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)[reply]

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)[reply]

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)[reply]

"(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)[reply]

GIYF: http://www.math.wisc.edu/~keisler/calc.html TomJF 00:37, 21 April 2006 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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 ${\frac {df}{dx}}(x)=f'(x)$ .

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#Symbol for differential and Wikipedia talk:WikiProject Mathematics/Archive2007#Upright d in math notation, 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)[reply]

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