calculus: Definition from Answers.com

calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit-the notion of tending toward, or approaching, an ultimate value. The English physicist Isaac Newton and the German mathematician G. W. Leibniz, working independently, developed the calculus during the 17th cent. The calculus and its basic tools of differentiation and integration serve as the foundation for the larger branch of mathematics known as analysis. The methods of calculus are essential to modern physics and to most other branches of modern science and engineering.

The Differential Calculus

The differential calculus arises from the study of the limit of a quotient, Δy/Δx, as the denominator Δx approaches zero, where x and y are variables. y may be expressed as some function of x, or f(x), and Δy and Δx represent corresponding increments, or changes, in y and x. The limit of Δy/Δx is called the derivative of y with respect to x and is indicated by dy/dx or D_xy:

The symbols dy and dx are called differentials (they are single symbols, not products), and the process of finding the derivative of y=f(x) is called differentiation. The derivative dy/dx=df(x)/dx is also denoted by y′, or f′(x). The derivative f′(x) is itself a function of x and may be differentiated, the result being termed the second derivative of y with respect to x and denoted by y″, f″(x), or d²y/dx². This process can be continued to yield a third derivative, a fourth derivative, and so on. In practice formulas have been developed for finding the derivatives of all commonly encountered functions. For example, if y=xⁿ, then y′=nx^{n − 1}, and if y=sin x, then y′=cos x (see trigonometry). In general, the derivative of y with respect to x expresses the rate of change in y for a change in x. In physical applications the independent variable (here x) is frequently time; e.g., if s=f(t) expresses the relationship between distance traveled, s, and time elapsed, t, then s′=f′(t) represents the rate of change of distance with time, i.e., the speed, or velocity.

Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative f′(t)=ds/dt, however, gives the velocity for any particular value of t, i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If y=f(x) is a real-valued function of a real variable, the ratio Δy/Δx=(y₂ − y₁)/(x₂ − x₁) represents the slope of a straight line through the two points P (x₁,y₁) and Q (x₂,y₂) on the graph of the function. If P is taken closer to Q, then x₁ will approach x₂ and Δx will approach zero. In the limit where Δx approaches zero, the ratio becomes the derivative dy/dx=f′(x) and represents the slope of a line that touches the curve at the single point Q, i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.

The Integral Calculus

The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes. For example, consider the problem of determining the area under a given curve y=f(x) between two values of x, say a and b. Let the interval between a and b be divided into n subintervals, from a=x₀ through x₁, x₂, x₃, … x_{i − 1}, x_i, … , up to x_n=b. The width of a given subinterval is equal to the difference between the adjacent values of x, or Δx_i=x_i − x_{i − 1}, where i designates the typical, or ith, subinterval. On each Δx_i a rectangle can be formed of width Δx_i, height y_i=f(x_i) (the value of the function corresponding to the value of x on the right-hand side of the subinterval), and area ΔA_i=f(x_i)Δx_i. In some cases, the rectangle may extend above the curve, while in other cases it may fail to include some of the area under the curve; however, if the areas of all these rectangles are added together, the sum will be an approximation of the area under the curve.

This approximation can be improved by increasing n, the number of subintervals, thus decreasing the widths of the Δx's and the amounts by which the ΔA's exceed or fall short of the actual area under the curve. In the limit where n approaches infinity (and the largest Δx approaches zero), the sum is equal to the area under the curve:

The last expression on the right is called the integral of f(x), and f(x) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.

An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if F(x) is a function whose derivative is f(x), then the area under the graph of y=f(x) between a and b is equal to F(b) − F(a). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols:

The function F(x), which is equal to the integral of f(x), is sometimes called an antiderivative of f(x), while the process of finding F(x) from f(x) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, a and b, are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function F(x) resulting from integration is determined only to within the addition of an arbitrary constant C, since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of f(x) is

The value of the constant C must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., x or t, but also functions of several variables. For example, if z=f(x,y) is a function of two independent variables, x and y, then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂z/∂x and ∂z/∂y or by D_xz and D_yz. Three different second derivatives are possible, ∂²z/∂x², ∂²z/∂y², and ∂²z/∂x∂y=∂²z/∂y∂x. Such derivatives are called partial derivatives. In any partial differentiation all independent variables other than the one being considered are treated as constants.

