28 votes
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How did Isaac Newton write the integral symbol?

Newton used both vertical bars ($\overset{|}{x}$) and rectangles ($\boxed{x}$) to denote integrals in his Quadratura curvarum published in 1704. Here, the bar notation is used on the bottom of page 9 ...
Scene's user avatar
  • 396
21 votes

Did Newton know about non-inertial frames?

TL;DR Yes, but... 1) Inertial frames To say that Newton had the modern conception of even inertial frames (based on the laws of motion), is an overstatement. Theoretically, he did not need them ...
Conifold's user avatar
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15 votes
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Riemann's Contribution to Integration

The following is a slightly edited version of my 31 January 2003 sci.math post archived at google groups. Riemann [6] introduced his integral in his December 1853 Habilitationsschrift thesis. In his ...
Dave L Renfro's user avatar
14 votes
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Where did Leibniz explore the product rule of differential calculus?

It is discussed in multiple manuscripts, letters and publications from 1675 to 1701. According to Fracois Ziegler's post on MO Did Leibniz really get the Leibniz rule wrong?, Leibniz originally ...
Conifold's user avatar
  • 76k
13 votes

Is Spivak right in what he says about Galileo?

Yes, indeed when trying to obtain the law of falling bodies, Galileo's first conjecture was that the speed is proportional to the distance traveled. After some contemplation, Galileo understood that ...
Alexandre Eremenko's user avatar
13 votes
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Did Benjamin Franklin know calculus?

Apparently, no. Boman's biography Benjamin Franklin's Numbers specifically focuses on Franklin's mathematical activities, and calculus is not among them. He was deep into magic squares, and ...
Conifold's user avatar
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12 votes
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Who did first use the Method of Characteristics?

[EDIT: check the (*) for a potential earlier reference in Lagrange] The first mathematician (who used the methods of characteristics for differential equations) seems to be Paul Charpit de Villecourt ...
Laurent Duval's user avatar
12 votes
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Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"?

"Did Rolle ever say/write any such thing (as that the calculus was 'a collection of ingenious fallacies')?" Michel Rolle (France, 1652-1719) certainly did attack the mathematical basis of the ...
terry-s's user avatar
  • 4,445
12 votes
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When did "neighbourhood of a point" first appear in the history of Taylor series?

The terminology "neighbourhood of a point" (in German, "Umgebung einer bestimmten Stelle") with its current meaning, dates back at least to 1841 when Weierstrass wrote his Zur ...
user6530's user avatar
  • 3,870
11 votes
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How did Newton prove the generalised binomial theorem?

According to William Dunham in his Journey Through Genius [Newton's Binomial Theorem is] not a "theorem" in the sense of Euclid or Archimedes in that Newton did not furnish a complete proof. Yet ...
nwr's user avatar
  • 6,859
11 votes
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Dirichlet integral's history

It arose in Dirichlet’s famous proof of the convergence of Fourier series (1829, p. 161),1 then again in his “discontinuous factor” method to compute integrals (1839; 1904, pp. 193-195, 353-385)2 and ...
Francois Ziegler's user avatar
11 votes
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What book did Maria Gaetana Agnesi write which contained both differential and integral calculus?

The book is a work in two volumes: Maria Gaetana Agnesi, "Instituzioni Analitiche ad uso della gioventù italiana", Milan 1748. The title roughly translates to "Foundations of Analysis ...
njuffa's user avatar
  • 6,516
10 votes
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What is rationale or history for using the symbol $\partial{U}$ to represent surface boundary?

There seem to be two different rationales, one for how it first appeared, and the other for why it stuck, see MO thread. Francois Ziegler dug up the following remark in Michèle Audin's Henri Cartan &...
Conifold's user avatar
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10 votes

What is the difference between Calculus of Newton and that of Leibniz?

You should definitely take a look at the second chapter of Arnold's Huygens & Barrow, Newton & Hooke. The late Prof. Arnold summarized therein the difference between Newton's approach to ...
José Hdz. Stgo.'s user avatar
10 votes

Who introduced the notation $y|_{x=a}$?

You can see : Giuseppe Peano , Lezioni di Analisi Infinitesimale, 2 vols., 1893, page 17 : $$[f(x)]_{x=a}=f(a).$$ Not sure it is the earliest... but Peano was a prolific "inventor of notations". ...
Mauro ALLEGRANZA's user avatar
10 votes
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Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?

