calculus

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OR:

Calculus, originally called infinitesimal calculus or "the calculus of
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
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 ca ...
is the study of shape, and
algebra Algebra () is one of the areas of mathematics, 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 mathem ...
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 operation (mathematics), operations on numbers—addition, subtraction, multiplication, division (mathematics), division, exponent ...
. 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 Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional object ...
s of
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
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 o ...
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 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 mo ...
s and
infinite series In mathematics, a series is, roughly speaking, a description of the operation of addition, 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 generalizat ...
to a well-defined limit. 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 mathema ...

. 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 Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...

,
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 specializ ...

, and
social science Social science is one of the branches of science, devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the o ...

.

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 method, scholarly research into the transfer of mathematical knowled ...
, ''calculus'' denotes courses of elementary
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, 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 Intimate relationship, intimacy or Term of endearment, endea ...
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 An oxide () is a chemical compound that contains at least one oxygen atom and one other chemical element, element in ...
,'' 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 history, ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the ado ...

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 (mathematics), connection. It is also the modern nam ...
,
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

,
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable ...
, and
process calculus In 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 scienc ...
. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as
felicific calculus The felicific calculus is an algorithm formulated by utilitarianism, utilitarian philosopher Jeremy Bentham (1747–1832) for calculating the degree or amount of pleasure that a specific action is likely to induce. Bentham, an ethics, ethical hed ...
, and the ethical calculus.

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 mathema ...

(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 o ...
, one goal of integral calculus, can be found in the Egyptian Moscow papyrus ( 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 Greece, ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though som ...
( – 337 BCE) developed the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by Inscribed figure, inscribing inside it a sequence of polygons whose areas limit (mathematics), converge to the area of the containing shape. If the sequence is correctly c ...
to prove the formulas for cone and pyramid volumes. During the
Hellenistic period In Classical antiquity, the Hellenistic period covers the time in History of the Mediterranean region, Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as sig ...
, this method was further developed by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, ...
( – ), who combined it with a concept of the
indivisibles In 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 o ...
—a precursor to
infinitesimals In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
—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 A ...
'' 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 m ...
of a solid
hemisphere Hemisphere refers to: * A half of a sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at th ...
, the center of gravity of a
frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a Polyhedron, solid (normally a Pyramid (geometry), pyramid or a Cone (geometry), cone) that lies between two parallel planes cutting this soli ...
of a circular
paraboloid In 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 ...
, and the area of a region bounded by a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
and one of its
secant line Secant is a term in mathematics derived from the 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 (th ...
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 o ...
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 Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1 ...
, 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 Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
.

Medieval

Middle East

In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( CE) derived a formula for the sum of
fourth power In arithmetic Arithmetic () is an elementary part of 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 ...
s. He used the results to carry out what would now be called an
integration Integration may refer to: Biology * Multisensory integration * Path integration * Pre-integration complex The pre-integration complex (PIC) is a nucleoprotein complex of viral genetic material and associated viral and host proteins which is capa ...
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 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 ...
.

India

In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the
Kerala School of Astronomy and Mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...
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 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 ...
'' or ''
infinite series In mathematics, a series is, roughly speaking, a description of the operation of addition, 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 generalizat ...
approximations''. However, they were not able to "combine many differing ideas under the two unifying themes of the
derivative In 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 ...
and the
integral 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 ...
, 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, German mathematician, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scienti ...
'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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathemati ...
, 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 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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. ...
, 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''
, which represented equality up to an infinitesimal error term. The combination was achieved by
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 of the United Kingdom, ...
,
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 o ...
, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670. The
product rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the stu ...
and
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, 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) ...
, 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 of a function, argument (input value). Derivatives are a fundamen ...
s and
Taylor series In 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 ...
, and of
analytic function In mathematics, an analytic function is a function (mathematics), function that is locally given by a convergent series, convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are ...
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 mathematics, 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 t ...
. 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 Rolling, rolls along a Line (geometry), straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette (curve), roulette, a curve g ...
, 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 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 ...
. 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 mathema ...

, 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 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the stu ...
and
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, 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) ...
, 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 Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
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 Isaac Newton, Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671, and published in 1 ...
''), 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, ...
. 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 Italians, Italian mathematician, philosopher, Theology, theologian, and humanitarianism, humanitarian. She was the first woman to write a mathematics handbook and the list of w ...
.

Foundations

In calculus, ''foundations'' refers to the rigorous development of the subject from
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s and definitions. In early calculus the use of
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned wi ...
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 "immateri ...
. 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, 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 mathematical analysis, analysis". Despite leaving university without a degree, ...
, 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathema ...
'', 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 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 mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mostly ...
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 (geometry), 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, ...
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 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 mod ...
, which contains full definitions and proofs 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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are con ...
invented
measure theory In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
, based on earlier developments by
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathe ...
, and used it to define integrals of all but the most
pathological Pathology is the study of the causal, 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 us ...
functions. Laurent Schwartz 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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are ...
's
non-standard analysis The history of calculus Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limit (mathematics), limits, continuity (mathematics), continuity, derivatives, integrals, and infinite series. Many elements of ...
. Robinson's approach, developed in the 1960s, uses technical machinery from
mathematical logic Mathematical logic is the study of formal logic within 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 quantiti ...
to augment the real number system with
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
and infinite 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 Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
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, categ ...
, smooth infinitesimal analysis views all functions as being continuous and incapable of being expressed in terms of
discrete Discrete may refer to: *Discrete particle or quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an fundamental interaction, interaction. The fundamental notion that a physi ...
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, Exclusive or, either this proposition or its negation is Truth value, true. It is one of the so-called Law of thought#The three tradit ...
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 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 mod ...
.

