HistoryModern calculus was developed in 17th-century Europe by and (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.
AncientThe ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian mathematics, Egyptian Moscow Mathematical Papyrus, Moscow papyrus (13th dynasty, BC); but the formulas are simple instructions, with no indication as to method, and some of them lack major components. From the age of Greek mathematics, Eudoxus of Cnidus, Eudoxus ( BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes ( BC) Archimedes' use of infinitesimals, developed this idea further, inventing heuristics which resemble the methods of integral calculus. The method of exhaustion was later discovered independently in Chinese mathematics, China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method that would later be called Cavalieri's principle to find the volume of a sphere.
MedievalIn 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 Integral, 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.Katz, V.J. 1995. "Ideas of Calculus in Islam and India." ''Mathematics Magazine'' (Mathematical Association of America), 68(3):163–174. 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 thereby stated components of calculus. A complete theory encompassing these components is now well known in the Western world as the ''Taylor series'' or ''infinite series approximations''. However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".
ModernIn Europe, the foundational work was a treatise written by Bonaventura Cavalieri, 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 of Mechanical Theorems, 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, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory (astronomer and mathematician), James Gregory, the latter two proving the Fundamental theorem of calculus, second fundamental theorem of calculus around 1670. The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by in an idiosyncratic notation which he applied to solve problems of mathematical physics. 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, and many other problems discussed in his ''Philosophiæ Naturalis Principia Mathematica, Principia Mathematica'' (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. 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 , who was originally accused of plagiarism by Newton.Leibniz, Gottfried Wilhelm. ''The Early Mathematical Manuscripts of Leibniz''. Cosimo, Inc., 2008. p. 228
FoundationsIn calculus, ''foundations'' refers to the Rigorous#Mathematical rigour, rigorous development of the subject from axioms and definitions. In early calculus the use of quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and George Berkeley, Bishop Berkeley. 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 Colin Maclaurin, Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Augustin Louis Cauchy, Cauchy and Karl Weierstrass, Weierstrass, 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'', we find a broad range of foundational approaches, including a definition of continuous function, continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. In his work Weierstrass formalized the concept of Limit of a function, 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 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 Euclidean space and the complex plane. In modern mathematics, the foundations of calculus are included in the field of real analysis, which contains full definitions and mathematical proof, proofs of the theorems of calculus. The reach of calculus has also been greatly extended. Henri Lebesgue invented measure theory and used it to define integrals of all but the most Pathological (mathematics), pathological functions. Laurent Schwartz introduced Distribution (mathematics), 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's non-standard analysis. Robinson's approach, developed in the 1960s, uses technical machinery from mathematical logic to augment the real number system with and Infinity, infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal numbers, 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.
SignificanceWhile many of the ideas of calculus had been developed earlier in Greek mathematics, Greece, Chinese mathematics, China, Indian mathematics, India, Islamic mathematics, Iraq, Persia, and Japanese mathematics, Japan, the use of calculus began in Europe, during the 17th century, when and built on the work of earlier mathematicians to introduce its basic principles. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves. Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and Mathematical optimization, optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work (physics), work, and pressure. More advanced applications include power series and Fourier series. 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 or sums of infinitely many numbers. These questions arise in the study of Motion (physics), motion and area. The ancient Greek philosopher Zeno of Elea gave several famous examples of such Zeno's paradoxes, paradoxes. Calculus provides tools, especially the Limit (mathematics), limit and the infinite series, that resolve the paradoxes.
Limits and infinitesimalsCalculus is usually developed by working with very small quantities. Historically, the first method of doing so was by 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. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols and were taken to be infinitesimal, and the derivative was simply 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. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals. In the late 19th century, infinitesimals were replaced within academia by the epsilon, delta approach to Limit of a function, limits. Limits describe the value of a function (mathematics), function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior in the context of the real number, real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits were thought to provide a more rigorous foundation for calculus, and for this reason they became the standard approach during the twentieth century.
Differential calculusDifferential calculus is the study of the definition, properties, and applications of the derivative 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 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 deriving 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. The most common symbol for a derivative is an apostrophe-like mark called prime (symbol), prime. Thus, the derivative of a function called is denoted by , pronounced "f prime". For instance, if is the squaring function, then is its derivative (the doubling function from above). This notation is known as Lagrange's notation. 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 of the ball. If a function is linear function, linear (that is, if the Graph of a function, graph 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: : 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 : This expression is called a ''difference quotient''. 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: : Geometrically, the derivative is the slope of the tangent line 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. : 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 notationA common notation, introduced by Leibniz, for the derivative in the example above is : 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: : In this usage, the in the denominator is read as "with respect to ". Another example of correct notation could be: 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.
