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Calculus, originally called infinitesimal calculus or "the calculus of
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
s", is the
mathematical 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 ...
study of continuous change, in the same way that
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
is the study of shape, and
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
is the study of generalizations of
arithmetic operations Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
. It has two major branches,
differential calculus In mathematics, differential calculus is a subfield of calculus that studies 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 ...
and
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
; the former concerns instantaneous rates of change, and the
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 use ...
s of
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s, while the latter concerns accumulation of quantities, and
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
s under or between curves. These two branches are related to each other by the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
, and they make use of the fundamental notions of
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
of
infinite sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
s and
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
to a well-defined
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
. Infinitesimal calculus was developed independently in the late 17th century by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, and
social science Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of s ...
.

# Etymology

In
mathematics education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although re ...
, ''calculus'' denotes courses of elementary
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, which are mainly devoted to the study of functions and limits. The word ''calculus'' is
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
for "small pebble" (the
diminutive A diminutive is a root word that has been modified to convey a slighter degree of its root meaning, either to convey the smallness of the object or quality named, or to convey a sense of intimacy or endearment. A ( abbreviated ) is a word-form ...
of ''
calx Calx is a substance formed from an ore or mineral that has been heated. Calx, especially of a metal, is now known as an oxide. According to the obsolete phlogiston theory, the calx was the true elemental substance, having lost its phlogiston in t ...
,'' meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...
arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton. In addition to the differential calculus and integral calculus, the term is also used for naming specific methods of calculation and related theories which seek to model a particular concept in terms of mathematics. Examples of this convention include
propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
,
Ricci calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
,
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
,
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation th ...
, and
process calculus In computer science, the process calculi (or process algebras) are a diverse family of related approaches for formally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions, communications, and ...
. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's
felicific calculus The felicific calculus is an algorithm formulated by utilitarian philosopher Jeremy Bentham (1747–1832) for calculating the degree or amount of pleasure that a specific action is likely to induce. Bentham, an ethical hedonist, believed the mo ...
, and the
ethical calculus An ethical calculus is the application of mathematics to calculate issues in ethics. Scope Generally, ethical calculus refers to any method of determining a course of action in a circumstance that is not explicitly evaluated in one's ethical code ...
.

# History

Modern calculus was developed in 17th-century Europe by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
(independently of each other, first publishing around the same time) but elements of it appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India.

## Ancient precursors

### Egypt

Calculations of
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
and
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
, one goal of integral calculus, can be found in the Egyptian
Moscow papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...
( BC), but the formulae are simple instructions, with no indication as to how they were obtained.

### Greece

Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...
( – 337 BCE) developed the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
to prove the formulas for cone and pyramid volumes. During the
Hellenistic period In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
, this method was further developed by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
( – ), who combined it with a concept of the indivisibles—a precursor to
infinitesimals In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...
—allowing him to solve several problems now treated by integral calculus. In ''
The Method of Mechanical Theorems ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is one of the major surviving works of the ancient Greek polymath Ar ...
'' he describes. for example, calculating the
center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...
of a solid
hemisphere Hemisphere refers to: * A half of a sphere As half of the Earth * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the (geocentric) celes ...
, the center of gravity of a
frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...
of a circular
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...
, and the area of a region bounded by a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
and one of its
secant line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...
s.

### China

The method of exhaustion was later discovered independently in
China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
by
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
in the 3rd century AD in order to find the area of a circle. In the 5th century AD,
Zu Gengzhi Zu Geng or Zu Gengzhi (; ca. 480 – ca. 525) was a Chinese mathematician, politician, and writer. His courtesy name was Jingshuo (). He was the son of the famous mathematician Zu Chongzhi. He is known principally for deriving and proving the for ...
, son of Zu Chongzhi, established a method that would later be called
Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
to find the volume of a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
.

## Medieval

### Middle East

In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( CE) derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...
.