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 with ...
,
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 north ...
, 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 cover ...
wrote of this work, Applications of differential calculus include computations involving
velocity Velocity is the directional derivative, directional speed of an physical object, object in motion as an indication of its time derivative, rate of change in position (vector), position as observed from a particular frame of reference and as m ...
and
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ...
, the
slope In mathematics, the slope or gradient of a line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional object ...
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 weight function, weighted relative position (vector), position of the distributed mass sums to zero. Thi ...
,
work Work may refer to: * Work (human activity) Work or labor (or labour in British English) is intentional activity people perform to support the needs and wants of themselves, others, or a wider community. In the context of economics, work ...
, 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 and e ...
power series In mathematics, a power series (in one variable (mathematics), 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 te ...
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 (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
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 (geometry), position with respect to time. Motion is mathematically described in terms of Displacement (geometry), displacement, distance, velocity, acceleration, speed ...
and area. The
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek Dark ...
philosopher
Zeno of Elea Zeno of Elea (; grc, wikt:Ζήνων, 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 i ...
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 and the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of addition, 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 generalizat ...

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 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
. 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), ''Limit'' (manga), a manga by Keiko Suenobu * Limit (film), ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 201 ...
. 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
(as a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
with the
least-upper-bound property In 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 mod ...
). 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 Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limit (mathematics), limits, continuity (mathematics), continuity, derivatives, integrals, and infinite series. Many elements of ...
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 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 ...
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 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 ...
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 In 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 ...
, 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 o ...
-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 wa ...
. 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 derivative, directional speed of an physical object, object in motion as an indication of its time derivative, rate of change in position (vector), position as observed from a particular frame of reference and as m ...
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 Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, ...
(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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebr ...
''. 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 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 (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...
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 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 ...
.

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 (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...
'', 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 (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distance ...
. The technical definition of the definite integral involves the limit 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 (mathematics), set of all antiderivatives of f(x) ...
.

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 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 mathema ...
, 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 o ...
, 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 (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...
s. It is also a prototype solution of a
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
. 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 language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
,
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 specializ ...

,
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
,
business Business is the practice of making one's living or making money by producing or Trade, buying and selling Product (business), products (such as goods and Service (economics), services). It is also "any activity or enterprise entered into for pr ...
,
medicine Medicine is the science and Praxis (process), practice of caring for a patient, managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment, Palliative care, palliation of their injury or disease, and Health promotion ...
,
demography Demography () is the statistics, statistical study of populations, especially human beings. Demographic analysis examines and measures the dimensions and Population dynamics, dynamics of populations; it can cover whole societies or groups ...
, and in other fields wherever a problem can be mathematically modeled and an optimal 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 matrix (mat ...
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 o ...
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) ...
. 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 engineerin ...
, 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 (mathematics), function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself ...
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 plane curve#Smooth plane curve, smooth plane curve at which the signed curvature, curv ...
. 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 re ...
, fixed point iteration, and linear approximation. 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 analysis, numerical procedure for solving ordinary differential equations (ODEs) with a given Initial value problem, initia ...
to approximate curved courses within zero gravity environments.
Physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
makes particular use of calculus; all concepts in
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, planets, stars, and galax ...
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 of a ...
are related through calculus. The
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
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 language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
, 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 accelera ...
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 Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...
with respect to time equals the net
force In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...
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 (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ...
, 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 of a ...
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 differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
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 In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral ...
, 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 A measuring instrument is a device to measurement, measure a physical quantity. In the physical sciences, quality assurance, and engineering, measurement is the activity of ob ...
, 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 aw ...
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 Cell growth#Disorders, abnormal cell growth with the potential to Invasion (cancer), invade or Metastasis, spread to other parts of the body. These contrast with benign tumors, which do not spread. Poss ...
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 r ...
and
marginal revenue Marginal revenue (or marginal benefit) is a central concept in microeconomics Microeconomics is a branch of mainstream economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the alloca ...
.

*
Glossary of calculus A glossary (from grc, γλῶσσα, ''glossa''; language, speech, wording) also known as a vocabulary or clavis, is an alphabetical list of terms in a particular domain of knowledge with the definition A definition is a statement of t ...
* List of calculus topics * List of derivatives and integrals in alternative calculi * List of differentiation identities * Publications in calculus * Table of integrals

References

* * * * * * Uses synthetic differential geometry 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 university, private Land-grant university, land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern t ...

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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, da ...
. *
Calculus training materials at imomath.com
*
The Excursion of Calculus
1772 {{Authority control