Talk:Leibniz's notation

Modern formalism
The modern formalism for derivative and integral is imprecise. Unless there is a reason not to do so, we should use that given in any real analysis textbook, namely the Riemann sum for an integral and the long functional forms of y(x) and x for the derivative. —Preceding unsigned comment added by SamuelRiv (talk • contribs) 04:48, 14 October 2007 (UTC)

On second thought, it can be argued that for a page dealing with notation, and not formal mathematics, a simplified, less precise form is acceptable as long as it is clear and unambiguous to someone with a semester of calculus (or limits). I guess the notation for integral is then acceptable, but I will change that for derivative. SamuelRiv 04:56, 14 October 2007 (UTC)

Needs work
This article needs a lot of work. The second paragraph (not the line, the paragraph) is nearly incoherent. The closing statement describing units merely hints at what I wanted to know. Maybe it's somewhere else; in that case, a link will be needed.

-Malakai


 * Does this look better now? Fresheneesz 00:17, 11 February 2006 (UTC)

"(One mathematician, Jerome Keisler, has gone so far as to write a first-year-calculus textbook according to Robinson's point of view.)" Why don't you tell us the name of the textbook, given that you tell us it exists? GangofOne 00:22, 11 February 2006 (UTC)
 * GIYF: http://www.math.wisc.edu/~keisler/calc.html TomJF 00:37, 21 April 2006 (UTC)

Merger
In the absence of any talk here, I have implemented the proposed merger by moving the content of Leibniz's notation for differentiation into this article. A while ago, I also adapted the notation here for coherence with that article. There is now a new (not fully developed) article at Notation for differentiation: the material here should ultimately inform this new article. I have made a similar move for the Newton's notation articles, where the issues are more straightforward, because Newton's notations for integration are not developed in wikipedia. Geometry guy 19:00, 26 March 2007 (UTC)


 * Good job. I have a question about this odd looking thing:


 * $$\frac{\mathrm{d} \Bigl(\frac{\mathrm{d} \left( \frac{\mathrm{d} \left(f(x)\right)} {\mathrm{d}x}\right)} {\mathrm{d}x}\Bigr)} {\mathrm{d}x}$$


 * Did Leibniz always write the function being differentiated on top of the line, instead of the nicer looking


 * $$\frac{\mathrm{d}}{\mathrm{d}x}\Bigl(\frac{\mathrm{d}}{\mathrm{d}x}\Bigl(\frac{\mathrm{d}f(x)}{\mathrm{d}x}\Bigr)\Bigr)$$? –Pomte 23:11, 26 March 2007 (UTC)

I suspect that the notation you suggest was not current in the 17th century, since it presumes the (later) idea of d/dx as an operator, and it was probably for precisely this reason that a more concise form was needed for multiple derivatives. However, I'm not an expert of 17th century notation, and they probably had a different way to write multiple fractions. Anyway, I would be happy if you would incorporate the notation you propose as an addition (but not a replacement) to the text, since it certainly makes sense, and is certainly now used. It could comfortably be inserted into the explanation for the origin of the d^ny/dx^n notation. Geometry guy 23:21, 26 March 2007 (UTC)

To answer your question: I have had a look into the correspondence of Leibniz:

Google Books "Mathematischer, naturwissenschaftlicher und technischer Briefwechsel von Gottfried Wilhelm Leibniz" Letter No. 44, Leibniz to Bernoulli, p.121, l.17 (3) $$\quad x = y\mathrm{d}y:\mathrm{d}x + a \overline{\overline{\overline{\mathrm{d}y}^2:\mathrm{d}x^2}+1}$$ Which would look in modern notation like $$x = y\frac{\mathrm{d}y}{\mathrm{d}x} + a (\frac{(\mathrm{d}y)^2}{\mathrm{d}x^2}+1)$$

I am not sure whether Leibniz thought the derivation as a functor $$\frac{d}{dx}$$ which would justify the notation

$$\frac{\mathrm{d}}{\mathrm{d}x}\Bigl(\frac{\mathrm{d}}{\mathrm{d}x}\Bigl(\frac{\mathrm{d}f(x)}{\mathrm{d}x}\Bigr)\Bigr)$$

And here is also my problem with this notation. The modern version regards $$\frac{d}{dx}$$ as functor applied to functions rather than to function values. Function values are considered as constants and so $$\frac{d(f(x))}{dx}$$ would evaluate to 0. I would prefer to write $$\frac{df}{dx}$$ for the derivation of the function f and $$\frac{df}{dx}(x)$$ as evaluation of $$\frac{df}{dx}$$ at x; hence $$\frac{df}{dx}(x)=f'(x)$$. —Preceding unsigned comment added by 138.253.184.200 (talk) 09:38, 16 July 2009 (UTC)