### India

In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions.
Madhava of Sangamagrama Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He ...
and the
Kerala School of Astronomy and Mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta S ...
thereby stated components of calculus. A complete theory encompassing these components is now well known in the Western world as the ''
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
'' or ''
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
approximations''. However, they were not able to "combine many differing ideas under the two unifying themes of the
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. ...
and the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
, show the connection between the two, and turn calculus into the great problem-solving tool we have today".

## Modern

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
's work ''Stereometrica Doliorum'' formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse. A significant work was a treatise, the origin being Kepler's methods, written by
Bonaventura Cavalieri Bonaventura Francesco Cavalieri ( la, Bonaventura Cavalerius; 1598 – 30 November 1647) was an Italian mathematician and a Jesuate. He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of in ...
, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in '' The Method'', but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the
calculus of finite differences A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
developed in Europe at around the same time.
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
, claiming that he borrowed from
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
, introduced the concept of
adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
,
Isaac Barrow Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem ...
, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670. The
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
and
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, the notions of
higher 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. F ...
s and
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, and of
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s were used by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
in an idiosyncratic notation which he applied to solve problems of
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cu ...
, and many other problems discussed in his ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
, who was originally accused of
plagiarism Plagiarism is the fraudulent representation of another person's language, thoughts, ideas, or expressions as one's own original work.From the 1995 '' Random House Compact Unabridged Dictionary'': use or close imitation of the language and though ...
by Newton. He is now regarded as an
independent inventor An independent inventor is a person who creates inventions independently, rather than for an employer. Many independent inventors patent their inventions so that they have rights over them, and hope to earn income from selling or licensing them. U ...
of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
and
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation. Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his ''
Method of Fluxions ''Method of Fluxions'' ( la, De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671, and published in 1736. Fluxion ...
''), but Leibniz published his " Nova Methodus pro Maximis et Minimis" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus " the science of fluxions", a term that endured in English schools into the 19th century. The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815. Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
was written in 1748 by
Maria Gaetana Agnesi Maria Gaetana Agnesi ( , , ; 16 May 1718 – 9 January 1799) was an Italian mathematician, philosopher, theologian, and humanitarian. She was the first woman to write a mathematics handbook and the first woman appointed as a mathematics profe ...
.

## Foundations

In calculus, ''foundations'' refers to the rigorous development of the subject from
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s and definitions. In early calculus the use of
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably
Michel Rolle Michel Rolle (21 April 1652 – 8 November 1719) was a French mathematician. He is best known for Rolle's theorem (1691). He is also the co-inventor in Europe of Gaussian elimination (1690). Life Rolle was born in Ambert, Basse-Auvergne. Ro ...
and
Bishop Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immater ...
. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book '' The Analyst'' in 1734. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today. Several mathematicians, including
Maclaurin Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
and
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, a way was finally found to avoid mere "notions" of infinitely small quantities. The foundations of differential and integral calculus had been laid. In Cauchy's ''
Cours d'Analyse ''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in de ...
'', we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an
(ε, δ)-definition of limit 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 z ...
in the definition of differentiation. In his work Weierstrass formalized the concept of
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
and eliminated infinitesimals (although his definition can actually validate nilsquare infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus".
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
with the development of
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. In modern mathematics, the foundations of calculus are included in the field of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, which contains full definitions and
proofs Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
of the theorems of calculus. The reach of calculus has also been greatly extended.
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
invented
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
, based on earlier developments by
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was ...
, and used it to define integrals of all but the most
pathological Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
functions.
Laurent Schwartz Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...
introduced distributions, which can be used to take the derivative of any function whatsoever. Limits are not the only rigorous approach to the foundation of calculus. Another way is to use
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
's
non-standard analysis 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 epsilon–delta ...
. Robinson's approach, developed in the 1960s, uses technical machinery from
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
to augment the real number system with
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
and
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
numbers, as in the original Newton-Leibniz conception. The resulting numbers are called
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
s, and they can be used to give a Leibniz-like development of the usual rules of calculus. There is also smooth infinitesimal analysis, which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations. Based on the ideas of F. W. Lawvere and employing the methods of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, smooth infinitesimal analysis views all functions as being
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
and incapable of being expressed in terms of discrete entities. One aspect of this formulation is that the
law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
does not hold. The law of excluded middle is also rejected in
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
, a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. Reformulations of calculus in a constructive framework are generally part of the subject of
constructive analysis In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more com ...
.

