Questions tagged [calculus]
For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.
185
questions
11
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-...
- 283
2
votes
0
answers
78
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, ...
3
votes
1
answer
2k
views
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 ...
- 201
1
vote
1
answer
108
views
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 ...
user18960
1
vote
0
answers
47
views
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....
- 253
2
votes
1
answer
147
views
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 ...
- 23
3
votes
0
answers
191
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 ...
- 31
4
votes
2
answers
257
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 ...
- 143
1
vote
1
answer
78
views
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 ...
- 111
0
votes
0
answers
52
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 ??
2
votes
1
answer
148
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 ...
- 163
2
votes
0
answers
94
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 ...
- 163
1
vote
0
answers
84
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 ...
- 491
1
vote
0
answers
140
views
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, ...
- 111
0
votes
0
answers
76
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 ...
- 101
2
votes
0
answers
92
views
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 ...
- 121
29
votes
2
answers
612
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 ...
- 401
2
votes
2
answers
220
views
Origin of Riemann-Stieltjes Integral
What need (if there was any) created Riemann-Stieltjes integral? What did Riemann-Stieltjes integral want to attain?
- 675
1
vote
0
answers
59
views
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:
$$\int_{x_1}^{x_2}f(x) + f^{-1}(x)dx = x_{2}^{2}...
- 409
1
vote
0
answers
52
views
In which work did Euler invent the Euler Substitutions for a quadratic composed into a radical?
A famous technique in the modification of integrands is the set of “euler substitutions” that provide substitutions for the structure
$$\sqrt{ax^2 +bx+c}$$
That is a fairly common occurence in ...
- 409
2
votes
2
answers
128
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 ...
- 409
3
votes
1
answer
174
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 ...
- 409
2
votes
1
answer
446
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. ...
- 23
2
votes
0
answers
118
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 ...
- 255
3
votes
2
answers
143
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, "...
- 283
3
votes
1
answer
347
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 ...
- 31
1
vote
1
answer
75
views
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?
- 125
2
votes
0
answers
92
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)+\...
- 21
2
votes
1
answer
245
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 ...
1
vote
0
answers
86
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 ...
4
votes
1
answer
638
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 ...
- 299
3
votes
0
answers
88
views
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 ...
- 1,801
0
votes
1
answer
243
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 ...
- 101
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 ...
user14943
10
votes
0
answers
242
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 ...
- 256
1
vote
2
answers
360
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 ...
- 121
0
votes
0
answers
221
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 ...
- 113
2
votes
1
answer
331
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 ...
- 121
5
votes
3
answers
1k
views
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 ...
- 337
0
votes
0
answers
156
views
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
4
votes
1
answer
305
views
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 ...
- 143
1
vote
0
answers
78
views
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 ...
- 642
0
votes
1
answer
82
views
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 $\...
- 642
4
votes
2
answers
455
views
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 ...
- 141
4
votes
1
answer
221
views
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) ...
- 249
1
vote
0
answers
105
views
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 ...
5
votes
2
answers
250
views
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 ...
1
vote
1
answer
313
views
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 ...
- 111
2
votes
1
answer
263
views
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 $\sin(x)$...
3
votes
0
answers
186
views
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 ...