Questions tagged [calculus]
For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.
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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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2answers
70 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 ...
5
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2answers
99 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?
5
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1answer
155 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 ...
6
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1answer
106 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 ...
6
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1answer
106 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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1answer
88 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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56 views
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 ...
5
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1answer
210 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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1answer
94 views
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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3answers
283 views
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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1answer
88 views
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 ...
10
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1answer
178 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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1answer
67 views
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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61 views
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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3answers
344 views
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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1answer
189 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 ...
4
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2answers
415 views
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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1answer
65 views
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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1answer
150 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 ...
4
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1answer
81 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 ...
4
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1answer
103 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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3answers
133 views
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 ...
5
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1answer
330 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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1answer
144 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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214 views
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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171 views
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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1answer
92 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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2answers
228 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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3answers
452 views
$\frac{dy}{dx}$ 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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2answers
376 views
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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1answer
393 views
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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228 views
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
$$
\frac{\mathrm{d}\, \mathrm{e}^\sqrt{s}}{\mathrm{d}\,s}=\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\...
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1answer
222 views
Name of the Gamma function
The Gamma function for positive arguments can be defined with the integral
$$ \Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx $$
The function $ x^{\alpha-1} e^{-x} $ is called the Gamma ...
3
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2answers
270 views
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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1answer
163 views
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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1answer
669 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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2answers
364 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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2answers
965 views
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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2answers
208 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) ...
4
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1answer
345 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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1answer
598 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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1answer
323 views
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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2answers
347 views
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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1answer
119 views
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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1answer
248 views
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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2answers
316 views
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}$$
to say "$y$ when $x$ has value $2$". This might be more familiar in the context of derivatives where we ...
3
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1answer
150 views
What is the name of the identity $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ and who derived it?
What is the name of this indentity and which mathematician did derive this?
$$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$
4
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1answer
98 views
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 ...
4
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1answer
886 views
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 ...