## Significance

While many of the ideas of calculus had been developed earlier in
Greece Greece,, or , romanized: ', officially the Hellenic Republic, is a country in Southeast Europe. It is situated on the southern tip of the Balkans, and is located at the crossroads of Europe, Asia, and Africa. Greece shares land borders wi ...
,
China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
,
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...
, Iraq, Persia, and
Japan Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the n ...
, the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles. The Hungarian polymath
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
wrote of this work, Applications of differential calculus include computations involving
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
and
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
, the
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 use ...
of a curve, and
optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
. Applications of integral calculus include computations involving area,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
,
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
,
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
,
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
, and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
and
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
. Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
or sums of infinitely many numbers. These questions arise in the study of
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
and area. The
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
philosopher
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known ...
gave several famous examples of such
paradoxes A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
. Calculus provides tools, especially the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
and the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

# Principles

## Limits and infinitesimals

Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
s. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols $dx$ and $dy$ were taken to be infinitesimal, and the derivative $dy/dx$ was their ratio. The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the epsilon, delta approach to
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
. Limits describe the behavior of a function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the
real number system In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Eve ...
(as a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
with the
least-upper-bound property In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if e ...
). In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for calculus, and for this reason they became the standard approach during the 20th century. However, the infinitesimal concept was revived in the 20th century with the introduction of
non-standard analysis 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 epsilon–delta ...
and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.

## Differential calculus

Differential calculus is the study of the definition, properties, and applications of the
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. ...
of a function. The process of finding the derivative is called ''differentiation''. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the ''derivative function'' or just the ''derivative'' of the original function. In formal terms, the derivative is a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by differentiating the squaring function turns out to be the doubling function. In more explicit terms the "doubling function" may be denoted by and the "squaring function" by . The "derivative" now takes the function , defined by the expression "", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function , as will turn out. In Lagrange's notation, the symbol for a derivative is an
apostrophe The apostrophe ( or ) is a punctuation mark, and sometimes a diacritical mark, in languages that use the Latin alphabet and some other alphabets. In English, the apostrophe is used for two basic purposes: * The marking of the omission of one ...
-like mark called a
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. Thus, the derivative of a function called is denoted by , pronounced "f prime" or "f dash". For instance, if is the squaring function, then is its derivative (the doubling function from above). If the input of the function represents time, then the derivative represents change with respect to time. For example, if is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of is how the position is changing in time, that is, it is the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the ball. If a function is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
(that is, if the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of the function is a straight line), then the function can be written as , where is the independent variable, is the dependent variable, is the ''y''-intercept, and: :$m= \frac= \frac = \frac.$ This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in divided by the change in varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let be a function, and fix a point in the domain of . is a point on the graph of the function. If is a number close to zero, then is a number close to . Therefore, is close to . The slope between these two points is :$m = \frac = \frac.$ This expression is called a ''
difference quotient In single-variable calculus, the difference quotient is usually the name for the expression : \frac which when taken to the limit as ''h'' approaches 0 gives the derivative of the function ''f''. The name of the expression stems from the fact ...
''. A line through two points on a curve is called a ''secant line'', so is the slope of the secant line between and . The secant line is only an approximation to the behavior of the function at the point because it does not account for what happens between and . It is not possible to discover the behavior at by setting to zero because this would require dividing by zero, which is undefined. The derivative is defined by taking the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
as tends to zero, meaning that it considers the behavior of for all small values of and extracts a consistent value for the case when equals zero: :$\lim_.$ Geometrically, the derivative is the slope of the
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the graph of at . The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function . Here is a particular example, the derivative of the squaring function at the input 3. Let be the squaring function. :$\beginf\text{'}\left(3\right) &=\lim_ \\ &=\lim_ \\ &=\lim_ \\ &=\lim_ \left(6 + h\right) \\ &= 6 \end$ The slope of the tangent line to the squaring function at the point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the ''derivative function'' of the squaring function or just the ''derivative'' of the squaring function for short. A computation similar to the one above shows that the derivative of the squaring function is the doubling function.

