Questions tagged [calculus]
For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.
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questions
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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 ...
5
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2answers
100 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
98 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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vote
1answer
77 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 ...
9
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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
495 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 ...
3
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0answers
130 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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82 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 $$\underbrace{\underset{x_n}\int\underset{x_{n-1}}\int\ldots\underset{x_1}\int}_{n} f(x_1,x_2,\...
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1answer
149 views
Did Cauchy ever deal with double or triple integrals?
Did Cauchy ever deal with double or triple integrals? Did he give rigorous proofs of multivariable integral calculus like what came to be called Stokes's theorem, the divergence theorem, etc.?
3
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1answer
102 views
From where the so-named “elastica problem” is coming from?
In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here).
The problem is presented as a system of ODEs,
$$
x' = \cos (\phi)
$$
$$
y' = \sin (\phi)
$$...
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1answer
118 views
Newton's calculus and the binomial theorem
I'm trying to understand the development of the calculus. Does this sound right as one of the stages?
Newton knows the binomial theorem, which gives
$$(x+y)^n={n\choose0}x^ny^0+...\;\;\;\;\;\;\;\;\;\...
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105 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,...
6
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1answer
208 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?
0
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1answer
75 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 ...
5
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1answer
212 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
251 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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53 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 $\ln(...
6
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1answer
157 views
Who introduced cylindrical coordinates?
Cylindrical coordinates$ 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?
1
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1answer
79 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
175 views
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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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-...
5
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1answer
248 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 ...
2
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0answers
110 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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142 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 ...
2
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2answers
244 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
183 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?
6
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1answer
601 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
569 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
171 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
118 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 ...
5
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1answer
375 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 ...
3
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1answer
145 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?
4
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3answers
376 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 ...
2
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1answer
131 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 ...
11
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1answer
436 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 ...
3
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1answer
83 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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82 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
663 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 ...
5
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1answer
251 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 ...
5
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2answers
641 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
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'Alembert’s Opuscules mathématiques, vol. 6 (1773), found in Mercure de France, April 1773, pp. 127-132?
It seems ...
2
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1answer
205 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
213 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
votes
1answer
144 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?) ...
4
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3answers
274 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
573 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
206 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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2answers
401 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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254 views
History of PDE's in the 19th Century (part 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 ...
