Questions tagged [calculus]
For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ??
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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Origin of Riemann-Stieltjes Integral
What need (if there was any) created Riemann-Stieltjes integral? What did Riemann-Stieltjes integral want to attain?
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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:
$$\int_{x_1}^{x_2}f(x) + f^{-1}(x)dx = x_{2}^{2}...
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In which work did Euler invent the Euler Substitutions for a quadratic composed into a radical?
A famous technique in the modification of integrands is the set of “euler substitutions” that provide substitutions for the structure
$$\sqrt{ax^2 +bx+c}$$
That is a fairly common occurence in ...
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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 ...
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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 ...
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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. ...
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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 ...
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The role of symmetry in mathematics and the half-angle formulas
It seems that the most impressive theorems of classical geometry always have to do with "half of something". Consider the following examples:
The three medians of a triangle meet at the ...
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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, "...
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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 ...
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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?
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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)+\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Notations for Laplacian: $\nabla^2$ vs. $\Delta$
For a (sufficiently smooth) function $f\colon \Bbb R^n\to\Bbb R$, the Laplacian of $f$ is defined to be $\sum_{j=1}^n \frac{\partial^2 f}{\partial x_j^2}$. There are two notations for the Laplacian ...
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What made Leibniz think about calculus?
We know that Sir Isaac Newton thought about calculus when he tried to efficiently describe his physical laws but what made Sir Gottfried Leibniz think about something which we know today as calculus?
user14661
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History of interpolation methods - Newton
I'm interested in reading more about how Newton developed his method of interpolation and also the proofs he developed to this topic. I'm currently reading "Analysis by its history" which ...
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Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.
Still, in my view there is fundamental difference between divergent ...
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Did anyone ever try to determine or propose the algebraic role of Euler-Mascheroni constant?
Both the constant $\pi$ and the constant $e$ have clear algebraic roles in complex numbers and in differential calculus.
But did anyone ever propose an algebraic role for Euler-Mascheroni constant $\...
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Yuktibhāṣā, 16th century, first modern proof of $\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$
It is an Indian (Kerala) text, it would be the first modern proof (based on earlier knowledge) of
$$\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$$
The integral would be a ...
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How did the idea of a formal derivation emerge?
Infinitesimal calculus and the introduction of derivatives is often linked to Newton and Leibniz.
I was wondering, when and why the idea of studying formal derivatives (e.g., of a formal polynomial) ...
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How did Fourier determine the coefficients of Fourier series?
I was reading a chapter of Fourier's seminal work "Analytic Theory of Heat". The third chapter of this book was translated by the famous Stephen Hawking in his book "God created the ...
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Has the heyday of mathematical formulae ended?
I have a strong emotional reaction when I read the works of Euler. I have seen many extremely beautiful and intriguing identities in the notebook of Ramanujan, so much so that I think he is indeed a ...
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Is there an English translation of Newton’s De Analysi?
I’m looking for an English translation of Newton’s De analysi. (Alas, my Latin is weak.) I’m rather dismayed by the fact that I can’t appear to find one. How is it possible that one of the most ...
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How to derive the power series of $\sin(x)$ and $\cos(x)$ followed the footstep of Euler
I am reading Euler's "Introduction to analysis of the infinite", chapter 8, page at the end of page 208, beginning of page 209 and came across his derivation of the power series for $\sin(x)$...
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How did Euler obtain this formula from a paper/work in 1748?
I am reading this book on trigonometric series, "Тригонометрические ряды от Эйлера до Лебега" (Trigonometric series from Euler to Lebesgue) , it is in Russian, and my Russian is abysmal. But ...
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How did definitions of a limit vary before the epsilon-delta definition?
My understanding is that before the epsilon-delta definition of a limit, the rigor and soundness of the definition of a limit was not good enough.
So, how did the definitions of a limit vary before ...
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Who first used the Completeness Axiom for real numbers?
I was studying calculus and the following question came to my mind: Who was the first person to use or suggest the use of the Completeness Axiom of the Real Numbers?
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