## Leibniz notation

A common notation, introduced by Leibniz, for the derivative in the example above is :$\begin y&=x^2 \\ \frac&=2x. \end$ In an approach based on limits, the symbol is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, being the infinitesimally small change in caused by an infinitesimally small change applied to . We can also think of as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example: :$\frac\left(x^2\right)=2x.$ In this usage, the in the denominator is read as "with respect to ". Another example of correct notation could be: :$\begin g\left(t\right) &= t^2 + 2t + 4 \\ g\left(t\right) &= 2t + 2 \end$ Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like and as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...
.

## Integral calculus

''Integral calculus'' is the study of the definitions, properties, and applications of two related concepts, the ''indefinite integral'' and the ''definite integral''. The process of finding the value of an integral is called ''integration''. The indefinite integral, also known as the ''
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
'', is the inverse operation to the derivative. is an indefinite integral of when is a derivative of . (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the
x-axis A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
. The technical definition of the definite integral involves the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of a sum of areas of rectangles, called a
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
. A motivating example is the distance traveled in a given time. If the speed is constant, only multiplication is needed: :$\mathrm = \mathrm \cdot \mathrm$ But if the speed changes, a more powerful method of finding the distance is necessary. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled. When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, travelling a steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with height equal to the velocity and width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve. This connection between the area under a curve and distance traveled can be extended to ''any'' irregularly shaped region exhibiting a fluctuating velocity over a given time period. If represents speed as it varies over time, the distance traveled between the times represented by and is the area of the region between and the -axis, between and . To approximate that area, an intuitive method would be to divide up the distance between and into a number of equal segments, the length of each segment represented by the symbol . For each small segment, we can choose one value of the function . Call that value . Then the area of the rectangle with base and height gives the distance (time multiplied by speed ) traveled in that segment. Associated with each segment is the average value of the function above it, . The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as approaches zero. The symbol of integration is $\int$, an elongated ''S'' chosen to suggest summation. The definite integral is written as: :$\int_a^b f\left(x\right)\, dx.$ and is read "the integral from ''a'' to ''b'' of ''f''-of-''x'' with respect to ''x''." The Leibniz notation is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width becomes the infinitesimally small . The indefinite integral, or antiderivative, is written: :$\int f\left(x\right)\, dx.$ Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function , where is any constant, is , the antiderivative of the latter is given by: :$\int 2x\, dx = x^2 + C.$ The unspecified constant present in the indefinite integral or antiderivative is known as the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
.

## Fundamental theorem

The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration. The fundamental theorem of calculus states: If a function is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
on the interval and if is a function whose derivative is on the interval , then :$\int_^ f\left(x\right)\,dx = F\left(b\right) - F\left(a\right).$ Furthermore, for every in the interval , :$\frac\int_a^x f\left(t\right)\, dt = f\left(x\right).$ This realization, made by both Newton and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, was key to the proliferation of analytic results after their work became known. (The extent to which Newton and Leibniz were influenced by immediate predecessors, and particularly what Leibniz may have learned from the work of
Isaac Barrow Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem ...
, is difficult to determine because of the priority dispute between them.) The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulae for
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s. It is also a prototype solution of a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

