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Questions tagged [calculus]

For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.

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When and how did the definition of continuity in terms of function's oscillation come into picture?

In this Wikipedia article, it is described how the oscillation of a function can be used to characterize the function having a limit. So, we say a function is continuous at a point when if its ...
Babu's user avatar
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Who was the first to understand a derivative and integral as both giving rise to new functions?

NOTE: Even if not called "functions" surely someone understood that returning something other than a numerical value was significant? If I understand it correctly, Newton, for example, was ...
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Earliest real-world uses of calculus and linear algebra

I want to illustrate in class that real-world applications of mathematics might take time to come to fruit. In this context, I want to find what the earliest real-world applications of Calculus and ...
Jaikrishnan's user avatar
3 votes
1 answer
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What is the origin of the method of undetermined coefficients?

This MSE post asked about a specific integration technique that appears to be attributed to Charles Hermite, per a comment. The OP's source calls the technique el método alemán, i.e. the German method....
user170231's user avatar
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1 answer
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Where does Oliver Heaviside fit in the ranks of physicists/mathematicians? [closed]

It seems to me that he was able to reformulate Maxwell's equations in a more understandable form and in fact come up with vector calculus without finishing high school would arguably cause him to be ...
releseabe's user avatar
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What can I read to learn the history of multivariable calculus?

People have been doing calculus of several variables since well before the concepts of vectors, matrices, and linear algebra were formalized. Where can I learn about the development of multivariable ...
Dominic Stewart-Guido's user avatar
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68 views

How exactly did Auguste Bravais come up with the regression line?

I am new to statistics and linear regression and I came across the face that auguste bravais discovered regression line but didn't realize it. Auguste Bravais (1811-1863), professor of astronomy and ...
Alexander Obidiegwu's user avatar
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38 views

What are the origins of calculating the area of a v/t graph to determine displacement?

I'm curious as to the origins of thinking of displacement as the area under a v/t curve. I assume that Newton (and/or Leibniz) was already familiar with the concept and knew that calculating the area ...
Physicator's user avatar
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1 answer
176 views

How good was Newton at definite integration?

On Math Stack Exchange, I am impressed by users' skill at finding closed form expressions for definite integrals. For example: Example 1: $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^...
Dan's user avatar
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8 votes
2 answers
973 views

What were the obstacles that made the discovery of calculus very late?

I wonder What were the obstacles that made the discovery of calculus very late ? Why the discovery of calculus took so long? I know that some of the ideas and techniques of calculus appeared in ...
pie's user avatar
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When did "neighbourhood of a point" first appear in the history of Taylor series?

I am trying to track down at what point mathematicians started to use the terminology of expanding a function "around a point" or in the "neighbourhood of a point". Neither Taylor ...
StormyTeacup's user avatar
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1 answer
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Which geometer first compared a length (one dimensional) to an area (two dimensional)?

What are sources placing a length (one dimensional) in proportion to an area (two dimensional)? The Greek geometers compared quantities of the same dimension: e.g. the area of a circle is in ...
SRobertJames's user avatar
12 votes
1 answer
2k views

Did Newton know about non-inertial frames?

When answering a Physics.SE question, I made a claim that Newton realized that $F=ma$ worked in some frames, which are called "inertial frames." Nowadays, we know that there are non-...
Cort Ammon's user avatar
2 votes
0 answers
89 views

Who evaluated the surface of the Torricelli solid/Gabriel's horn

The Torricelli solid/Gabriel's Horn is defined as the rotation-invariant solid delimited by a hyperbola. It appears in De solido hyperbolico acuto where Torricelli proves that it has a finite volume, ...
Antoine Chambert-Loir's user avatar
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1 answer
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What were the "weird" things people were doing in calculus at the time of Marx?

I was reading the preface of Marx's Mathematical Manuscripts. They explain the situation of calculus in the time of Marx, it seems that at the time analysis as we know today was still being forged by ...
Red Banana's user avatar
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1 answer
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Where can I find the Royal Society report on the controversy over the invention of differential calculus?

Where can I find the report on the Leibniz–Newton calculus controversy mentioned in this article? In 1712 the Royal Society in England wrote a report purporting to settle the matter — except, the ...
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Dissemination of Calculus in China

Much has already been written about the dissemination of Euclidean geometry into China: https://www.maa.org/press/periodicals/convergence/mathematical-treasure-euclid-in-china, https://academic.oup....
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Can the so-called completeness of real numbers be understood as closure under limits in the real number system?

