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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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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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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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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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How - Historically- René Descartes works affect the invention of calculus?

When "Cartesian coordinate system" been discovered By René Descartes, Algebra and Geometry become connected and vice versa , but how that exactly affect Newton and Gottfried Leibniz to invent what we ...
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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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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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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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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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142 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 ...
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160 views

How did Newton and Leibniz interpret the integral?

How did Newton and Leibniz think about the integral? Did they only see it as an anti-derivative or did they also think of it as the area under a curve?
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283 views

What algebra problem did Serge Lang give to calculus students?

Joel Spolsky tells this story: Serge Lang, a math professor at Yale, used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could ...
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251 views

Why is differentiation under the integral sign named the Leibniz rule?

The question here asked why differentiation under the integral sign is named "Feynman's trick". That is a comparatively recent name for the method. Aside from the name "differentiation under the ...
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124 views

The exhausting Greek fear of infinity

Every serious source I consulted, be it Cajori, Struik, Edwards,... discusses the method of exhaustion as the means used by ancient Greeks to avoid “taking limits”, because they “disliked infinity”. ...
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106 views

Level of maths of engineers in the Industrial Revolution

Did engineers like I.K. Brunel and his contemporaries employ calculus in their constructions? Or did they work just with 'rules of the thumb' and useful 'laws' like the square-cube...? What was the ...
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Questions about the history of proofs of the divergence theorem

The wikipedia article on the divergence theorem states that it was first discovered by Lagrange in 1762, Gauss in 1813, Ostrogradsky in 1826 - who also gave a proof of the general theorem, Green in ...
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301 views

Did Michel Rolle say that the calculus is “a collection of ingenious fallacies”?

It is probable that this quote was popularized by the writings of Morris Kline, e.g. in Mathematics for Liberal Arts (1967): [Michel Rolle] taught that the calculus was a collection of ingenious ...
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Original document of the Gaussian integral

The Gaussian integral $$\intop_{-\infty}^{\infty} dx \exp(-x^2) = \sqrt{\pi} $$ is done in a very smart way. But where is the original document?
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What brought about the need for real analysis and formal logic in recent years?

I can't seem to find a clear, definitive, non-circular answer on this. For centuries and centuries, we've been doing mathematics in one form or another, be it geometry and pictures, or inventing ...
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English equivalent for a German idiom concerning integration

In German there is phrase concerning the complexity of finding the integral of a given function in contrast to the simplicity of finding its derivative. It has various slightly different formulations ...
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188 views

When and why did $\frac{dy}{dx}$ become $\frac{d}{dx}y$?

It's obvious for us, that $\frac{dy}{dx}$ can also be written as $\frac{d}{dx}y$, but skimming through Leibniz or Eulers writings I couldn't see them write the latter. I speculate that this change ...
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Gregory's integration of $\sec\theta$

The integral of the secant function was first correctly conjectured by Henry Bond in the 1640s, and Isaac Newton was aware of his conjecture in 1665, although no proof was published until 1668. Of ...
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Books on the history of influential Treatises on Calculus and Analysis

I'm interested in the history of calculus & analysis and looking for books that examine in some detail the history of writings on these subjects, mainly the history of the 17th-century "Treatise ...
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Was English mathematics behind Europe by many years because of Newton's notation?

Below are several quotes suggesting that Newton's notation had the effect of retarding English mathematics by 50 years, 100 years, or even centuries. Here is my simplistic two-sentence historical ...
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When was Euler's log-sine integral first computed by real methods?

In Sec. 2.4 of Inside Interesting Integrals (2015), Paul J Nahin says of $$I:=\int_0^{\pi/2}\ln (a\sin x)dx=\int_0^{\pi/2}\ln (a\cos x)dx$$that: For many years it was commonly claimed in textbooks ...
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How influential was the Kerala school to European development in Calculus?

Did it influence the work of Newton or Leibniz, i have often heard that Europeans "stole" calculus from the Kerala school, these are views often parroted by Indian nationalists, but how accurate is it?...
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Author of a review in *Mercure de France*

Would anyone know a way to figure out who wrote the (rather dithyrambic) review of D'Alembert’s Opuscules mathématiques, vol. 6 (1773), found in Mercure de France, April 1773, pp. 127-132? It seems ...
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156 views

D'Alembertian symbol $\Box$

The D'Alembertian is a generalization of the Laplacian operator to a space of arbitrary dimension and metric. Where does the D'Alembertian symbol $\Box$ come from? According to Wikipedia it has to ...
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117 views

What is the origin of q-calculus notation?

It is kind of cute that q-analogues are used in physics (see this link for example), but it is also kind of confusing because the 'q' does not stand for 'quantum'. It predates that use! So, where ...
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116 views

Was Newton's method of finding derivatives of his fluents based on applying the chain rule?