Bibliography

See R. Courant and F. John, Introduction to Calculus and Analysis, Vol. I (1965); M. Kline, Calculus: An Intuitive and Physical Approach (2 vol., 1967); G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry (7th ed. 2 vol., 1988).

This article is about the branch of mathematics. For other uses, see Calculus (disambiguation).

Topics in Calculus

Fundamental theorem
Limits of functions
Continuity
Mean value theorem

Differential calculus
Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Identities Rules: Power rule, Product rule, Quotient rule, Chain rule

Integral calculus
Integral Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order

Vector calculus
Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem

Multivariable calculus
Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian

Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.

Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus and join calculus.

History

Sir Isaac Newton is one of the most famous contributors to the development of calculus, with, among other things, the use of calculus in his laws of motion and gravitation.

Main article: History of calculus

Ancient

The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1820 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.^[1]^[2] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.^[3] The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi used what would later be called Cavalieri's principle to find the volume of a sphere.^[2]

Medieval

Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhacen) was the first to derive the formula for the sum of the fourth powers of an arithmetic progression, using a method that is readily generalizable to finding the formula for the sum of any higher integral powers, which he used to perform an integration.^[4] In the 11th century, the Chinese polymath Shen Kuo developed 'packing' equations that dealt with integration. In the 12th century, the Indian mathematician, Bhāskara II, developed an early derivative representing infinitesimal change, and he described an early form of Rolle's theorem.^[5] Also in the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī discovered the derivative of cubic polynomials, an important result in differential calculus.^[6] In the 14th century, Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,^[7] which are treated in the text Yuktibhasa.^[8]^[9]^[10]

Modern

This section does not cite any references or sources.
Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2009)

In the modern period, independent discoveries relating to calculus were being made in early 17th century Japan, by mathematicians such as Seki Kowa, who expanded upon the method of exhaustion.

In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimal thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieri's work was not well respected since his methods can lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

The formal study of calculus combined Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1675.

The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. In his publications, 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 Principia Mathematica. 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.

Gottfried Wilhelm Leibniz was originally accused of plagiarizing Sir Isaac Newton's unpublished work (only in Britain, not in continental Europe), but is now regarded as an independent inventor of and contributor to calculus.

These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism 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 manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism – he often spent days determining appropriate symbols for concepts.

Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics 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, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.

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, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental 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. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions".

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass (see (ε, δ)-definition of limit). It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue generalized the notion of the integral so that virtually any function has an integral, while Laurent Schwartz extended differentiation in much the same way.

Calculus is a ubiquitous topic in most modern high schools and universities around the world.^[11]

Significance

While some of the ideas of calculus were developed earlier in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz 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 optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.

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 and area. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.

Foundations

In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. 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.

There is more than one rigorous approach to the foundation of calculus. The usual one today is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalizations such as measure theory and distribution theory.

Principles

Limits and infinitesimals

Main articles: Limit of a function and Infinitesimal

Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, ½, ⅓, ... and less than any positive real number. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This 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 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary 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 are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.

Differential calculus

Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.

Main article: Differential calculus

Differential 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 mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. 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 it produces turns out to be the doubling function.)

The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f′, pronounced "f prime." For instance, if f(x) = x² is the squaring function, then f′(x) = 2x is the doubling function.

If the input of the function represents time, then the derivative represents change with respect to time. For example, if f 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 f is how the position is changing in time, that is, it is the velocity of the ball.

If a function is linear (that is, if the graph of the function is a straight line), then the function can be written y = mx + b, where:

$m= \frac{\mbox{rise}}{\mbox{run}}= {\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}}.$

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 y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f. (a, f(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore (a + h, f(a + h)) is close to (a, f(a)). The slope between these two points is

$m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.$

This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The secant line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

$\lim_{h \to 0}{f(a+h) - f(a)\over{h}}.$

Geometrically, the derivative is the slope of the tangent line to the graph of f at a. 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 f.

Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x² be the squaring function.

The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is f(x) = x³ − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.

