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History Of The Function Concept
The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope \operatorname\!y/\operatorname\!x of a graph at a point was regarded as a function of the ''x''-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme. Mathematicians of the 18th century typically regarded a function as being defined by an analytic expression. In the 19th century, the demands of the rigorous development of analysis by Weierstrass and others, the reformulation of geometry in terms of analysis, and the invention of set theory by Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set to another. Functions before the 17th century Already in the 12th century, mathematician Sharaf al-Din al-Tusi analyzed the equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicole Oresme
Nicole Oresme (; c. 1320–1325 – 11 July 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a French philosopher of the later Middle Ages. He wrote influential works on economics, mathematics, physics, astrology and astronomy, philosophy, and theology; was Bishop of Lisieux, a translator, a counselor of King Charles V of France, and one of the most original thinkers of 14th-century Europe. Life Nicole Oresme was born c. 1320–1325 in the village of Allemagnes (today's Fleury-sur-Orne) in the vicinity of Caen, Normandy, in the diocese of Bayeux. Practically nothing is known concerning his family. The fact that Oresme attended the royally sponsored and subsidised College of Navarre, an institution for students too poor to pay their expenses while studying at the University of Paris, makes it probable that he came from a peasant family. Oresme studied the "arts" in Paris, together with Jean Buridan (the so-called founder of the French school of na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Institutiones Calculi Differentialis
''Institutiones calculi differentialis'' (''Foundations of differential calculus'') is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus. It consists of a single volume containing two internal books; there are 9 chapters in book I, and 18 in book II. writes that "this is the first textbook on the differential calculus which has any claim to be both complete and accurate, and it may be said that all modern treatises on the subject are based on it." See also * ''Institutiones calculi integralis'' *List of important publications in mathematics This is a list of important publications in mathematics, organized by field. Some reasons why a particular publication might be regarded as important: *Topic creator – A publication that created a new topic *Breakthrough – A publi ... References * * External links Full textin Latin available from e-rara.ch. German translation''Vollständige ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship between given quantities. The plural of ''formula'' can be either ''formulas'' (from the most common English plural noun form) or, under the influence of scientific Latin, ''formulae'' (from the original Latin). In mathematics In mathematics, a formula generally refers to an identity which equates one mathematical expression to another, with the most important ones being mathematical theorems. Syntactically, a formula (often referred to as a ''well-formed formula'') is an entity which is constructed using the symbols and formation rules of a given logical language. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. However, having do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expression (mathematics)
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax. Many authors distinguish an expression from a ''formula'', the former denoting a mathematical object, and the latter denoting a statement about mathematical objects. For example, 8x-5 is an expression, while 8x-5 \geq 5x-8 is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to ''true'' or ''false'', depending on the values that are given to the variables occurring in the expressions. For example 8x-5 \geq 5x-8 takes the value ''false'' if is given a value less than –1, and the value ''true'' otherwise. Examples The use of exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introductio In Analysin Infinitorum
''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introductio'' contains 18 chapters in the first part and 22 chapters in the second. It has Eneström numbers E101 and E102. Carl Boyer's lectures at the 1950 International Congress of Mathematicians compared the influence of Euler's ''Introductio'' to that of Euclid's '' Elements'', calling the ''Elements'' the foremost textbook of ancient times, and the ''Introductio'' "the foremost textbook of modern times". Boyer also wrote: :The analysis of Euler comes close to the modern orthodox discipline, the study of functions by means of infinite processes, especially through infinite series. :It is doubtful that any other essentially didactic work includes as large a portion of original material that survives in the college courses today...Can be read ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Function
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function ''f'' assigns an output ''f''(''x'') to every input ''x''. We say that the function has a limit ''L'' at an input ''p,'' if ''f''(''x'') gets closer and closer to ''L'' as ''x'' moves closer and closer to ''p''. More specifically, when ''f'' is applied to any input ''sufficiently'' close to ''p'', the output value is forced ''arbitrarily'' close to ''L''. On the other hand, if some inputs very close to ''p'' are taken to outputs that stay a fixed distance apart, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. 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. Derivatives can be generalized to functions of several real variables. In this generalization, the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexis Claude Clairaut
Alexis Claude Clairaut (; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had outlined in the '' Principia'' of 1687. Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newton's theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as "Clairaut's theorem". He also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precession of the Moon's orbit. In mathematics he is also credited with Clairaut's equation and Clairaut's relation. Biography Childhood and early life Clairaut was born in Paris, France, to Jean-Baptiste and Catherine Petit Clairaut. The couple had 20 children, however only a few of them survived childbirth. His father taught mathematics. Alexis was a prodig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth. Biography Early life Johann was born in Basel, the son of Nicolaus Bernoulli, an apothecary, and his wife, Margarethe Schongauer, and began studying medicine at University of Basel. His father desired that he study business so that he might take over the family spice trade, but Johann Bernoulli did not like business and convinced his father to allow him to study medicine instead. Johann Bernoulli began studying mathematics on the side with his older brother Jacob Bernoulli. Throughout Johann Bernoulli's education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "''y'' = ''mx'' + ''c''". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The ''steepness'', incline, or grade of a line is measured by the absolu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
