Calculus

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Calculus is one of the most important breakthroughs in modern mathematics, answering questions that had puzzled mathematicians, scientists, and philosophers for more than two thousand years. In particular, calculus gave a clear and precise definition of infinity, both in the case of the infinitely large and the infinitely small. It provided an answer to Zeno's paradoxes and gave the first clear definition of what Aristotle called "the quality of motion".

Calculus was not discovered all at once. In the ancient world, Eudoxus and Archimedes proposed mathematical ideas that can now be seen as similar to calculus. In Twelfth Century India, Bhaskara conceived of differential calculus and two centuries later, Madhava and the Kerala school studied infinite series, convergence, differentiation and other concepts of calculus. In the 17th century, Kowa Seki in Japan elaborated some of the fundamental principles of integral calculus. At roughly the same time, in Europe, Wallis and Barrow proposed ideas that would now be considered integrals, derivatives, and the fundamental theorem of calculus, which was first proved by James Gregory. But it is Isaac Newton and Gottfried Leibniz who are credited with bringing all these ideas together, and they are usually credited with the independent and nearly simultaneous creation of calculus. Even so, it was generations after Newton and Leibniz that Cauchy and other mathematicians finally put calculus on a rigorous basis, with the definition of the limit.

From a mathematical standpoint, calculus gives the definitions and properties of three linear operators, the limit, the derivative, and the integral. All of these depend on the definition of the limit. Roughly speaking, the limit allows us to control an otherwise uncontrollable output, the derivative is the slope of a graph, and the integral is the area under a curve. In scientific applications, the derivative is often used to find a changing velocity given a changing position, and the integral is often used to find a changing position given a changing velocity. The fundamental theorem of calculus roughly states that the derivative and the integral are inverse operators.

Today, calculus is used in every branch of science and engineering, in business, in medicine, and in virtually every human endeavor where the goal is an optimum solution to a problem that can be given in mathematical form.

History

Though the origins of integral calculus are generally regarded as going back no farther than to the time of the ancient Greeks, circa 200 B.C., there is some evidence that the ancient Egyptians may have had some hint of the idea at a much earlier date. The Hellenic mathematician Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the area of regions and the volume of solids. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts in calculus. Of all the mathematicians of the ancient world, he was the closest to discovering calculus, but he never made the breakthrough, and after him study of calculus did not advance appreciably for more than a thousand years.

An Indian mathematician, Bhaskara (1114-1185), developed a number of ideas that can now be seen to be forerunners of calculus and gave the basic idea of what is now known as Rolle's theorem. He was the first to conceive the idea of differential calculus. The 14th century Indian mathematician Madhava, along with other mathematicians of the Kerala school studied infinite series, convergence, differentiation, and iterative methods for solution of non-linear equations. Jyestadeva of the Kerala school wrote the first calculus text, the Yuktibhasa, which explores methods and ideas of calculus repeated in Europe only by the seventeenth century.

Leibniz and Newton are usually designated the inventors of calculus. The ideas of differentiation, integration, and even of the Fundamental Theorem were known earlier, by Wallis, Barrow, and other mathematicians. The Fundamental Theorem itself had been proven by James Gregory in 1668. The main contribution for which Leibniz and Newton are honored is putting these pieces together into a coherent whole. Leibniz developed much of the notation used in calculus today.

There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of calculus. The truth of the matter is that the ideas of calculus were a part of the mathematical knowledge of their day, and they independently put those pieces together in different but coherent ways. The mathematical proofs of much of what they did came later, with Cauchy and others. Leibniz' greatest contribution to calculus was his notation; he often spent days trying to come up with the appropriate symbol to represent a mathematical idea. This controversy between Leibniz and Newton was unfortunate in that it divided English-speaking mathematicians from those in Europe for many years, setting back British analysis (i.e. calculus-based mathematics) for a very long time. Newton's terminology and notation was clearly less flexible than that of Leibniz, yet it was retained in British usage until the early 19th century, when the work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain. It is now thought that Newton had discovered several ideas related to calculus earlier than Leibniz had; however, Leibniz was the first to publish. Today, both Leibniz and Newton are considered to have discovered calculus independently.