$\begin{align}f'(3) &=\lim_{h \to 0}{(3+h)^2 - 9\over{h}} \\ &=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\ &=\lim_{h \to 0}{6h + h^2\over{h}} \\ &=\lim_{h \to 0} (6 + h) \\ &= 6. \end{align}$

The slope of 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 similar computation to the one above shows that the derivative of the squaring function is the doubling function.

Leibniz notation

Main article: Leibniz's notation

A common notation, introduced by Leibniz, for the derivative in the example above is

$\begin{align} y=x^2 \\ \frac{dy}{dx}=2x. \end{align}$

In an approach based on limits, the symbol dy/dx 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, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:

$\frac{d}{dx}(x^2)=2x.$

In this usage, the dx in the denominator is read as "with respect to x". Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy 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

Main article: Integral

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 is the antiderivative, the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of upper- and lower-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 area between the graph of the input and the x-axis. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum.

A motivating example is the distances traveled in a given time.

$\mathrm{Distance} = \mathrm{Speed} \cdot \mathrm{Time}$

If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. 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.

Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).

If f(x) in the diagram on the left represents speed as it varies over time, the distance traveled (between the times represented by a and b) is the area of the shaded region s.

To approximate that area, an intuitive method would be to divide up the distance between a and b into a number of equal segments, the length of each segment represented by the symbol Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base Δx and height h gives the distance (time Δx multiplied by speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x)=h. 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 Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx approaches zero.

The symbol of integration is $\int \,$ , an elongated S (the S stands for "sum"). The definite integral is written as:

$\int_a^b f(x)\, dx.$

and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. In a formulation of the calculus based on limits, the notation $\int_a^b \ldots\, dx$ is to be understood as an operator that takes a function as an input and gives a number, the area, as an output; dx is not a number, and is not being multiplied by f(x).

The indefinite integral, or antiderivative, is written:

$\int f(x)\, dx.$

Functions differing by only a constant have the same derivative, and therefore the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x² + C, where C is any constant, is y′ = 2x, the antiderivative of the latter is given by:

$\int 2x\, dx = x^2 + C.$

An undetermined constant like C in the antiderivative is known as a constant of integration.

Fundamental theorem

Main article: Fundamental theorem of calculus

The 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 f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval (a, b), then

$\int_{a}^{b} f(x)\,dx = F(b) - F(a).$

Furthermore, for every x in the interval (a, b),

$\frac{d}{dx}\int_a^x f(t)\, dt = f(x).$

This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the massive 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.

Applications

The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus

Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired.

Physics makes particular use of calculus; all concepts in classical mechanics are interrelated 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. In the subfields of electricity and magnetism calculus can be used to find the total flux of electromagnetic fields. A more historical example of the use of calculus in physics is Newton's second law of motion, it expressly uses the term "rate of change" which refers to the derivative: The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Even the common expression of Newton's second law as Force = Mass × Acceleration involves differential calculus because acceleration can be expressed as the derivative of velocity. Maxwell's theory of electromagnetism and 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.

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.

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.

In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow.

In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, concavity and inflection points.

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.

Calculus can be used to find approximate solutions to equations, in 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.

References

Notes

^ There is no exact evidence on how it was done; some, including Morris Kline (Mathematical thought from ancient to modern times Vol. I) suggest trial and error.
^ ^a ^b Helmer Aslaksen. Why Calculus? National University of Singapore.
^ Archimedes, Method, in The Works of Archimedes ISBN 978-0-521-66160-7
^ Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), pp. 163-174.
^ Ian G. Pearce. Bhaskaracharya II.
^ J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2), pp. 304-309.
^ "Madhava". Biography of Madhava. School of Mathematics and Statistics University of St Andrews, Scotland. http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html. Retrieved 2006-09-13.
^ "An overview of Indian mathematics". Indian Maths. School of Mathematics and Statistics University of St Andrews, Scotland. http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html. Retrieved 2006-07-07.
^ "Science and technology in free India" (PDF). Government of Kerala — Kerala Call, September 2004. Prof.C.G.Ramachandran Nair. http://www.kerala.gov.in/keralcallsep04/p22-24.pdf. Retrieved 2006-07-09.
^ Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland.
^ UNESCO-World Data on Education [1]

Books

Larson, Ron, Bruce H. Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning. ISBN 9780547167022
McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 9781891389245
Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 9780495011668
Thomas, George B., Maurice D. Weir, Joel Hass, Frank R. Giordano (2008), "Calculus", 11th ed., Addison-Wesley. ISBN 0-321-48987-X

