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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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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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153 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 maths at any point behind Europe by 50, 100, or more 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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187 views

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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143 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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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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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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301 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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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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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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218 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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555 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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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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207 views

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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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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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 ...
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First appearance of the sine function [duplicate]

I was wondering when was the sine function (I suppose cosine too) first introduced or defined and what was the need for?
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Looking for an online version of Archimedes' “The Method” (in Greek)

To me, one of the most exciting mathematical achievements of antiquity is Archimedes' The Method. It is crazy to think that, had it not been for its miraculous recovery in the early 20th century, it ...
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Taylor's Theorem and Newton's Method of Divided Differences

While reading Chandrashrkhar's edition of Principia , I came to know that Newton's Method of Divided Differences can be used to prove Taylor's Theorem. Could some one help me in knowing how this is ...
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Who introduced the notation $y|_{x=a}$?

When a variable $y$ depends on other variables, say $y=c x^3$, one often writes $$y|_{x=2}$$ to say "$y$ when $x$ has value $2$". This might be more familiar in the context of derivatives where we ...
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What is the name of the identity $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ and who derived it?

What is the name of this indentity and which mathematician did derive this? $$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$
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Were integrals really called solution curves (or vice versa)?

For some reason I recall hearing that around the time Euler wrote his Calculus books (1768-1770), or even before then, what we call integrals now were called solution cuvres (or even possibly the ...
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Who invented the gradient?

Who is responsible for coming up with the gradient and why did they do so? In which work was it first described? I have Googled this extensively, to no avail, and Boyer's History of Calculus does ...
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How did newton APPROXIMATE THE AREA UNDER THESE PARTICULAR CURVES [duplicate]

To find - How did Newton derive the general binomial theorem. I know he approximated the area under functions. 1) But how did he approximate the area under CURVES like ( 1/((1-x^2)^(1/2))) or ( 1/((...
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When was the convention for the indefinite integral $\int\frac{1}{x}dx$ changed?

In Europe, in the 20th century, $\int\frac{1}{x}dx$ equalled $\ln{x}+C$. (I have references from Poland for 1930-1947 and the UK for the 1960s and 1970s). Now, if one mentions $\int\frac{1}{x}dx=\ln{...
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on the classification of singular points

After reading this question and the answers to it, I am interested o know who were the first mathematicians who started classifying singular points of curves: i.e. different kind of nodes, of cusps ...
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How did Newton prove the generalised binomial theorem?

Do we have any idea of how Newton proved the generalised binomial theorem?
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Based on what criteria one could say Leibniz “invented” the differential calculus?

I am already aware of the notation differences. But is it the only criterion?
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History of the derivative/tangent of a curve

I just want to know the history of the derivative. Whenever I Google for it, I find the history of calculus or the tangent of a curve. However, they barely touch upon what happened before Leibniz and ...
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Who did first use the Method of Characteristics

Which mathematician did introduce the Method of Characteristics for solving linear Partial differential equations? I some papers I saw that Saint Venant, Lagrange, Cauchy or Riemann were attributed ...
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306 views

Who derived $\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$?

I want to know who derived $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$ In school, our book mentioned that Euler proved this result. But on Math Stack Exchange, some people say ...
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When was the inverse relationship between tangents and quadrature/area first identified?

Problems concerning tangents and quadrature have a long history predating the Newton/Leibniz formulation of calculus; indeed, they are amongst the oldest problems in mathematics. It seems reasonable ...