Cantor 1895 is predated at least by Dedekind in §2 of Was sind und was sollen die Zahlen? (1888) (translation): 21. Erklärung *). Unter einer Abbildung $\varphi$ eines Systems $S$ wird ein Gesetz ...
Francois Ziegler's user avatar
10 votes
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Is the prime notation for derivatives $f'$ due to Euler?

I agree, as there is further evidence that Lagrange got his primes from Euler: 1. Everyone since Cajori (1923, p. 6; 1929, p. 207) quotes Théorie des fonctions analytiques (1797) for the (sic) “new” ...
Francois Ziegler's user avatar
10 votes

$\frac{dy}{dx}$ versus $\frac{{\mathrm d}y}{{\mathrm d}x}$

One already finds upright d's in Lacroix's Traité élémentaire de calcul différentiel et de calcul intégral (1802). I don't now if this is the earliest, but it is interesting to note that the ...
Michael Bächtold's user avatar
10 votes

How influential was the Kerala school to European development in Calculus?

In my work with primary sources in such authors as Fermat and Leibniz, I have occasionally come across references to earlier works by Arab mathematicians, but have never seen references to work by the ...
Mikhail Katz's user avatar
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10 votes

Was English mathematics behind Europe by many years because of Newton's notation?

Several factors come together to suggest that the idea that "English mathematics [was] ever significantly behind -- by say 50 years, 100 years, or even centuries" (i.e. in the post-Newtonian 18th or ...
terry-s's user avatar
  • 4,445
10 votes
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Why is differentiation under the integral sign named the Leibniz rule?

This rule is, indeed, due to Leibniz, although it was Johann Bernoulli who realized its broader implications, and there is an interesting story to its discovery. It is told in Chapter 3 of Families of ...
Conifold's user avatar
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10 votes
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What algebra problem did Serge Lang give to calculus students?

In 1969 Lang wrote an article for the Columbia Daily Spectator, Don't Blame Us if You Flunk Math (Volume III, Number 4, December 8, 1969). The phrasing of the subline illustrates how much the times ...
Conifold's user avatar
  • 76k
10 votes
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What was the real need of divergence and curl operators?

These operations arose from the study of quaternions see e.g. Thomson and Tait's Treatise on Natural Philosophy, that should probably have information on the sort of math. Stokes's theorem originated ...
Sam Gallagher's user avatar
10 votes

Who discovered the indeterminate forms like 0/0?

Special cases were handled algebraically even before the "l'Hopital's" rule, which appears in l'Hopital's 1696 transcription of tips on calculus he purchased (literally) from Johann Bernoulli in 1694, ...
Conifold's user avatar
  • 76k
10 votes

How did the idea of a formal derivation emerge?

The motivation for applying derivatives to polynomials over general fields is their use in detecting multiple roots: if $K$ is a general field, a polynomial $f(x)$ in $K[x]$ has no repeated roots (...
KCd's user avatar
  • 5,607
9 votes

What is the difference between Calculus of Newton and that of Leibniz?

Beyond the issue of notation, Newton experimented with a number of foundational approaches. One of the earliest ones involved infinitesimals, whereas later he shied away from them because of ...
Mikhail Katz's user avatar
  • 5,772
9 votes
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How were double integrals calculated before Fubini's Theorem?

Double integrals were calculated in the 18th and 19th centuries in the same way they teach you in calculus now, using special cases of what later became Fubini's theorem. Most of these special cases ...
Alexandre Eremenko's user avatar
9 votes
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Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?

Gibbs (1889, p. 140): $ \qquad \dfrac{d\,\log\mathrm V}{d\,\log p} = - \dfrac{d\,\log n}{d\,\log\lambda} $ Riemann (1868, p. 89): $ \qquad \dfrac{d^2y}{dx^2}-\dfrac1{\alpha\alpha}\dfrac{d^2y}{dt^2}=...
Francois Ziegler's user avatar
9 votes
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Was Newton's method of finding derivatives of his fluents based on applying the chain rule?

The answer is more of a yes, but with many buts. Newton did not have the modern concept of function, it was introduced by Dirichlet in the 19th century, or even its predecessor as assignment of values ...
Conifold's user avatar
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9 votes
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Who invented the gradient descent algorithm?

The "gradient descent" algorithm was invented before the gradient. It is described in equivalent form by Cauchy in a 3-page paper in Comptes Rendus, Méthode générale pour la résolution des systèmes d'...
Conifold's user avatar
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