# Applications

Calculus is used in every branch of the physical sciences, actuarial science,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
,
business Business is the practice of making one's living or making money by producing or buying and selling products (such as goods and services). It is also "any activity or enterprise entered into for profit." Having a business name does not separ ...
,
medicine Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...
,
demography Demography () is the statistical study of populations, especially human beings. Demographic analysis examines and measures the dimensions and dynamics of populations; it can cover whole societies or groups defined by criteria such as ed ...
, and in other fields wherever a problem can be mathematically modeled and an
optimal Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other. Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
to find the "best fit" linear approximation for a set of points in a domain. Or, it can be used in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
to determine the
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of a continuous random variable given a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
. In
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope,
concavity In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
and
inflection points In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
. Calculus is also used to find approximate solutions to equations; in practice it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...
, fixed point iteration, and
linear approximation In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving o ...
. For instance, spacecraft use a variation of the
Euler method In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit m ...
to approximate curved courses within zero gravity environments.
Physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
makes particular use of calculus; all concepts in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
and
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
are related through calculus. The
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
of an object of known
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
of objects, and the potential energies due to gravitational and electromagnetic forces can all be found by the use of calculus. An example of the use of calculus in mechanics is
Newton's second law of motion Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at re ...
, which states that the derivative of an object's
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
with respect to time equals the net
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
upon it. Alternatively, Newton's second law can be expressed by saying that the net force is equal to the object's mass times its
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
, which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path. Maxwell's theory of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
and
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and in studying radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orie ...
, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a
planimeter A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape. Construction There are several kinds of planimeters, but all operate in a similar way. The precise way in whic ...
, which is used to calculate the area of a flat surface on a drawing. For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property. In the realm of medicine, calculus can be used to find the optimal branching angle of a
blood vessel The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also take waste and carbon dioxide awa ...
so as to maximize flow. Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a
cancer Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumors, which do not spread. Possible signs and symptoms include a lump, abnormal b ...
ous tumour grows. In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both
marginal cost In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it ...
and
marginal revenue Marginal revenue (or marginal benefit) is a central concept in microeconomics that describes the additional total revenue generated by increasing product sales by 1 unit.Bradley R. chiller, "Essentials of Economics", New York: McGraw-Hill, Inc., ...
.

* Glossary of calculus *
List of calculus topics {{Calculus This is a list of calculus topics. Limits * Limit (mathematics) * Limit of a function ** One-sided limit * Limit of a sequence * Indeterminate form * Orders of approximation * (ε, δ)-definition of limit *Continuous function Differe ...
*
List of derivatives and integrals in alternative calculi There are many alternatives to the Calculus, classical calculus of Isaac Newton, Newton and Gottfried Wilhelm Leibniz, Leibniz; for example, each of the infinitely many non-Newtonian calculi. Occasionally an alternative calculus is more suited th ...
* List of differentiation identities * Publications in calculus *
Table of integrals Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not ...

# References

* * * * * * Uses
synthetic differential geometry In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic ...
and nilpotent infinitesimals. * * * * * * Keisler, H.J. (2000). ''Elementary Calculus: An Approach Using Infinitesimals''. Retrieved 29 August 2010 fro
http://www.math.wisc.edu/~keisler/calc.html
* * * * * * * * * * * *

* * *
Calculus Made Easy (1914) by Silvanus P. Thompson
Full text in PDF *
Calculus.org: The Calculus page
at University of California, Davis – contains resources and links to other sites

* ttp://www.ericdigests.org/pre-9217/calculus.htm The Role of Calculus in College Mathematicsfrom ERICDigests.org
OpenCourseWare Calculus
from the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of th ...

Infinitesimal Calculus
nbsp;– an article on its historical development, in ''Encyclopedia of Mathematics'', ed.
Michiel Hazewinkel Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book ''Formal groups and a ...
. *
Calculus training materials at imomath.com
*
The Excursion of Calculus
1772 {{Authority control