But if it is a functor, the argument should be anonymous, which does not match the d/dx notation very well. However, f(x) in df(x)/dx is not constant, because x is not constant. (If x is constant, you are dividing by zero, which is not allowed. You have to fill in the value of "x" after computing the derivative, similar as the same way you do with limits calculations.) --zzo38(✉) 06:24, 23 May 2010 (UTC)

The d debate
For those like me wondering whether the d should be upright or italic, this has been discussed at length on Wikipedia before: see Wikipedia talk:WikiProject Mathematics/Archive 20 and Wikipedia talk:WikiProject Mathematics/Archive2007, etc. In summary: italic d is usual in mathematics, though not universal; Wikipedia consensus is that each page should keep its existing usage, and one form should not be turned into the other solely for consistency. --82.36.30.34 22:34, 25 June 2007 (UTC)


 * What did Leibniz himself use??? — DIV (128.250.204.118 10:10, 31 August 2007 (UTC))


 * Roger Penrose used upright "d" --zzo38(✉) 06:24, 23 May 2010 (UTC)
 * For Leibniz's own usage, the best place to check would be the book by Margaret Baron on calculus. Tkuvho (talk) 07:41, 23 May 2010 (UTC)

unexplained deletion of material
The following material was recently deleted from the lede: ''In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed rigorous mathematical explanations for Leibniz' intuitive notion of the "infinitesimal," and developed non-standard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based to Robinson's approach.'' What is the reason for the deletion? Tkuvho (talk) 10:06, 4 January 2012 (UTC)

Needs detail on Leibniz's notation for integration
Notation for integration is mentioned at the very end of the article, but it wasn't introduced anywhere. We should add details with examples, as Leibniz notation is also used to describe integration along a path.


 * $$\int_\gamma f d\gamma = \int_a^b f'(\gamma(t))\gamma'(t) dt$$

Logical Gentleman (talk) 21:25, 7 October 2016 (UTC)

Fix the year
The year is wrong, it must be fixed but I'm not allowed to do so, can someone else look into the year issue? Vyvek (talk) 11:52, 6 February 2017 (UTC)
 * Regarding and : the cited source seems to support 1684. And in 1884 Leibniz was long dead. Comments from others? - DVdm (talk) 11:57, 6 February 2017 (UTC)

Non-standard analysis
I have recently been cleaning up this page and adding new material and references. As I look at the structure of the page, however, I am a bit concerned by the amount of material on non-standard analysis that appears in the lead. I am considering the following changes:
 * 1) Put into the lead the simple statement that non-standard analysis can be used to legitimize Leibniz's original conception of differentials.
 * 2) Pull all the rest of the non-standard analysis out of the lead and the history section and create a new section with that material and place it before the last section on other notations.

I think that these changes might restore some balance to this article. Whether or not I do this, I do plan on expanding the history section to include some of the alternative notations that Leibniz played around with (based on Cajori's treatment). Comments? --Bill Cherowitzo (talk) 05:26, 10 February 2017 (UTC)

Having gotten no comments in a week and a half, I will proceed to make these changes and I am sure that if I ruffle any feathers I will hear about it. --Bill Cherowitzo (talk) 22:26, 21 February 2017 (UTC)

clarification
I really don't get why the derivative is supposedly not a fraction since it is said to be the limit of a fraction. Non standard analysis is interesting but it seems it allows to precise the meaning of the derivative, not contradicting its "nature". Several times in the article, it's repeated a derivative is not a fraction without any counter example, and then it's explained it's the limit of a fraction.I have nothing against subtlety, but I would be grateful if the subtleties were clearly exposed, and not let "as a trivial exercice for the reader" Klinfran (talk) 23:21, 28 March 2019 (UTC)

Changes to lede
I removed the statement that nonstandard analysis gives a new ratio interpretation or justification of dy/dx. What it provides are infinitesimal numbers, and thus infinitesimal increments (differentials) of functions, but the derivative does not become more of a ratio than it was using limits. You can argue that it is the ratio "up to infinitesimal error", but justifications in that weaker sense were well known and widely taught before nonstandard analysis (e.g., as "up to an error tending to 0", up to o(1) error, up to higher-order terms in a series, up to a 1+o(1) factor, equal to leading asymptotic order, ...). The formalism where dy/dx is, exactly and not up to small error, the derivative, is differential forms, though there it is done by definition.

Added also that Leibniz notation is used in integrals, which is where it is rather important. 73.89.25.252 (talk) 04:20, 13 June 2020 (UTC)