In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve; the primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value; the process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
Differentiation has applications to nearly all quantitative disciplines. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, the derivative of velocity with respect to time is acceleration; the derivative of the momentum of a body with respect to time equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. Derivatives are used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, abstract algebra. Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, there is a corresponding value of y.
This relationship can be written. If f is the equation for a straight line there are two real numbers m and b such that y = mx + b. In this "slope-intercept form", the term m is called the slope and can be determined from the formula: m = change in y change in x = Δ y Δ x, where the symbol Δ is an abbreviation for "change in", it follows that Δy = m Δx. A general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a; this is denoted f ′ in Lagrange's notation or dy/dx|x = a in Leibniz's notation. Since the derivative is the slope of the linear approximation to f at the point a, the derivative determines the best linear approximation, or linearization, of f near the point a. If every point a in the domain of f has a derivative, there is a function that sends every point a to the derivative of f at a. For example, if f = x2 the derivative function f ′ = dy/dx = 2x. A related notion is the differential of a function.
When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, denoted ∂y/∂x; the linearization of f in all directions at once is called the total derivative. The concept of a derivative in the sense of a tangent line is a old one, familiar to Greek geometers such as Euclid and Apollonius of Perga. Archimedes introduced the use of infinitesimals, although these were used to study areas and volumes rather than derivatives and tangents; the use of infinitesimals to study rates of change can be found in Indian mathematics as early as 500 AD, when the astronomer and mathematician Aryabhata used infinitesimals to study the orbit of the Moon.
The use of infinitesimals to compute rates of change was developed by Bhāskara II. The Islamic mathematician, Sharaf al-Dīn al-Tūsī, in his Treatise on Equations, established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials, he proved, for example, that the maximum of the cubic ax2 – x3 occurs when x = 2a/3, concluded therefrom that the equation ax2 — x3 = c has one positive solution when c = 4a3/27, two positive solutions whenever 0 < c < 4a3/27. The historian of science, Roshdi Rashed, has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known; the modern develo
Adolph Strasser, born in the Austro-Hungarian empire, was an American trade union organizer. Strasser is best remembered as a founder of the United Cigarmakers Union and the American Federation of Labor. Strasser was additionally the president of the Cigar Makers' International Union for a period of 14 years, heading the union during the period in which it introduced its successful union label and gained substantial organizational strength. Adolph Strasser was born in the Austro-Hungarian empire in part of today's Hungary in 1843, he was a native speaker of German. Strasser emigrated to the United States in 1871 or 1872. After his arrival in America, Strasser worked at the craft of cigar making, taking up residence and employment in New York City. In his posthumous memoirs fellow cigarmaker Samuel Gompers recalled his impressions of Strasser from the time he met him in 1872: "Strasser was a man of extraordinary mentality, he came to America some months before, traveling through the South before settling in New York City.
He had been identified with the IWA and became a leader of American Section 5. For a while he was exceedingly active in the work of the Social Democratic Party, he shifted all his energy to the trade union movement when he came to understand the unsoundness and impracticability of Socialist Party policy and philosophy or, as Strasser called it,'sophistry.' * * * "Strasser had a keen practical mind... No one knew Strasser's early life and no one asked him questions for Strasser had a terse bluntness of expression in English and in German that made the most venturesome hesitate to take liberties. Whether he learned cigarmaking in Europe or the United States I do not know, but he did not make cigars as one who had learned the trade in his youth.... It was said that Strasser came of a well-to-do cultured Hungarian family. At any rate, he looked the part, he must have had some little means, for he dressed well when he gave all his time to the International for a salary of $250 a year." While Gompers is unclear about the date of Strasser's break with the socialist movement, it is known that in 1874 Strasser helped to organize the Social Democratic Workingmen's Party of the United States, one of the first International Socialist political parties in North America.
Moreover, continued his activity in its successor organizations, which culminated as the Socialist Labor Party at the end of 1877. In the course of this activity, Strasser became involved in the trade union movement helping to found the United Cigarmakers Union, concentrating for its members upon those tenement-based workers excluded from membership in the Cigar Makers International Union. Strasser soon joined forces with the CMIU, editing the monthly magazine established by that union in 1875, the Cigar Makers' Official Journal. In 1876 and 1877 Strasser was instrumental in helping to establish a central body bringing together New York City's various local trade unions. Strasser was elected vice president of the Cigar Makers' International Union in 1876 and president in 1877, he continued to serve in that capacity until stepping down from the job in 1891. During Strasser's term as head of the CMIU the organization began to win strikes which it had lost. Between 1871 and 1875 the union had waged 78 strikes, winning just 12, but in the years from 1876 to 1881 a total of 69 strikes had been fought, with 58 resolved in favor of the striking cigar workers.
Strasser was a close ally of Samuel Gompers, siding with him in the early 1880s against the Socialist "Progressive" faction of the Cigarmakers' Union in a split of the union. Strasser fought against New York's District Assembly 49 of the Knights of Labor for its support of the Progressive Cigarmakers. In 1886 Strasser was one of five signatories to a call for a convention in Columbus, to formally establish the American Federation of Labor. Gompers and Strasser were outspoken opponents of the tenement system of production, in which raw materials were provided to workers for manufacture at home. Under their leadership the CMIU attempted to outlaw the practice of home work outright rather than making any effort to organize cigar workers engaged in that form of production. In 1881 the CMIU adopted use of a special "Blue Label" to denote union-made cigars. Following his retirement as CMIU president in 1891 he continued to work for the union as an organizer and auditor, he was active in the American Federation of Labor as a lecturer and arbitrator of jurisdictional disputes between competing craft unions.
Strasser left the trade union movement in 1914, becoming a real estate agent in Buffalo, New York, for the next five years. In 1919 Strasser retired, living first in Chicago through 1929. In 1930 Strasser moved to Daytona Beach, located on the Atlantic coast of the state of Florida, where he lived out the last decade of his life. Adolph Strasser died on January 1939, in Lakeland, Florida, he was 95 years old at the time of his death. Patricia A. Cooper, "Whatever Happened to Adolph Strasser?" Labor History, Summer 1979. H. M. Gitelman, "Adolph Strasser and the Origins of Pure and Simple Unionism." Labor History, vol. 6, no. 1, pp. 71–83
Enkhbatyn Amartuvshin, is a Mongolian operatic baritone and Honoured Artist of Mongolia. Known as Amartuvshin Enkhbat, he has been a soloist in the State Academic Opera and Ballet Theatre of Mongolia since 2008. Amartüvshin was born on 23 March 1986 in Sukhbaatar in Mongolia. In 2009 he graduated from State University of Arts and Culture, Mongolia. Amartüvshin won numerous national and international opera competitions, including the Mongolian National Competition for Young Opera Singers, international opera competition BAIKAL in Ulan-Ude, Russia, XIV Tchaikovsky Competition in St. Petersburg, Operalia Competition, China, he won the Dame Joan Sutherland Audience Prize at the 2015 BBC Cardiff Singer of the World competition. His stage roles have included Escamillo in Carmen by Bizet, Tonio in I Pagliacci by Leoncavallo, Aleko in Aleko by Rachmaninov, Genghis Khan in Genghis Khan by Sharav, Onegin in Eugene Onegin by Tchaikovsky, Prince Yeletsky in The Queen of Spades by Tchaikovsky, Amonasro in Aida by Verdi, Count di Luna in Il Trovatore by Verdi, Giorgio Germont in La Traviata by Verdi and Rigoletto in Rigoletto by Verdi.
Official website Amartüvshin's page on the site of SMOLART International Artist Management Agency. Amartüvshin in the list of Operalia 2012 winners