Someone suggested (please see the comments below) that I post this question on hsm.stackexchange. There is a connection to the history of mathematics in this, regarding the relationship between the ...
bokabokaboka's user avatar
3 votes
0 answers
197 views

Did the Maclaurin series for sine and cosine unsettle Indian mathematicians?

As many of you may know, sometime around the 14/15th centuries an Indian mathematician by the name of Madhava of Sangamagrama derived the Maclaurin series for sine and cosine for the first time in ...
voltamatron's user avatar
4 votes
2 answers
282 views

Who was the first person in history to calculate the limit $\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n$?

At the beginning of a calculus course, we encounter two famous limits. They are $$\lim_{x\to0}\frac{\sin x}{x}\qquad\text{and}\qquad\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n.$$ I'm not ...
HGF's user avatar
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Did any mathematicians of the time (the 17th Century) try out an intermediary between Bernoulli's and Nieuwentijdt’s infinitesimals?

In §4 of the Stanford Encyclopedia of Philosophy article on continuity and infinitesimals, the author (John L. Bell) mentions that: ... Johann Bernoulli (1667–1748) [in a] letter of his to Leibniz ...
Kristian Berry's user avatar
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80 views

Ab-initio method (First principle of Mathematics)

Who was the first one to give proof of 1st Principle of Mathematics in calculus (also known as ab-initio method) ,was he newton or someone else ??
πααρτθ Σαρθι's user avatar
2 votes
1 answer
162 views

When did derivative mean not only "slope of tangent" but also "instantaneous rate of change"?

When did derivative mean not only "slope of tangent" but also "instantaneous rate of change"? Fermat was interested in minima and maxima, and realized these occur when the tangent ...
SRobertJames's user avatar
2 votes
0 answers
118 views

How was the Fundamental Theorem of Calculus discovered?

How was the Fundamental Theorem of Calculus discovered? The FTC is at once simple enough that Math.SE is full of questions asking "why is it such a big deal" and yet avoided discovery for ...
SRobertJames's user avatar
1 vote
0 answers
96 views

What physical problems required the invention of the derivative?

I know that Fermat had a method of adequality in order to solve certain optimization. One such problem was: "Suppose that you have a rectangle of material and need to cut corners into it such ...
Addem's user avatar
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When were the foundations of vector calculus laid?

Upon some browsing I find from many online sources that vector calculus was created in the time of late 19th century by Gibbs and Heaviside, but Gauss, Green, Stokes, etc., who lived much before that, ...
nickbros123's user avatar
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0 answers
105 views

Surface Integrals History

Can't find any information about who and when first used surface/surface area integrals. What was the original motivation? In it's modern form it depends on some relatively modern notations like ...
simd's user avatar
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2 votes
0 answers
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A result on convergence of series of positive terms

Just recently I learned from a highschooler I help in her Calculus class about a convergence test for series with positive terms, and and of which I was not aware. The test is a hybrid between the ...
Oliver Díaz's user avatar
29 votes
2 answers
1k views

How did Isaac Newton write the integral symbol?

Isaac Newton is known as the discoverer of the FTC (Fundamental Theorem of Calculus), so maybe he wrote the integral symbol and derivative symbol. I know he wrote the derivative symbol as $\dot y$ but ...
MIKANkankitsu's user avatar
2 votes
2 answers
273 views

Origin of Riemann-Stieltjes Integral

What need (if there was any) created Riemann-Stieltjes integral? What did Riemann-Stieltjes integral want to attain?
Sensebe's user avatar
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Who discovered the rule for the definite integration of a function summed with its inverse function across a fixed limit?

I saw this rule used in an MIT integration bee that gives a result for the definite integral of a function summed with its inverse across a fixed limit: $$\int_{x_1}^{x_2}f(x) + f^{-1}(x)dx = x_{2}^{2}...
Hisham's user avatar
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0 answers
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In which work did Euler invent the Euler Substitutions for a quadratic composed into a radical?

A famous technique in the modification of integrands is the set of “euler substitutions” that provide substitutions for the structure $$\sqrt{ax^2 +bx+c}$$ That is a fairly common occurence in ...
Hisham's user avatar
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2 votes
2 answers
151 views

What are the various manuscripts/transmissions of Newton's book "The Method of Fluxions"?

I am looking for which manuscripts and if available, through what chains of transmission copies of Newton's book "TheMethod of Fluxions" have reached us today So far I could not find ...
Hisham's user avatar
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3 votes
1 answer
199 views

What is the title of the 1676 Memoir in which Leibniz first used the Chain Rule?

On Wikipedia it says: "The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative. He first mentioned it in a 1676 memoir [ Chain Rule ...
Hisham's user avatar
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2 votes
1 answer
498 views

When did the error function get its modern definition?

