Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.
Edwin Abbott Abbott was an English schoolmaster, theologian, and Anglican priest, best known as the author of the novella Flatland (1884).
Giuseppe Peano was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin. He also created an international auxiliary language, Latino sine flexione, which is a simplified version of Classical Latin. Most of his books and papers are in Latino sine flexione, while others are in Italian.
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using limits rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.
Project Gutenberg (PG) is a volunteer effort to digitize and archive cultural works, as well as to "encourage the creation and distribution of eBooks." It was founded in 1971 by American writer Michael S. Hart and is the oldest digital library. Most of the items in its collection are the full texts of books or individual stories in the public domain. All files can be accessed for free under an open format layout, available on almost any computer. As of 13 February 2024, Project Gutenberg had reached 70,000 items in its collection of free eBooks.
Karl Theodor Wilhelm Weierstrass was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics, botany and gymnastics. He later received an honorary doctorate and became professor of mathematics in Berlin.
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-eth" item in a sequence.
Walter William Rouse Ball, known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding president of the Cambridge Pentacle Club in 1919, one of the world's oldest magic societies.
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.
The Harvard Classics, originally marketed as Dr. Eliot's Five-Foot Shelf of Books, is a 50-volume series of classic works of world literature, important speeches, and historical documents compiled and edited by Harvard University President Charles W. Eliot. Eliot believed that a careful reading of the series and following the eleven reading plans included in Volume 50 would offer a reader, in the comfort of the home, the benefits of a liberal education, entertainment and counsel of history's greatest creative minds. The initial success of The Harvard Classics was due, in part, to the branding offered by Eliot and Harvard University. Buyers of these sets were apparently attracted to Eliot's claims. The General Index contains upwards of 76,000 subject references.
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.
Silvanus Phillips Thompson was an English professor of physics at the City and Guilds Technical College in Finsbury, England. He was elected to the Royal Society in 1891 and was known for his work as an electrical engineer and as an author. Thompson's most enduring publication is his 1910 text Calculus Made Easy, which teaches the fundamentals of infinitesimal calculus, and is still in print. Thompson also wrote a popular physics text, Elementary Lessons in Electricity and Magnetism, as well as biographies of Lord Kelvin and Michael Faraday.
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of contemporary mathematics education. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
Oskar Bolza was a German mathematician, and student of Felix Klein. He was born in Bad Bergzabern, Palatinate, then a district of Bavaria, known for his research in the calculus of variations, particularly influenced by Karl Weierstrass' 1879 lectures on the subject.
In physics and mathematics, the differential coefficient of a function f(x) is what is now called its derivative df(x)/dx, the (not necessarily constant) multiplicative factor or coefficient of the differential dx in the differential df(x).
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal , the unique real infinitely close to it, i.e. is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity.
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by , the infinity symbol.
In a publishing career spanning 80 years (1930–2010), popular mathematics and science writer Martin Gardner (1914–2010) authored or edited over 100 books and countless articles, columns and reviews.
In mathematical analysis, the staircase paradox is a pathological example showing that limits of curves do not necessarily preserve their length. It consists of a sequence of "staircase" polygonal chains in a unit square, formed from horizontal and vertical line segments of decreasing length, so that these staircases converge uniformly to the diagonal of the square. However, each staircase has length two, while the length of the diagonal is the square root of 2, so the sequence of staircase lengths does not converge to the length of the diagonal. Martin Gardner calls this "an ancient geometrical paradox". It shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve.
