Questions tagged [calculus]
For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.
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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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991 views
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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77 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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142 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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141 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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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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129 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 ...
4
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75 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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159 views
Why do we use Leibniz's āversionā 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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116 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
80 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
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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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508 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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133 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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57 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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Who first proved Fubini's theorem $n$th order integrals?
Who first proved a generalized Fubini theorem for integrals of order $ā„3$?
An $n$th order integral is $$\underbrace{\underset{x_n}\int\underset{x_{n-1}}\int\ldots\underset{x_1}\int}_{n} f(x_1,x_2,\...
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1answer
158 views
Did Cauchy ever deal with double or triple integrals?
Did Cauchy ever deal with double or triple integrals? Did he give rigorous proofs of multivariable integral calculus like what came to be called Stokes's theorem, the divergence theorem, etc.?
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From where the so-named “elastica problem” is coming from?
In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here).
The problem is presented as a system of ODEs,
$$
x' = \cos (\phi)
$$
$$
y' = \sin (\phi)
$$...
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131 views
Newton's calculus and the binomial theorem
I'm trying to understand the development of the calculus. Does this sound right as one of the stages?
Newton knows the binomial theorem, which gives
$$(x+y)^n={n\choose0}x^ny^0+...\;\;\;\;\;\;\;\;\;\...
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107 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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272 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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82 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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230 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? ...
3
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1answer
270 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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56 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 $\ln(...
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1answer
159 views
Who introduced cylindrical coordinates?
Cylindrical coordinates$ x=r\cos Īø, y=r\sin Īø, 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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83 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 ...
3
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1answer
204 views
How - Historically- René Descartes works affect the invention of calculus?
When "Cartesian coordinate system" been discovered By RenƩ 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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89 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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269 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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114 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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154 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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321 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
votes
2answers
186 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
657 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 ...
7
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1answer
657 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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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 ātaking limitsā, because they ādisliked infinityā.
...
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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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387 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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158 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
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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 ...
2
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1answer
133 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 ...
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438 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
86 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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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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711 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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254 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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711 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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67 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'Alembertās Opuscules mathĆ©matiques, vol. 6 (1773), found in Mercure de France, April 1773, pp. 127-132?
It seems ...
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1answer
217 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 ...
