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Calculus

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Calculus is an important branch of mathematics. The word stems from the
   ancient Greeks' use of pebbles arranged in patterns to study arithmetic
   and geometry. The Latin word for " pebble" is "calculus." Two
   complementary disciplines comprise calculus, both of which rely on the
   concept of a limit. The first is differential calculus, which is
   concerned with the instantaneous, as opposed to average, rate of change
   of a quantity. This can be illustrated by the slope of a function's
   graph at a particular point. The second is integral calculus, which
   studies the accumulation of infinitely small quantities, summing to
   areas under a curve, linear distance travelled, or volume displaced.
   These two processes act inversely to each other, as shown by the
   fundamental theorem of calculus.

   Differential calculus typically provides a way to derive the
   acceleration and velocity of a body at a particular moment while
   integral calculus problems are used to compute areas and volumes, to
   find the amount of a liquid pumped by a pump with a set power input but
   varying conditions of pumping losses and pressure, or to find the
   amount of parking lot plowed by a snowplow of given power with varying
   rates of snowfall.

   In Europe, fundamental advances in calculus during the 17th and 18th
   century had a deep impact on the ensuing development of physics. Today,
   calculus is used in every branch of the physical sciences, in computer
   science, in statistics, and in engineering; in economics, business, and
   medicine; and as a general method whenever the goal is an optimal
   solution to a problem that can be given in mathematical form.

History

   Historians generally regard integral calculus as going back no further
   than to the time of the ancient Greeks, circa 200 BC. Modern sources
   generally credit Hellenic mathematician Eudoxus with the method of
   exhaustion, which made it possible to compute the areas of regions and
   the volumes of solids. Archimedes developed this idea further,
   inventing heuristics which resemble integral calculus. After him, the
   development of calculus did not advance appreciably for over 500 years.

   In India, the mathematician-astronomer Aryabhata in 499 used
   infinitesimals and expressed an astronomical problem in the form of a
   basic differential equation. Manjula, in the 10th century, elaborated
   on this differential equation in a commentary. This equation eventually
   led Bhaskara in the 12th century to develop independently a number of
   fundamental ideas in calculus, perhaps including an early form of the
   theorem now known as " Rolle's theorem". He was also the first to
   define the notion of the derivative as a limit. In the 14th century,
   Madhava, along with other mathematician-astronomers of the Kerala
   School, studied infinite series, power series, Taylor series,
   differentiation, integration, and the mean value theorem. Yuktibhasa,
   which some consider to be the first text on calculus, summarizes these
   results. These developments would not be duplicated in Europe until
   much later.
   Sir Isaac Newton
   Enlarge
   Sir Isaac Newton
   Gottfried Wilhelm Leibniz
   Enlarge
   Gottfried Wilhelm Leibniz

   Calculus started making great strides in Europe towards the end of the
   early modern period and into the first years of the eighteenth century.
   This was a time of major innovation in Europe. Calculus provided a new
   method in mathematical physics to solve long-standing problems. Several
   mathematicians contributed to these breakthroughs, notably John Wallis
   and Isaac Barrow. James Gregory proved a special case of the second
   fundamental theorem of calculus in 1668. In Japan at around this time,
   Seki Kowa expanded further upon Eudoxus's method of exhaustion.

   Leibniz and Newton pulled these ideas together into a coherent whole
   and they are usually credited with the probably independent and nearly
   simultaneous "invention" of calculus. Newton was the first to apply
   calculus to general physics and Leibniz developed much of the notation
   used in calculus today; he often spent days determining appropriate
   symbols for concepts. The fundamental insight that both Newton and
   Leibniz had was the fundamental theorem of calculus. Virtually all
   modern methods of symbolic integration follow from this theorem, and it
   proved indispensable in the development of modern mathematics and
   physics. For example, see Integration by parts and Integration by
   substitution.

   When Newton and Leibniz first published their results, whether
   Leibniz's work was independent of Newton's was somewhat controversial.
   While Newton had derived his results years before Leibniz, Newton
   published only some time after Leibniz published in 1684. Later, Newton
   would claim that Leibniz got the idea from Newton's notes on the
   subject; however examination of the papers of Leibniz and Newton show
   they arrived at their results independently, with Leibniz starting
   first with integration and Newton with differentiation. This
   controversy between Leibniz and Newton divided English-speaking
   mathematicians from those in Europe for many years, which slowed the
   development of mathematical analysis. Today, both Newton and Leibniz
   are given credit for developing calculus independently. It is Leibniz,
   however, who is credited with giving the new discipline the name it is
   known by today: "calculus". Newton's name for it was "the science of
   fluxions". Some others who contributed important ideas are Descartes,
   Barrow, Fermat, Huygens, and Wallis.

   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 Cauchy,
   Riemann, Weierstrass, and others. It was also during this time period
   that the ideas of calculus were generalized to Euclidean space and the
   complex plane. Calculus continues to be further generalized, such as
   with the development of the Lebesgue integral in 1900.

Derivatives and Differentiation

   The derivative measures the sensitivity of one variable to small
   changes in another variable. Consider the formula:

          \mathrm{Speed} = \frac{\mathrm{Distance}}{\mathrm{Time}}

   for an object moving at constant speed. The speed of a car, as measured
   by the speedometer, is the derivative of the car's distance traveled as
   a function of time. Calculus is a mathematical tool for dealing with
   this complex but natural and familiar situation.
   The derivative of a curve defined by the function f(x) can be thought
   of as the slope, or angle, of the secant between two points on the
   curve at x and x+h, then letting the separation h between them shrink
   to zero.
   Enlarge
   The derivative of a curve defined by the function f(x) can be thought
   of as the slope, or angle, of the secant between two points on the
   curve at x and x+h, then letting the separation h between them shrink
   to zero.