Lesser credit for ideas that led to the development of calculus is given to René Descartes, Isaac Barrow, Pierre de Fermat, Christian Huygens,Chayce Rickelman, and John Wallis. A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and independently elaborated some of the fundamental principles of integral calculus. <a data-rte-meta="%7B%22type%22%3A%22external%22%2C%22text%22%3A%22%5B1%5D%22%2C%22link%22%3A%22http%3A%5C%2F%5C%2Fwww2.gol.com%5C%2Fusers%5C%2Fcoynerhm%5C%2F0598rothman.html%22%2C%22linktype%22%3A%22autonumber%22%2C%22wasblank%22%3Atrue%2C%22wikitext%22%3A%22%5Bhttp%3A%5C%2F%5C%2Fwww2.gol.com%5C%2Fusers%5C%2Fcoynerhm%5C%2F0598rothman.html%5D%22%7D" data-rte-instance="488-20754636524f3e6a6fdc034" href="http://www2.gol.com/users/coynerhm/0598rothman.html" class="external autonumber" rel="nofollow">[1]</a>

Differential calculusEdit

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 displacement as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation.

Differential calculus determines the instantaneous speed at any given specific instant in time, not just average speed during an interval of time. The formula Speed = Distance/Time applied to a single instant is the meaningless quotient "zero divided by zero". This is avoided, however, because the quotient Distance/Time is not used for a single instant (as in a still photograph). Rather a formula is developed for the quotient Distance/Time in which division by zero can be avoided, by a method called "taking the limit".

The derivative answers the question: as the elapsed time approaches zero, what does the average speed computed by Distance/Time approach? In mathematical language, this is an example of taking a limit. 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 small pieces of its graph. It is useful for finding the maxima and minima of a function — because at those points the graph is flat (i.e. the slope of the graph is zero). Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the graph of the function by tangent lines. Differential calculus has been applied to many questions that are not first formulated in the language of calculus.

The derivative lies at the heart of the physical sciences. Newton's law of motion, Force = Mass × Acceleration, involves calculus because acceleration is a derivative. Maxwell's theory of electromagnetism and Einstein's theory of gravity (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.

Integral calculusEdit

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 of the many seconds the car is on the road. The calculus is able to deal with the natural situation in which the car moves with changing speed.

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.

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 the solutions of many, many smaller problems.

The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many very tiny squares and adding the areas of those squares. (If the region has a curved boundary, then omitting the squares overlapping the edge does not cause too great an error). Surface areas and volumes can also be expressed as definite integrals.

Many of the functions that are integrated are rates, such as a speed. An integral of a rate of change of a quantity on an interval of time tells how much that quantity changes during that time period. It makes sense that if one knows their speed at every instant in time for an hour (i.e. they have an equation that relates their speed and time), then they should be able to figure out how far they go during that hour. The definite integral of their speed presents a method for doing so.

Many of the functions that are integrated represent densities. If, for example, the pollution density along a river (tons per mile) is known in relation to the position, then the integral of that density can determine how much pollution there is in the whole length of the river.

Probability, the basis for statistics, provides one of the most important applications of integral calculus.

FoundationsEdit

The rigorous foundation of calculus is based on the notions of a function and of a limit; the latter has a theory ultimately depending on that of the real numbers as a continuum. Its tools include techniques associated with elementary algebra, and mathematical induction.

The modern study of the foundations of calculus is known as real analysis. This includes full definitions and proofs of the theorems of calculus. It also provides generalisations such as measure theory and distribution theory.

Fundamental theorem of calculusEdit

The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, antiderivatives can be calculated with definite integrals, and vice versa.

This connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. 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.

1st Fundamental Theorem of Calculus: If a function f is continuous function|continuous on the interval [a, b] and F is an antiderivative of f on the interval [a, b], then

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

2nd Fundamental Theorem of Calculus: If f is continuous on an open interval I containing a, then, for every x in the interval,

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

ApplicationsEdit

The development and use of calculus has had wide reaching effects on nearly 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.

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