Questions tagged [calculus]

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

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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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4 votes
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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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Who first used exponential function as a solution of second order differential equations? [duplicate]

You can refer this question from math SE which explains why exponential function is used as a solution of second order differential equations. However, I am interested to know who was the ...
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3 votes
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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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How old is the idea of an Oscullating Circle? [duplicate]

In the second volume of Spivak's Comprehensive introduction to differential geometry, he begins the discussion of curvature by discussing the oscullating circle of a curve in the plane. This leads me ...
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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?
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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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5 votes
2 answers
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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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1 vote
1 answer
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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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2 votes
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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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3 votes
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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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Founders of undetermined forms [duplicate]

Who are the people actually 1st to mention about undetermined forms like 0^∞ or 0/0 or ∞/∞ etc . In which books these forms are mentioned and if the people were not known specifically , mention the ...
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What does an 100 year old calculus exam look like?

I wonder whether the questions on a calculus exam at university were easier or harder 100 years ago. Nowadays we have all these aids and different learning methods. I would love to see an old exam.
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4 votes
1 answer
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When did the Notion of "Limit" Arise and for What Purpose?

It is my understanding that Cauchy was the first to incorporate the notion of a $\delta$-$\epsilon$ limit in his proofs, although a definition was not formulated until Weierstrass did so. How far ...
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8 votes
2 answers
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Where did Leibniz explore the product rule of differential calculus?

In what book/letter did Gottfried Wilhelm Leibniz explore the product rule as part of differential calculus?
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Where did the contour integral sign appear for the first time?

A simple question: Where did the contour integral sign appear for the first time? Wikipedia says that it was introduced by physicist Arnold Sommefield in 1917 ( Table of mathematical symbols by ...
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What did Newton's teacher contribute to the Fundamental Theorem of Calculus?

Isaac Barrow was one of the professors who taught Isaac Newton at Cambridge. According to this page, he is said to have made contributions to the Fundamental Theorem of Calculus that was devised by ...
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Why is the "universal trigonometric substitution" called "Weierstrass's substitution"?

The universal trigonometric substitution converts a rational function in $\sin(x), \cos(x) $ into a rational function of a new variable $t$ by the substitution $t = \tan(x/2) $. It therefore enables ...
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Newton's evaluation of 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11

I had asked the following question on MSE here, and I was directed to this exchange. There is a nice solution in the comments there, but perhaps someone here can add some additional insight? How ...
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1 vote
1 answer
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What steps did Richard Feynman take to devise his Integral Trick?

Richard Feynman is considered to be one of the greatest minds in physics, and has won many accolades as a result of his research in areas such as quantum mechanics and particle physics. However, I am ...
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5 votes
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Who introduced the comma notation for partial derivatives?

In general relativity, it is common to use the comma notation for partial derivatives $$\frac{\partial g_{\mu\nu}}{\partial x_\rho} = g_{\mu\nu_,\rho}$$ Where did this notation first appear? Was it ...
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5 votes
2 answers
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Why do we use Leibniz's “version” of calculus instead of Newton's?

I understand that they invented calculus independently at roughly the same time, but why do we use Leibniz's terminology/notation rather than Newton's? For example, why don't we use "fluxion"...
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Who discovered the indeterminate forms like 0/0?

Who discovered the indeterminate forms and how did they discover them? How did someone come to know that a particular form (fraction, product, sum/difference, exponent) is indeterminate? For example, $...
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1 vote
1 answer
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Where did Euler prove 'his' theorem on homogeneous functions?

Where in Eulers writings can I find a proof of his homogeneous function theorem: $y$ is a homogeneous function of degree $k$ in $x_1,\ldots,x_n$ iff $ky = \sum_{i=1}^n x_i\frac{\partial y}{\partial ...
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9 votes
1 answer
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Did Benjamin Franklin know calculus?

My sense is that Franklin was a scientist more like Faraday than Maxwell. Given also that Calculus was fairly new when Franklin was in school (and as far as I know, Benjamin Franklin did not get very ...
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What is the history of staircase or 𝜋=4 paradox?

The staircase 'paradox' has been discussed here and elsewhere a few times (search for staircase + paradox). My question is whether this puzzle has been discussed in the academic literature or ...
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Before differential calculus was discovered, why were mathematicians interested in tangents?

I think it is often said that one great motivation for the invention of calculus was to have a tool allowing to calculate the slope of a tangent to a curve $C$ at a given point $P$, and even, to find ...
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Were fluxions the only infinitesimals?

The Encyclopaedia Britannica in its history of Science article states that Newton integrals were made of infinitesimals, whereas Leibniz' were made of sticks and that the former's theory prevailed. ...
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Who first proved Fubini's theorem $n$th order integrals?

Who first proved a generalized Fubini theorem for integrals of order $≥3$? An $n$th order integral is $$\underbrace{\underset{x_n}\int\underset{x_{n-1}}\int\ldots\underset{x_1}\int}_{n} f(x_1,x_2,\...
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6 votes
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Did Cauchy ever deal with double or triple integrals?

Did Cauchy ever deal with double or triple integrals? Did he give rigorous proofs of multivariable integral calculus like what came to be called Stokes's theorem, the divergence theorem, etc.?
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From where the so-named "elastica problem" is coming from?

In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here). The problem is presented as a system of ODEs, $$ x' = \cos (\phi) $$ $$ y' = \sin (\phi) $$...
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Newton's calculus and the binomial theorem

I'm trying to understand the development of the calculus. Does this sound right as one of the stages? Newton knows the binomial theorem, which gives $$(x+y)^n={n\choose0}x^ny^0+...\;\;\;\;\;\;\;\;\;\...
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Two questions about Gauss's contributions to capillarity and the calculus of variations

In the last page of the abstract of Gauss's paper on capillarity "Principia generalia theoriae figurae fluidorum in statu aequilibrii" (1829), the author (who is he?) mentions (Gauss's werke, volume V,...
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7 votes
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Who invented the gradient descent algorithm?

In connection to the question "Who invented the gradient?", I want to know who invented the gradient descent algorithm?
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Was there a more intuitive early proof of the generalized mean value theorem?

I am interested in the early proofs of the theorem. It is often called Cauchy mean value theorem, so perhaps Cauchy proved it first. In all the proofs that I have seen we construct a contrived ...
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What problem led to the discovery of Calculus?

As far as I remember, Calculus was invented/discovered/founded by Newton. What was he trying to achieve that made him find the limit of differences approaching zero? How far did he get into Calculus? ...
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