# Questions tagged [calculus]

For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change.

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### 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 ...
120 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 ...
137 views

### How did Euler obtain this formula from a paper/work in 1748?

I am reading this book on trigonometric series, "Тригонометрические ряды от Эйлера до Лебега" (Trigonometric series from Euler to Lebesgue) , it is in Russian, and my Russian is abysmal. But ...
100 views

### 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 ...
141 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 ...
95 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,\...
218 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 ...
127 views

### Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?

Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?
60 views

### Source of L’Hôpital’s 1696 Calculus textbook

A calculus textbook I’m using references a calculus book of L’Hôpital in which he illustrates his rule, which is taught in many calculus classes. Does anyone have a source as a scanned PDF? I’d love ...
61 views

### Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?

There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ...
70 views

### How did Fourier determine the coefficients of Fourier series?

I was reading a chapter of Fourier's seminal work "Analytic Theory of Heat". The third chapter of this book was translated by the famous Stephen Hawking in his book "God created the ...
129 views

### How to derive the power series of $\sin(x)$ and $\cos(x)$ followed the footstep of Euler

I am reading Euler's "Introduction to analysis of the infinite", chapter 8, page at the end of page 208, beginning of page 209 and came across his derivation of the power series for $\sin(x)$...
73 views

### How did definitions of a limit vary before the epsilon-delta definition?

My understanding is that before the epsilon-delta definition of a limit, the rigor and soundness of the definition of a limit was not good enough. So, how did the definitions of a limit vary before ...
115 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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### 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. ...
266 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 ...
109 views

We know that Sir Isaac Newton thought about calculus when he tried to efficiently describe his physical laws but what made Sir Gottfried Leibniz think about something which we know today as calculus?
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### 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. ...