Questions tagged [calculus]
For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.
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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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3answers
166 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 ...
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109 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?
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131 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 ...
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61 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 ...
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1answer
74 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 $\...
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1answer
106 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 ...
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1answer
81 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) ...
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70 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 ...
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2answers
189 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 ...
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1answer
81 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 ...
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1answer
129 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 $\sin(x)$...
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137 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 ...
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73 views
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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1answer
167 views
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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1answer
263 views
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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1answer
257 views
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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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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979 views
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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1answer
178 views
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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200 views
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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100 views
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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1answer
232 views
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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120 views
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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2answers
277 views
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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1answer
207 views
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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1answer
89 views
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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1answer
2k views
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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2answers
552 views
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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141 views
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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61 views
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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95 views
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 $$\underbrace{\underset{x_n}\int\underset{x_{n-1}}\int\ldots\underset{x_1}\int}_{n} f(x_1,x_2,\...
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1answer
224 views
Did Cauchy ever deal with double or triple integrals?
Did Cauchy ever deal with double or triple integrals? Did he give rigorous proofs of multivariable integral calculus like what came to be called Stokes's theorem, the divergence theorem, etc.?
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1answer
107 views
From where the so-named "elastica problem" is coming from?
In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here).
The problem is presented as a system of ODEs,
$$
x' = \cos (\phi)
$$
$$
y' = \sin (\phi)
$$...
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1answer
182 views
Newton's calculus and the binomial theorem
I'm trying to understand the development of the calculus. Does this sound right as one of the stages?
Newton knows the binomial theorem, which gives
$$(x+y)^n={n\choose0}x^ny^0+...\;\;\;\;\;\;\;\;\;\...
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115 views
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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500 views
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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1answer
121 views
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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1answer
399 views
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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1answer
333 views
Source of claim that Leibniz discovered separation of variables for ODEs in 1691?
Claims I'm evaluating
I've read in multiple sources that Leibniz formulated separation of variables for ODEs in 1691. A couple example sources are below.
Mathematical Thought from Ancient to Modern ...
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67 views
How was logarithm discovered? [duplicate]
How was the concept of $\ln(x)$ found before the man knows that it is the area under hyperbola or it is related to the power of $e$ (base of logarithm). How did Napier compute the value of $e$ or $\ln(...
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1answer
189 views
Who introduced cylindrical coordinates?
Cylindrical coordinates$ x=r\cos θ, y=r\sin θ, z=w$ seem to be a simple generalization of polar coordinates. When did they appear first? Also, who came up with the name?
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1answer
102 views
Was multivariable calculus particularly prominent in Italy?
From my classes I don't hear about a lot of italian mathematicians, but two of them, Fubini and Tonelli, are both related to multivariable calculus. Is there a reason for this? Just a coincidence? Or ...
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333 views
Historically, how did René Descartes's works affect the invention of calculus?
When the "Cartesian coordinate system" was discovered By René Descartes, Algebra and Geometry were connected. How exactly did that affect Newton and Leibniz in the invention of what we know ...
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95 views
Is it the 'd' or 'D' operator?
Philip J. Davis' article on the history of the gamma function (PDF) mentions how Leibniz proposed the iterated differential operator (p. 851 in the upper right corner, or p. 3 of the PDF, about half-...
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1answer
502 views
Earliest Instances of a Slope/Direction Field for a First-Order ODE
Background
When first encountering slope fields in calculus or elementary differential equations, students often ask "What is the purpose?"
A concise answer is that slope fields provide a way to ...
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128 views
At what point did Calculus become a required field of study for aspiring scientists?
Nowadays, it's nearly impossible to obtain a university degree in a scientific discipline without completing at least some basic coursework in Calculus, and often times advanced courses are required. ...
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218 views
The Integral as a Uniform Limit of Step Functions
Who first realized that it is possible to define the integral of a function as the limit of the integrals of a sequence of step functions that converge uniformly to the given function?
This is ...
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2answers
418 views
Riemann's Contribution to Integration
What did Riemann do for the theory of integration?
I am asking because I hear his name a lot in relation to integration and it is often implied that he made large contributions, but I do not know ...