy that article wholesale, since planetmath is GFDL?) and q-derivative. -lethe talk June 30, 2005 19:04 (UTC)

Nice article, more needed
Hey you people, nice work! But would be good if this article did more than just listing the generalizations. Since this is by definition a more advanced article than derivative, maybe more can be said about the modern mathematics view of the concept, what the various types of derivative share, etc. I know you will say "go ahead and do it", but I don't have any time at the moment, so I will limit myself to ranting. :) Oleg Alexandrov 1 July 2005 02:32 (UTC)


 * you're right about what my response will be. Be bold!  Snarkiness aside, I like the idea of making this article have some coherence of the overarching concept behind all the derivatives.  I tried to give some flavor of that by specifying which derivatives were abstractions or extensions of other derivatives.  A lot of my descriptions are very half-assed, though.  I think one change I could make right away is in multivariable calculus; the assumption of real-valuedness is not necessary, multivariable calculus would look largely the same over Cn, and instead of "real-valued" functions, I could just say "scalar functions".  I would also like to add something in the section on complex analysis about how the complex derivative differs from the real derivative.  And maybe in the section on algebraic geometry we could mention how derivatives of polynomials can be defined algebraically (the algebraic derivative of polynomials is use, for example, to decide the multiplicity of a root). -lethe talk July 1, 2005 04:33 (UTC)

algebraic exterior derivative

 * I claim in this article that something like the exterior derivative exists for any exterior algebra. I'm not sure this is true.  Is it? -lethe talk July 1, 2005 04:43 (UTC)


 * I'm pretty sure it's true. On second thought, I'm not so sure anymore. You'd need to define the exterior derivative of scalars and vectors. Or at least for scalars. I guess you can always use the dual space to do that. Well, there doesn't seem any reason for it not to exist. --MarSch 1 July 2005 08:51 (UTC)


 * Right, for a vector space the exterior derivative annihilates the scalars, so you cannot form a dx basis, for scalars x. So how do you define dv, for a vector v?--MarSch 1 July 2005 09:08 (UTC)


 * I'm going to check some sources and see if I can figure out where I might have heard such a thing. -lethe talk July 1, 2005 15:03 (UTC)

Allow me to speak out of my backside for a while. It should be possible do define exterior derivatives for the exterior algebra A of a vector space V, except that d doesn't map A back to itself, it needs to map to something else. I'm not sure exactly how to set up the formal construction, but I think you take B to be the set of "wedge products" of objects of the form "v" or "dv", where you're only allowed terms like u \wedge v \wedge dw that have a single d term, and then d : A \to B using all the rules that d should satisfy. Does something like this work? Dmharvey Talk 11:44, 12 July 2005 (UTC)
 * I don't know about that, but I think I figured out what I was thinking when I made that mistake: the inner derivative is defined for any exterior algebra, not just the algebra of differential forms. So I might move that one to the algebra section. -lethe talk 22:18, July 14, 2005 (UTC)

"still need descriptions"

 * Dini derivative
 * Matrix calculus
 * Pincherle derivative
 * Parametric derivative
 * Semi-differentiability

OK, these have been here for a while now, but nobody has worked them in properly. I don't think we want a section called "still need descriptions", so I'll put them here and someone can hopefully find a home for them. Dmharvey 16:32, 3 March 2006 (UTC)

multivariable derivatives
Would this be an accurate account of these derivatives? A map from R2 to V has a Gâteaux derivative if the derivative along each direction in R2 exists, a Fréchet derivative if this derivative depends linearly on the direction, and a complex derivative if it is independent of the direction. -lethe talk [ +] 05:30, 24 March 2006 (UTC)

Fractional Derivatives
I have added a section on fractional derivatives directly below higher order derivatives with a link to fractional calculus. I thought it was important to mention fractional derivatives on this page and direct those interested to the more complete explanation on the fractional calculus page.--Nick Y. 21:04, 11 April 2006 (UTC)
 * The links were already there, two lines above where you added them. I reverted. -lethe talk [ +] 21:06, 11 April 2006 (UTC)

Well, there they are. I guess this may just be an organization thing but I did not look there because it is labeled as "higher order derivatives" which I assumed to mean those of order greater than 1 and most likely whole number order. I would suggest a separate fractional section but if you want them to be combined the section should at least be labeled something like "other order derivatives" or "derivatives of orders other than one" or "derivatives of alternative order". Especially since the section includes negative order too.--Nick Y. 21:17, 11 April 2006 (UTC)
 * Fine by me. Rename, split, if you prefer.  We'll see how it turns out.  Obviously having the same thing twice is not good though. -lethe talk [ +] 21:32, 11 April 2006 (UTC)

strictly differentiable
I just added this for its uses in p-adic analysis. I suspect that the term is also used in classical analysis, but I don't know anything about that, so if anyone has anything to add, please go right ahead. Dmharvey 12:51, 9 June 2006 (UTC)

Fox derivative
Anybody can tell where this belongs in this article? :) By the way, nice job, this article has grown tremendously since I remember it. Oleg Alexandrov (talk) 02:35, 12 April 2007 (UTC)

Another type of discrete derivative
This generalization agrees sometimes with the regular derivative:

If $$\Delta $$ is the forward difference operator,

and $$\Delta ^{n}$$ is the iterated form of it,

then the generalization of the derivative is: $$\frac{\Delta ^{1}}{1}-\frac{\Delta ^{2}}{2}+\frac{\Delta ^{3}}{3}-\frac{\Delta ^{4}}{4}+\ldots $$

--77.125.14.68 (talk) 08:51, 7 July 2009 (UTC)

Still needed
It would be nice to have here also: Dini derivative, weak derivative (there are distributions though) described; derived subgroup, derived set are a little bit different notions but do they belong in here? konradek (talk) 18:09, 26 November 2009 (UTC)

Canonical form for Fréchet derivative?
The definitions of Fréchet derivative in this article and at the main article Fréchet derivative disagree in notation. $$A_x(h)$$ is used in Fréchet derivative and $$A(x)h$$ is this article. $$A_x(h)$$ is defined to be a bounded linear operator, whereas $$A(x)$$ is simply stated to be a linear operator. I am no analyst, but we should probably get these consistent across articles. Is there a preferred definition in the literature? --Mark viking (talk) 20:36, 11 April 2013 (UTC)

Padegogical alternative
Hi guys, nice article. Interesting I have thought about what is written on this article for a long. But what I found striking was too much abstraction. is there no way of forking a more pedogigical article without too much abstraction. For eg frechnet derivative is just jakobian matrix in my book, and debaux just looks like a directional derivative. Infact in my private study I envisioned a derivative in each major field of mathematics, functional calculus has variations, DG has connection, multivariate has Jacobians, exterior calculus has exterior derivatives, tensors has covariant derivative, even elementary mathematics has ordinary derivative(Infact ironically all derivatives reduce to this), more abstract category theory has derived functors(if I'm not wrong). There is paper by some guy who showed derivatives are Turing complete. And my take is the derivate is essential component in any mathematical language that purports to create models(which I am interested in). 41.74.49.21 (talk) 08:58, 16 December 2021 (UTC)