Other resources

Online books

Crowell, B. (2003). "Calculus" Light and Matter, Fullerton. Retrieved 6 May 2007 from http://www.lightandmatter.com/calc/calc.pdf
Garrett, P. (2006). "Notes on first year calculus" University of Minnesota. Retrieved 6 May 2007 from http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf
Faraz, H. (2006). "Understanding Calculus" Retrieved 6 May 2007 from Understanding Calculus, URL http://www.understandingcalculus.com/ (HTML only)
Keisler, H. J. (2000). "Elementary Calculus: An Approach Using Infinitesimals" Retrieved 6 May 2007 from http://www.math.wisc.edu/~keisler/keislercalc1.pdf
Mauch, S. (2004). "Sean's Applied Math Book" California Institute of Technology. Retrieved 6 May 2007 from http://www.cacr.caltech.edu/~sean/applied_math.pdf
Sloughter, Dan (2000). "Difference Equations to Differential Equations: An introduction to calculus". Retrieved 17 March 2009 from http://synechism.org/drupal/de2de/
Stroyan, K.D. (2004). "A brief introduction to infinitesimal calculus" University of Iowa. Retrieved 6 May]] 2007 from http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm (HTML only)
Strang, G. (1991). "Calculus" Massachusetts Institute of Technology. Retrieved 6 May 2007 from http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
Smith, William V. (2001). "The Calculus" Retrieved 4 July 2008 [2] (HTML only).

Web pages

Weisstein, Eric W., "Calculus" from MathWorld.
Topics on Calculus at PlanetMath.
Calculus Made Easy (1914) by Silvanus P. ThompsonFull text in PDF
Calculus.org: The Calculus page at University of California, Davis — contains resources and links to other sites
COW: Calculus on the Web at Temple University - contains resources ranging from pre-calculus and associated algebra
Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
Online Integrator (WebMathematica) from Wolfram Research
The Role of Calculus in College Mathematics from ERICDigests.org
OpenCourseWare Calculus from the Massachusetts Institute of Technology
Infinitesimal Calculus — an article on its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. .
Elements of Calculus I and Calculus II for Business, OpenCourseWare from the University of Notre Dame with activities, exams and interactive applets.
Calculus for Beginners and Artists by Daniel Kleitman, MIT

v • d • e Major fields of mathematics

Logic · Set theory · Category theory · Algebra (elementary – linear – abstract) · Number theory · Analysis/Calculus · Geometry · Topology · Dynamical systems · Combinatorics · Game theory · Information theory · Numerical analysis · Optimization · Computation · Probability · Statistics · Trigonometry

Category · Mathematics portal · Outline · Topics

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 1994-2009 Encyclopædia Britannica, Inc. All rights reserved. Read more
		Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more
		Dental Dictionary. Mosby's Dental Dictionary. Copyright © 2004 by Elsevier, Inc. All rights reserved. Read more
		Spotlight. © 1999-2009 by Answers Corporation. All rights reserved. Read more
		Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Word Tutor. Copyright © 2004-present by eSpindle Learning, a 501(c) nonprofit organization. All rights reserved. eSpindle provides personalized spelling and vocabulary tutoring online; free trial. Read more
		Sign Language Videos. Copyright © 2009 Signing Savvy, LLC. All rights reserved. Read more
		Science Dictionary. The New Dictionary of Cultural Literacy, Third Edition Edited by E.D. Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by Houghton Mifflin Company. Published by Houghton Mifflin. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Calculus". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

calculus

calculus

Calculus

calculus

calculus

calculus

calculus

calculus

calculus

calculus

Calculus

History

Ancient

Medieval

Modern

Significance

Foundations

Principles

Limits and infinitesimals

Differential calculus

Leibniz notation

Integral calculus

Fundamental theorem

Applications

See also

Lists

Related topics

References

Notes

Books

Other resources

Further reading

Online books

Web pages

calculus

calculus

	How do you do calculus? Read answer...
	How do get into calculus? Read answer...
	Who is the calculus pioneer? Read answer...

	Who pioneered calculus?
	Why is calculus interesting?
	When was calculus discover?