Can Newton’s process for finding the derivative of y as a function of x resulting in $\frac{dy}{dx}$ be thought of as an application of the chain rule? My thinking is: If y is a (continuous?) ...
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What did the typical German student know before reading/studying Courant's Calculus when it was published?

I've been reading Courant's Integral and Differential Calculus for a bit, at some sections, there are problems which do not seem to be answerable with previously given material in the book. Several ...
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420 views

Dirichlet integral's history

I was calculating the so-called Dirichlet Integral, $\displaystyle \int_{0}^{\infty}\frac{\sin x}{x} \text{d} x$ and then I wondered about his name and history: Why it has the name of Dirichlet? ...
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180 views

Galileo and normal distribution discovery

If differential equation theory was known and also studied by Galileo, so why he didn't manage to discover a normal distribution (its discovery had to wait for Laplace and Gauss)?
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A question on Gauss' “Vicimus GEGAN”

The 43rd entry(Oct. 1796) of Gauss' mathematical diary "Vicimus GEGAN" remained a mystery for a long time. K. R. Biermann found evidence that GEGAN is related the famous arithmetic-geometric mean(AGM) ...
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History of PDE's in the 19th Century 2

This is a follow up to this question: History of PDE's in the 19th Century The question I have been given to answer is: The history of partial differential equations in the 19th Century belongs ...
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110 views

The Origin of the Jacobian

In what work did Jacobi introduce the jacobian, and what was his motivation for doing so?
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244 views

Is Spivak right in what he says about Galileo?

On chapter 9 of M. Spivak's book on calculus there is an exercise in which Spivak asks the reader to prove that Galileo "got his facts wrong". More specifically, Spivak asks one to to show if a body ...
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$\frac{dy}{dx}$ versus $\frac{{\mathrm d}y}{{\mathrm d}x}$

When I first learned calculus a few decades ago, the books I read used italicized letter "d"s in derivatives (like this: $\frac{dy}{dx}$). But a few years ago, I started seeing upright "d"s (like ...
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History of PDE's in the 19th Century

I've been asked to write an essay on whether the work on PDE's in the 19th century belonged to applied or pure mathematics. I was wondering if anyone knows of any useful sources I could use?
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Difficulty in Understanding Newton's Principia

I see in many essays about Newton's Principia how it was very difficult to read and follow, so I was wondering if any of you know of any quotes from mathematicians of the time to that effect.
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Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?

This answer, to a question on teaching the chain rule, suggests writing something like this $$ \frac{\mathrm{d}\, \mathrm{e}^\sqrt{s}}{\mathrm{d}\,s}=\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\...
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Name of the Gamma function

The Gamma function for positive arguments can be defined with the integral $$ \Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx $$ The function $ x^{\alpha-1} e^{-x} $ is called the Gamma ...
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How did Newton & Raphson's version of the N-R method differ?

To quote Wikipedia, Raphson's most notable work... contains a method, now known as the Newton–Raphson method... Newton had developed a very similar formula in his Method of Fluxions, written in ...
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History of the origins and development of problems of finding maximum and minimum values of quantities

I am aware that perhaps the earliest source concerning problems of maximum and minimum values occurs in Euclid's Elements. After Euclid, Archimedes of Syracuse and Apollonius of Perga seem to consider ...
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874 views

Why is differentiation under the integral sign named Feynman's trick?

It's a simple enough result I would have been unsurprised if it weren't named for anyone at all. I certainly find it odd it's named for a relatively modern physicist rather than an early-calculus ...
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448 views

Is the prime notation for derivatives $f'$ due to Euler?

Cajori, the website on Earliest Uses of Symbols of Calculus and many other sources claim that Lagrange introduced the notation $f'(x)$ for the derivative of $f(x)$ with respect to $x$. But I see Euler ...
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Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?

The question is in the title, but allow me to provide some background. I’m aware that Leibniz introduced the word “function” into mathematics (around 1673) and that Johann Bernoulli or Euler ...
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Notational change with Integrals

A little over 50 years ago I took my first Calculus class and learned the conventional form of an integral as: $$ \int f(x)\,\, \textrm{d}x $$ That is, the integral sign (definite or indefinite) ...
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419 views

History of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concepts of differentiation and integration together. How did mathematicians of the past see the link between these two concepts? Integration is used to ...
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807 views

How were double integrals calculated before Fubini's Theorem?

In my multi-variable calculus class we have been learning how to calculate double integrals over regions. To accomplish this task we usually make use of Fubini's Theorem because it relates the double ...
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Newton , Binomial Series and Power Series , and James Gregory

After 2 month long search on the net, I was lucky enough to find a pdf on how Newton found the series for sine. It was a beautiful derivation mostly geometrical. But he used the Binomial Series. Now ...