I am currently writing an essay on the error function and after researching its historical origin, I found out who first defined it: J.W.L. Glaisher. But his definition is different from today's form. ...
Nogi Biggi's user avatar
2 votes
0 answers
119 views

Euler's "comfortable" series

I am reading Proofs and Confirmations by David Bressoud. On page $150$ is a long excerpt by Richard Askey, from "How can mathematicians and mathematical historians help each other?" There is ...
Somos's user avatar
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3 votes
2 answers
161 views

History of Speed - is it really new? [duplicate]

I was writing a paper on the basics of calculus, and of course the study of velocities plays a big part in that. In introducing the problem statement, I started with a classic word problem, "...
Cort Ammon's user avatar
3 votes
1 answer
388 views

Why did Clairaut's theorem take so long to prove?

I was reading the Wikipedia on Clairaut's Theorem (Symmetry of second derivatives) and the article accounts the significant amount of time and failed proofs occurred before the theorem was made fully ...
szeits's user avatar
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1 vote
1 answer
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When did Abel publish his test for the convergence of series?

Did Abel published of testing the convergence of series? If so, when did he published it. Also, did he offer a proof of the test? Or did he simply stated the test?
user926356's user avatar
2 votes
0 answers
105 views

How the asymptotic expansions of the Dawson integral and $\exp(x^2)\operatorname {erfc}(x)$ were originally obtained?

There are two well known asymptotic expansions of the Dawson integral $F(x)$ and the function $\exp(x^2)\operatorname {erfc}(x)$ as $x \rightarrow \infty$: $$ F(x)\sim (1/2)(1/x+1/(2x^3)+ 3/(4x^5)+\...
Leszek's user avatar
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2 votes
1 answer
395 views

What is the history of the Basel Problem before Euler and how did it inform him?

I am interested in the history of the Basel problem. More specifically, I'm interested in knowing the history of failed attempts before Euler's crack of it, so as to know what bits of evidence ...
melembroucarlitos's user avatar
1 vote
0 answers
90 views

What is Weierstrass' "simple diagram" in the calculus of variations?

In An essay on the psychology of invention in the mathematical field Hadamard said this about Weierstrass: The two German mathematicians whom Poincaré compares are Weierstrass and Riemann. That, as ...
MaudPieTheRocktorate's user avatar
4 votes
1 answer
650 views

What book did Maria Gaetana Agnesi write which contained both differential and integral calculus?

Wikipedia says the following about Maria Gaetana Agnesi: She is credited with writing the first book discussing both differential and integral calculus and was a member of the faculty at the ...
Galen's user avatar
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3 votes
0 answers
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The term "constant" in "integration by parts" ("partielle Integration")

In Riemann's "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe", Riemann mentions taking a factor as "constant" in "partial integration", which ...
Michael E2's user avatar
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0 votes
1 answer
279 views

Did Newton know the chain rule?

I heard someone say recently that Newton didn't know the chain rule. Is that true? I know Newton didn't share our current conception of functions, the real line, limits, etc., so if he did use ...
greatBigDot's user avatar
0 votes
0 answers
78 views

What was the difference between Number and Magnitude in Ancient Greece [duplicate]

I've been reading Infinite Powers by Steven Strogatz and in it, he writes about how the greeks differentiated between numbers as being discrete and magnitudes as being continuous. However, all of ...
user avatar
11 votes
0 answers
278 views

Origin of the special Finnish notation for difference of antiderivative

Apologies for a question that is specific to one country (but perhaps others find it a curious example of how mathematical notation can vary between countries). In Finnish calculus texts, if $F$ is an ...
Jukka Kohonen's user avatar
1 vote
2 answers
439 views

Did Berkeley's criticism of infinitesimals hobble calculus pedagogy?

I recently read an article that discussed--rather briefly--the issues of infinitesimals and the criticism of them by Berkeley. The author of the article (which, of course, I cannot find, as I read it ...
Eric Snyder's user avatar
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0 answers
242 views

Where can I find a copy of Dieudonné's 'Infinitesimal Calculus'?

I found a copy of the French version 'Calcul infinitésimal' online but the English edition seems to only be available on Amazon for a very hefty price, or in American libraries which I do not have ...
fnalkj's user avatar
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2 votes
1 answer
379 views

Source of L’Hôpital’s 1696 Calculus textbook

A calculus textbook I’m using references a calculus book of L’Hôpital in which he illustrates his rule, which is taught in many calculus classes. Does anyone have a source as a scanned PDF? I’d love ...
Thomas Davis's user avatar