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''. In technical language, integral calculus studies two related linear operators. The ''indefinite integral'', also known as the ''antiderivative'', 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. The technical definition of the definite integral involves the limit of a sum of areas of rectangles, called a Riemann sum. A motivating example is the distances traveled in a given time. : If the speed is constant, only multiplication is needed, 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) 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. In the diagram on the left, when constant velocity and time are graphed, these two values form 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 in the diagram on the right represents speed as it varies over time, the distance traveled (between the times represented by and ) is the area of the shaded region . 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 , an long s, elongated ''S'' (the ''S'' stands for "sum"). The definite integral is written as: : 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 . In a formulation of the calculus based on limits, the notation : is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, , is not a number, and is not being multiplied by , although, serving as a reminder of the limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: : 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: : The unspecified constant present in the indefinite integral or antiderivative is known as the constant of integration.
Fundamental theoremThe fundamental theorem of calculus 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 function, continuous on the interval and if is a function whose derivative is on the interval , then : Furthermore, for every in the interval , : This realization, made by both Isaac Newton, Newton and Gottfried Leibniz, Leibniz, who based their results on earlier work by Isaac Barrow, was key to the proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
ApplicationsCalculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, , , business, medicine, demography, and in other fields wherever a problem can be mathematical model, mathematically modeled and an optimization (mathematics), 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. Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. An example of the use of calculus in mechanics is Newton's laws of motion, Newton's second law of motion: historically stated it expressly uses the term "change of motion" which implies the derivative saying ''The'' change ''of momentum of a body is equal to the resultant force acting on the body and is in the same direction.'' Commonly expressed today as Force = Mass × acceleration, it implies differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path. Maxwell's theory of electromagnetism and Albert Einstein, Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes. Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. Or it can be used in probability theory to determine the probability of a continuous random variable from an assumed density function. In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, Concave function, concavity and inflection points. Green's Theorem, 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, 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. Discrete Green's Theorem, which gives the relationship between a double integral of a function around a simple closed rectangular curve ''C'' and a linear combination of the antiderivative's values at corner points along the edge of the curve, allows fast calculation of sums of values in rectangular domains. For example, it can be used to efficiently calculate sums of rectangular domains in images, in order to rapidly extract features and detect object; another algorithm that could be used is the summed area table. In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. From the decay laws for a particular drug's elimination from the body, it is used to derive dosing laws. In nuclear medicine, it is used to build models of radiation transport in targeted tumor therapies. In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue. 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, fixed point iteration, and linear approximation. For instance, spacecraft use a variation of the Euler method to approximate curved courses within zero gravity environments.
VarietiesOver the years, many reformulations of calculus have been investigated for different purposes.
Non-standard calculusImprecise calculations with infinitesimals were widely replaced with the rigorous (ε, δ)-definition of limit starting in the 1870s. Meanwhile, calculations with infinitesimals persisted and often led to correct results. This led Abraham Robinson to investigate if it were possible to develop a number system with infinitesimal quantities over which the theorems of calculus were still valid. In 1960, building upon the work of Edwin Hewitt and Jerzy Łoś, he succeeded in developing non-standard analysis. The theory of non-standard analysis is rich enough to be applied in many branches of mathematics. As such, books and articles dedicated solely to the traditional theorems of calculus often go by the title non-standard calculus.
Smooth infinitesimal analysisThis is another reformulation of the calculus in terms of s. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous function, continuous and incapable of being expressed in terms of Discrete mathematics, discrete entities. One aspect of this formulation is that the law of excluded middle does not hold in this formulation.
Constructive analysisConstructive mathematics is 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. As such constructive mathematics also rejects the law of excluded middle. Reformulations of calculus in a constructive framework are generally part of the subject of constructive analysis.
Lists* Glossary of calculus * List of calculus topics * List of derivatives and integrals in alternative calculi * List of differentiation identities * List of publications in mathematics#Calculus, Publications in calculus * Table of integrals
Other related topics* Calculus of finite differences * Calculus with polynomials * Complex analysis * Differential equation * Differential geometry and topology, Differential geometry * ''Elementary Calculus: An Infinitesimal Approach'' * Discrete calculus * Fourier series * Integral equation * Mathematical analysis * Multivariable calculus * Non-classical analysis * Non-standard analysis * Non-standard calculus * Precalculus (Mathematics education, mathematical education) * Product integral * Stochastic calculus * Taylor series
Books* Carl Benjamin Boyer, Boyer, Carl Benjamin (1949)
Online books* * Crowell, B. (2003). "''Calculus''". Light and Matter, Fullerton. Retrieved 6 May 2007 fro
External links* * *