# Questions tagged [calculus]

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

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### Is there a 'lost calculus'?

Are there any 'lost' theorems of calculus that could be used to 'simplify' it? For example, are there ways to calculate derivatives without using limits, maybe by some forgotten methods in calculus?
1k 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 ...
10k views

### What is the difference between Calculus of Newton and that of Leibniz?

Are there any differences between the study of Calculus done by Newton as compared to that done by Leibniz? If yes, please mention point by point.
3k views

### Who discovered the power rule for derivatives?

Who discovered the general rule for differentiating polynomials, in particular that the derivative of $x^n$ is $n x^{n-1}$, and when? I appreciate the answer may not be a clear-cut individual and year,...
691 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 ...
738 views

### Who invented the Leibnitz notation $\frac{d^2y}{dx^2}$ for the *second* derivative?

This MSE question made me wonder where the Leibnitz notation $\frac{d^2y}{dx^2}$ for the second derivative comes from. It does not arise immediately as the obvious generalization of $\frac{dy}{dx}$. ...
2k views

### Electromagnetics and vector calculus

A friend of mine claims that vector calculus was invented to do electrodynamics. I'm dubious. I know that Maxwell first wrote down the so-called Maxwell's equations in scalar form and only later ...
2k views

### Why is calculus missing from Newton's Principia?

I'm not suggesting that Newton did not discover calculus - the question is written this way to express my surprise that the Principia does not use the methods of calculus (or 'fluxions'). He instead ...
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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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### 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? ...
385 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 ...
780 views

### Are there any theorems that become “lost” and discarded over time?

I read that Descartes and some other mathematician figured out a 'double tangent' method (as I think it was called) for calculating a derivative of a conic or some curve without using the concepts of ...
296 views

### What is the origin of the cut and weigh method of integration, is it Galileo's?

I recently heard a story of a clever method of approximating an integral which, instead of using numerical techniques, relied on physically drawing out the graph of a function, cutting it out, and ...
279 views

### 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 ...
423 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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### Discovery of Sine and Cosine

Discovery of Sine and Cosine of an angle, the intuition behind it is always intriguing. Apart from "that is the way they were defined", could someone explain how the discovery happened? I have read "...
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### Who calculated for the first time the volume (and surface area) of the sphere exactly?

As we know, even Archimedes did soon some experimental calculations. My question were, who calculated first time the exact formulas ($V=\frac{4\pi}{3}r^3$, $A=4\pi r^2$)? As I know, these formulas ...
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### Historical development of power series

I'm very interested to know about historical development of power series, i.e. $$\sum_{n=0}^{\infty}a_n(x-c)^n=a_0+a_1(x-c)+a_2(x-c)^2+\dots$$ What was the situation and historical context that ...
506 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 ...
605 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 ...
622 views

### How were vector calculus nabla ∇ identities first derived?

(Math Stack Exchange suggested that the same question I posted there be migrated here; The one at Math Stack Exchange was thus deleted. The recommendation message of migration can be found here, ...
912 views

### How was curvature originally defined and calculated?

I am interested in the early history of curvature. Who defined it first and when, who came up with the name, how was it calculated before mathematicians used calculus to define $k=|α''(s)|$? Are there ...
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### 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 ...
542 views

### Is it true that Leibniz introduced “constant,” “variable,” and “function”?

I read in a not always reliable source (David Foster Wallace's Everything and More, p.104), that Leibniz introduced the terms constant, variable, and function, the latter as an alternative to Newton's ...
379 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 ...
636 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 ...
438 views

### Did Newton and Leibniz lack rigour?

I've read that the original approach to demonstrating the efficacy of calculus left something to be desired among mathematicians. What exactly was the problem? Does the later accepted definition ...
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### How were vector quantities developed?

I'm very interested to know how the concept of vectors came in mathematics and physics. How were vector quantities discovered in physics, and how and by whom were they developed?
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### When/How were the product and chain rules first proved?

Pretty much every proof of the product or chain rules presented today revolve around the definition of the derivative as a limit (e.g. this post). However, when Newton/Leibniz were developing ...
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### How were derivatives of trigonometric functions first discovered?

When proving them the "modern" way (from first principles) it seems impossible to get around proving the identities $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$ and the related $\cos$ limit. This itself ...
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### Was the Riemann Integral the first integration theory?

My sketchy understanding of the (no doubt long) history of integration theory is that the first integration theory was created by Riemann as part of his work on trigonometric series ("Über die ...
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### Who first drew the Weierstrass function?

As we know, it was Weierstrass who gave the first (published) example, in 1872, of a function which is continuous but everywhere non-differentiable. However, in his paper "Über continuirliche ...
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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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### 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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### 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 ...