   Differential calculus can be used to determine the instantaneous speed
   at any given instant, while the formula "speed = distance divided by
   time" only gives the average speed, and cannot be applied to an instant
   in time because it then gives an undefined quotient zero divided by
   zero. Calculus avoids division by zero by using the concept of the
   limit which, roughly speaking, is a method of controlling an otherwise
   uncontrollable output, such as division by zero or multiplication by
   infinity. More formally, differential calculus defines the
   instantaneous rate of change (the derivative) of a mathematical
   function's value, with respect to changes of the variable. The
   derivative is defined as a limit of a difference quotient.

   The derivative of a function, if it exists, gives information about its
   graph. It is useful for finding optimum solutions to problems, called
   maxima and minima of a function. It is proved mathematically that these
   optimum solutions exist either where the tangent of the graph is flat,
   so that the slope is zero; or where the graph has a sharp turn ( cusp)
   where the derivative does not exist; or at the endpoints of the graph.
   Another application of differential calculus is Newton's method, a
   powerful equation solving algorithm. Differential calculus has been
   applied to many questions that were first formulated in other areas,
   such as business or medicine.

   The derivative lies at the heart of the physical sciences. Newton's
   second law of motion expressly uses the term "rate of change" which is
   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 is the derivative of velocity. (See Differential
   equation.) Maxwell's theory of electromagnetism and Einstein's theory
   of general relativity are also expressed in the language of
   differential calculus, as is the basic theory of electrical circuits
   and much of engineering. It is also applied to problems in biology,
   economics, and many other areas.

   The derivative of a function y = f(x) with respect to x is usually
   expressed as either y ′ (read "y-prime"), f ' (x) (read "f-prime of x")
   or using Leibniz notation to write:

          \frac{d}{dx}(y)

   which is commonly shortened to:

          \frac{dy}{dx}

Integrals and Integration

   There are two types of integral in calculus, the indefinite and the
   definite. The indefinite integral is simply the antiderivative. That
   is, F is an antiderivative of f when f is a derivative of F. (This use
   of capital letters and lower case letters is common in calculus. The
   lower case letter represents the derivative of the capital letter.)

   The definite integral evaluates the cumulative effect of many small
   changes in a quantity. The simplest instance is the formula

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

   for calculating the distance a car moves during a period of time when
   it is traveling at constant speed. The distance moved is the cumulative
   effect of the small distances moved in each instant. Calculus is also
   able to deal with the natural situation in which the car moves with
   changing speed.
   Integration can be thought of as measuring the area under a curve,
   defined by f(x), between two points (here a and b) by subdividing the
   area into ever-smaller slices and then adding them all up.
   Enlarge
   Integration can be thought of as measuring the area under a curve,
   defined by f(x), between two points (here a and b) by subdividing the
   area into ever-smaller slices and then adding them all up.

   Integral calculus determines the exact distance traveled during an
   interval of time by creating a series of better and better
   approximations, called Riemann sums, that approach the exact distance
   as a limit. More formally, we say that the definite integral of a
   function on an interval is a limit of Riemann sum approximations.

   Applications of integral calculus arise whenever the problem is to
   compute a number that is in principle (approximately) equal to the sum
   of a large number of small quantities. The classic geometric
   application is to area computations. In principle, the area of a region
   can be approximated by chopping it up into many pieces (typically
   rectangles, or, in polar coordinates, circular sectors), and then
   adding the areas of those pieces. The length of an arc, the area of a
   surface, and the volume of a solid can also be expressed as definite
   integrals. Probability, the basis for statistics, provides another
   important application of integral calculus.

   The symbol of integration is ∫, a stretched s (which stands for "sum").
   The precise meanings of expressions involving integrals can be found in
   the main article Integral. The definite integral, written as:

          \int_a^b f(x)\, dx

   is read "the integral from a to b of f-of-x dx".

Foundations

   There is more than one rigorous approach to the foundation of calculus.
   The usual one 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. The
   tools of calculus include techniques associated with elementary
   algebra, and mathematical induction. The foundations of calculus are
   included in the field of real analysis, which contains all full
   definitions and proofs of the theorems of calculus as well as
   generalisations such as measure theory and distribution theory.

Fundamental theorem

   The fundamental theorem of calculus states that differentiation and
   integration are, in a certain sense, inverse operations. More
   precisely, if one defines one function as the integral of another
   continuous function, then differentiating the newly defined function
   returns the function you started with. Furthermore, if you want to find
   the value of a definite integral, you usually do so by evaluating an
   antiderivative.

   Here is the mathematical formulation of the Fundamental Theorem of
   Calculus: 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).

          Also, 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, 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 development and use of calculus has had wide reaching effects on
   all areas of modern living. It underlies nearly all of the sciences,
   especially physics. Virtually all modern developments such as building
   techniques, aviation, and other technologies make fundamental use of
   calculus. Many algebraic formulas now used for ballistics, heating and
   cooling, and other practical sciences were worked out through the use
   of calculus. In a handbook, an algebraic formula based on calculus
   methods may be applied without knowing its origins. The success of
   calculus has been extended over time to differential equations, vector
   calculus, calculus of variations, complex analysis, and differential
   topology. Calculus has been used on a broad field of subjects, even
   linguistics.

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